Spectra of noncommutative spaces.

Alexander Rosenberg

Introduction

Noncommutative spectral theory was started by P. Gabriel [Gab] with the definition of the injective spectrum of a locally noetherian abelian category and application of this notion to the reconstruction of a noetherian scheme from its category of quasi-coherent sheaves. Elements of the Gabriel’s spectrum of an abelian category are isomorphism classes of its indecomposable injectives. The Gabriel’s spectrum of the category of modules over a commutative noetherian ring is naturally isomorphic to the prime spectrum of this ring.

Trying to find a spectrum of an associative ring which does not rely on the noetherian hypothesis and is more closely related with representation theory, I run into a notion of the left spectrum of a ring (see [R1], [R, Ch.1]) which is a certain set of left ideals endowed with a specialization preorder, and containing left maximal ideals as closed points. Reformulated in terms of (quotient) modules, the left spectrum gave rise to a spectrum of an abelian category (see [R2], or [R, Ch.3]). This spectrum can be realized as a certain family of Serre subcategories. In particular, to each point of the spectrum, there corresponds a localization at this point with a remarkable property: the spectrum of the quotient category has only one closed point. Such categories are called local. If a local category has simple objects, then all of them are isomorphic to each other. Local categories suggested the notion of S-spectrum (in [R, Ch.6] it is called flat spectrum). Its points are Serre subcategories such that the corresponding quotient category is local. If the category is locally noetherian, then the S-spectrum is naturally isomorphic to the Gabriel’s spectrum. But, even for locally noetherian categories, the S-spectrum is, usually, much bigger than the spectrum of [R2].

In [R4], several other spectra appeared. Those are related with exact localizations of arbitrary (not necessarily abelian) categories and triangulated categories. An abundance of new spectra made me suspect that the previously found spectra are not unique, as they made an impression at the beginning, and there exist pattern producing these spectra and the spectra of [R4], and, probably, some other spectra which make sense. Such pattern do exist, indeed, and explaining them is the main purpose of this text.

For the readers’ convenience, original definitions of the left spectrum of a ring, and the spectrum and S-spectrum of an abelian category are gathered in Appendix, together with a couple of other notions from [R] which reappear in the main text in a different disguise.

The main construction is presented in Section 1. Roughly, it runs as follows. With any category, ℌ, we associate two spectra, 𝔖𝔭𝔢𝔠^{0}(ℌ) and 𝔖𝔭𝔢𝔠^{1}(ℌ). These spectra are subcategories of ℌ. And, by construction, there is a natural functor 𝔖𝔭𝔢𝔠^{0}(ℌ) ⟶ ℌ (different from the inclusion functor). Given a functor 𝔊 ⟶^{F} ℌ, we define (in 1.6) two spectra, 𝔖𝔭𝔢𝔠^{0}(𝔊, F) and 𝔖𝔭𝔢𝔠^{1}(𝔊, F), of the pair (𝔊, F) as pullbacks of 𝔖𝔭𝔢𝔠^{0}(ℌ) and 𝔖𝔭𝔢𝔠^{1}(ℌ) along F. If ℌ is a preorder (which is the case of our main examples), then there exists a canonical morphism 𝔖𝔭𝔢𝔠^{0}(𝔊, F) ⟶ 𝔖𝔭𝔢𝔠^{1}(𝔊, F). In particular, there is a canonical morphism 𝔖𝔭𝔢𝔠^{0}(ℌ) ⟶ 𝔖𝔭𝔢𝔠^{1}(ℌ).

Taking as F the inclusion map of the preorder of Serre subcategories to the preorder of topologizing subcategories of an abelian category, we recover the spectrum of [R2] mentioned above. If ℌ is the preorder of saturated multiplicative systems of a category (resp. triangulated category) and F is the identical functor, we recover the basic spectra of an arbitrary category (resp. triangulated category) introduced in [R4]. These and some other applications of the general construction are sketched in Section 2.

Spectra considered in Section 2 are related with saturated (left and right) multiplicative systems, or what is the same, with exact (i.e. preserving finite limits and colimits) localizations. In the case of an abelian or triangulated category, they correspond to thick subcategories. There are categories with only trivial saturated multiplicative systems. A fundamental example is the category Sets of sets which belong to a given universe. It has no non-trivial right multiplicative systems, but has plenty of saturated left multiplicative systems. The latter are in bijective correspondence with right exact (– preserving colimits) localizations. In Section 3 we apply the pattern of Section 1 to the preorder of saturated left multiplicative systems of a category and obtain, as a result, left versions of the spectra discussed in Section 2 (and in [R4]).

In Section 4, we look at injective objects and related localizations and spectra, in particular, the Gabriel’s spectrum. Injective objects play important role not only in abelian (Grothendieck) categories, but also in a large class of non-additive categories which includes toposes. Therefore, the exposition here is not restricted to abelian or even additive categories. We define a left exact multiplicative system as a saturated left multiplicative systems such that the corresponding localization functor preserves strict monomorphisms. In the case of abelian categories, left exact multiplicative systems are precisely saturated (left and right) multiplicative systems. On the other hand, in the case of the category Sets, every saturated left system is left exact, but, as it is mentioned above, there are no non-trivial right saturated multiplicative systems.

To any injective object E of a category C, there corresponds a left exact multiplicative system Σ_{E} which consists of all arrows s such that Hom_{C}(s, E) is an isomorphism. Injective spectra, in particular (the non-additive version of) the Gabriel’s spectrum, are obtained by applying to this correspondence the general formalism of Section 1.

It is worth to mention that left exact multiplicative systems are usually more important, at least from spectral prospective, than injective objects. For instance, if C is the category Sets^{1} of non-empty sets, then there are only trivial left multiplicative systems of the form Σ_{E}. In particular, injective spectra are trivial. On the other hand, the preorder of left exact multiplicative systems is isomorphic to the order of infinite cardinals. Both spectra, 𝔖𝔭𝔢𝔠^{0} and 𝔖𝔭𝔢𝔠^{1}, of this preorder are naturally isomorphic to the order of non-limit infinite cardinals.

The purpose of this work is to explain what stands behind the known constructions of spectra and give a couple of curious examples. There is no attempt to make the list of applications and examples complete (i.e. include all applications which seem to be important ones) and, with more reason, no attempt to impose choices. The reader might make a different choice of the functor 𝔊 ⟶^{F} ℌ and use the ’spectral cuisine’ of Section 1 to produce other spectra which could be appropriate for something.

Main constructions of this work were guessed in the process of writing [R4] and are partly motivated by [R4], [R2], and [R]. Section 1.5, where ’local’ properties of spectra are discussed, is related to [R3]. In ’Complementary facts’, we discuss associated points following the lead of the corresponding part of [R4]. A considerable part of the paper might be regarded as comments to [R4] and [R].

The work was partially supported by the NSF grant DMS-0070921.

1. General pattern.

Fix a category ℌ. Let ℌ_{0} denote the full subcategory of ℌ whose objects are initial objects of ℌ. Thus, ℌ_{0} is either empty, or a groupoid. Let ℌ^{1} denote the full subcategory of ℌ defined by Obℌ^{1} = Obℌ − Obℌ_{0}.

1.1. Definition. We call ℌ local if the category ℌ^{1} has an initial object.

1.1.1. Note. It follows that if ℌ is local, than ℌ has initial objects, i.e. ℌ_{0} ≠ ∅.

1.1.2. Example. The preorder {x → y} is local, since ℌ^{1} has only one object, y, and one morphism, id_{y}.

1.1.3. Example. Let R be an associative commutative unital ring, and let IR denote the set of its ideals. The preorder (IR, ⊇) is local iff the ring R is local, i.e. there exists a maximal ideal in R which contains all other proper ideals.

1.2. The spectrum 𝔖𝔭𝔢𝔠^{1}(ℌ). We denote by 𝔖𝔭𝔢𝔠^{1}(ℌ) the full subcategory of the category ℌ generated by all x ∈ Obℌ such that the category x\ℌ is local. We call 𝔖𝔭𝔢𝔠^{1}(ℌ) the local spectrum of ℌ.

In other words, an object x of ℌ belongs to 𝔖𝔭𝔢𝔠^{1}(ℌ) iff there exists an object x^{⋆} of ℌ and an arrow x ⟶^{γ_{x}} x^{⋆} such that γ_{x} is not an isomorphism and if x ⟶^{f} y is not an

isomorphism, then there exists a unique arrow x^{⋆} ⟶^{f̄} y such that f = f̄∘γ_{x}. The morphism x ⟶^{γ_{x}} x^{⋆} (in particular, the object x^{⋆}) is determined by these conditions uniquely up to isomorphism.

1.2.1. Note. It follows from this definition and 1.1.1 that ℌ is local iff it has initial objects and they belong to 𝔖𝔭𝔢𝔠^{1}(ℌ).

1.2.1.1. Example. Let ℌ = Sets. The category Sets has one initial object – ∅. It is local: every one element set is an initial object of the category Sets^{1}. Notice that the spectrum 𝔖𝔭𝔢𝔠^{1}(Sets^{1}) is empty. Therefore 𝔖𝔭𝔢𝔠^{1}(Sets) consists of one point which is the initial object ∅.

1.2.2. Maximal proper objects and 𝔖𝔭𝔢𝔠^{1}(ℌ). We call an object x of the category ℌ proper if there exists an arrow x ⟶ y which is not an isomorphism. We call a proper object, x, maximal if any two proper morphisms, y_{1} ⟵^{s} x ⟶^{t} y_{2}, are isomorphic; that is there exists an isomorphism y_{1} ⟶^{u} y_{2} such that t = u ∘ s.

We denote by Max(ℌ) the full subcategory of ℌ generated by maximal proper objects. It follows that Max(ℌ) is a groupoid which is connected iff all maximal proper objects are isomorphic to each other.

If for every proper object, y, there is an arrow from y to a maximal proper object, then the groupoid Max(ℌ) is connected iff ℌ^{op} is a local category.

1.2.2.1. Proposition. Max(ℌ) ⊆ 𝔖𝔭𝔢𝔠^{1}(ℌ).

Proof. In fact, if x is an object of Max(ℌ), then the category x\ℌ is equivalent to the preorder {x → y}, hence it is local (cf. 1.1.2).

1.2.2.2. Example. If ℌ is the preorder (I_{ℓ}R, ⊆) of left ideals of an associative unital ring R, then Max(ℌ) coincides with the set Max_{ℓ}R of left maximal ideals of R regarded as a discrete category. The category Max_{ℓ}R is connected iff R has only one left maximal ideal, μ. Notice that in this case the left ideal μ is two-sided, because for every r ∈ R − μ, the ideal (μ : r) = {a ∈ R | ar ∈ μ} is a maximal left ideal, hence it coincides with μ.

1.2.3. Minimal proper objects. We call x ∈ Obℌ a minimal proper object of the category ℌ if x is a maximal proper object of ℌ^{op}. We denote by Min(ℌ) the full subcategory of ℌ generated by minimal proper objects. By definition, Min(ℌ) is isomorphic to Max(ℌ^{op}). In particular, Min(ℌ) is a groupoid which is connected iff ℌ is a local category. By 1.2.2.1, Min(ℌ) ⊆ 𝔖𝔭𝔢𝔠^{1}(ℌ^{op}).

1.2.3.1. Example. Let C_{X} be a category with an initial object, and let C_{𝔐(X)} be the subcategory of C_{X} formed by all monoarrows of C_{X}. Then Min(C_{𝔐(X)}) is the groupoid of all simple objects of the category C_{X}. Isomorphism classes of simple objects can be regarded as a naive spectrum of C_{X}.

The groupoid Min(C_{𝔐(X)}) is connected (that is the category C_{𝔐(X)} is local) iff the category C_{X} has a unique, up to isomorphism, simple object.

1.2.3.2. Note. A useful version of 1.2.3.1 is obtained by taking instead of C_{𝔐(X)} the subcategory C_{𝔐_{𝔰}(X)} generated by all strict monomorphisms of the category C_{X}.

Recall that a monomorphism L ⟶^{h} M is called strict if every arrow L′ ⟶ M which

equalizes all pairs of arrows M \overrightarrow{⟶}^{g_{1}}_{g_{2}}N equalized by h is represented as the composition of

h and an arrow L′ ⟶ L uniquely determined by this property. If an arrow L ⟶^{h} M is such that there exists a fibred coproduct M ∐_{L} M, then h is a strict monomorphism iff

the canonical diagram L ⟶^{h} M \overrightarrow{⟶}^{π_{1}}_{π_{2}} M ∐_{L} M is exact.

The groupoid Min(C_{𝔐_{𝔰}(X)}) is generated by objects which are simple in a ”strict” sense. For instance, if C_{X} is the category of continuous representaions of a topological algebra in topological vector spaces, objects of Min(C_{𝔐_{𝔰}(X)}) are topologically irreducible representations of this algebra.

1.2.3.3. Example: injectives and the Gabriel’s spectrum. Let C_{X} be a category with finite limits. An object E of the category C_{X} is called injective if the functor C_{X}(−, E) : C_{X}^{op} ⟶ Sets preserves strict epimorphisms (in other words, C_{X}(𝔧, E) is a surjective map for any strict monomorphism 𝔧). We denote by C_{ℑ(X)} the subcategory of C_{X} formed by injective objects and strict monomorphisms (see 1.2.3.2) If follows that if E is an injective object, than any strict monomorphism E ⟶^{g} M is a split monomorphism; i.e. h ∘ g = id_{E} for some M ⟶^{h} E.

We call an arrow in C_{X} a zero morphism if it factors through an initial object (if any). We call an injective object E of the category C_{X} indecomposable if the only nonzero

idempotent E ⟶ E is the identical morphism. Equivalently, any strict monomorphism E_{1} ⟶ E with E_{1} injective and non-initial, is an isomorphism.

Objects of the groupoid Min(C_{ℑ(X)}) are precisely indecomposable injectives of the full subcategory C_{X}^{1} of the category C_{X} formed by non-initial objects. Isomorphism classes of indecomposable injectives are points of the Gabriel’s spectrum.

1.2.4. Functorial properties. Let ℌ ⟶^{F} ℌ̃ be a functor. For any x ∈ Obℌ, the functor F induces a functor x\ℌ ⟶^{F_{x}} F (x)\ℌ̃. Suppose that the functor F is such that F_{x} is an equivalence of categories. Then F induces a functor 𝔖𝔭𝔢𝔠^{1}(ℌ) ⟶ 𝔖𝔭𝔢𝔠^{1}(ℌ̃).

A typical example is the functor

y\ℌ ⟶^{f_{⋆}} z\ℌ, (y, y →^{g} v) ⟼ (z, z →^{gf} v),

corresponding to a morphism z ⟶^{f} y, or the canonical functor y\ℌ ⟶ ℌ.

1.3. Supports. For any x ∈ Obℌ, we denote by 𝔖𝔲𝔭𝔭_{ℌ}(x) the full subcategory of ℌ generated by all y ∈ Obℌ such that ℌ(x, y) = ∅. We call 𝔖𝔲𝔭𝔭_{ℌ}(x) the support of x in ℌ.

1.3.1. Proposition. (a) For any two objects, x and y, of the category ℌ, there exists an arrow x → y iff 𝔖𝔲𝔭𝔭_{ℌ}(x) ⊆ 𝔖𝔲𝔭𝔭_{ℌ}(y).

(b) Let {x_{i} | i ∈ J} be a set of objects of ℌ such that there exists a coproduct, ∐_{i∈J} x_{i}.

Then

𝔖𝔲𝔭𝔭_{ℌ}( ∐_{i∈J} x_{i}) = ⋃_{i∈J} 𝔖𝔲𝔭𝔭_{ℌ}(x_{i}).  (1)

Proof. (a) If there exists a morphism x ⟶ y and ℌ(x, z) = ∅, then, obviously, ℌ(y, z) = ∅, hence 𝔖𝔲𝔭𝔭_{ℌ}(x) ⊆ 𝔖𝔲𝔭𝔭_{ℌ}(y).

If ℌ(x, y) = ∅, i.e. y ∈ 𝔖𝔲𝔭𝔭_{ℌ}(x), then, since y ∉ Ob𝔖𝔲𝔭𝔭_{ℌ}(y), the inclusion 𝔖𝔲𝔭𝔭_{ℌ}(x) ⊆ 𝔖𝔲𝔭𝔭_{ℌ}(y) does not hold.

(b) Since ℌ(∐_{i∈J} x_{i}, z) ≃ ∏_{i∈J} ℌ(x_{i}, z), ℌ(∐_{i∈J} x_{i}, z) = ∅ iff ℌ(x_{i}, z) = ∅ for some i ∈ J, whence the equality (1).

1.3.2. Support in 𝔖𝔭𝔢𝔠^{1}(ℌ). For any x ∈ Obℌ, we denote the intersection 𝔖𝔲𝔭𝔭_{ℌ}(x) ⋂ 𝔖𝔭𝔢𝔠^{1}(ℌ) by 𝔖𝔲𝔭𝔭_{ℌ}^{1}(x) and call it the support of x in 𝔖𝔭𝔢𝔠^{1}(ℌ). Evidently, 1.3.1(b) is still true if 𝔖𝔲𝔭𝔭_{ℌ}(x) is replaced by 𝔖𝔲𝔭𝔭_{ℌ}^{1}(x), as well as a half of 1.3.1(a): if ℌ(x, y) is not empty, then 𝔖𝔲𝔭𝔭_{ℌ}^{1}(x) ⊆ 𝔖𝔲𝔭𝔭_{ℌ}^{1}(y).

1.4. The spectrum 𝔖𝔭𝔢𝔠^{0}(ℌ). We denote by 𝔖𝔭𝔢𝔠^{0}(ℌ) the full subcategory of ℌ generated by x ∈ Obℌ such that 𝔖𝔲𝔭𝔭_{ℌ}(x) is not empty and has a final object, x̂.

1.4.1. Proposition. Let ℌ be local. Then initial objects of ℌ^{1} belong to 𝔖𝔭𝔢𝔠^{0}(ℌ).

Proof. Let ℌ_{0} be the full subcategory (groupoid) of ℌ generated by all initial objects of ℌ. If x is an initial object of the category ℌ^{1}, then 𝔖𝔲𝔭𝔭_{ℌ}(x) coincides with ℌ_{0}.

In fact, suppose that there is an arrow, x ⟶^{f} y, for some y ∈ Obℌ_{0}. Since y is an initial object of the category ℌ, there exists a unique morphism y ⟶^{g} x. By the universal property of y, the composition y ⟶^{fg} y is the identical morphism. Since x is an initial object of the category ℌ^{1}, the composition x ⟶^{gf} x is the identical morphism too. This means that the morphism x ⟶^{f} y is an isomorphism which contradicts to the fact that x is not an initial object of the category ℌ.

Thus, ℌ_{0} is a subcategory of 𝔖𝔲𝔭𝔭_{ℌ}(x). Since for every z ∈ Obℌ^{1} = Obℌ − Obℌ_{0} there is a (unique) morphism x ⟶ z, the subcategory 𝔖𝔲𝔭𝔭_{ℌ}(x) is contained in ℌ_{0}; i.e. 𝔖𝔲𝔭𝔭_{ℌ}(x) = ℌ_{0}.

Since ℌ_{0} is a connected groupoid, every object of ℌ_{0} is final.

1.4.2.1. Example. The spectrum 𝔖𝔭𝔢𝔠^{0}(Sets) coincides with the subcategory Sets^{1} of all non-empty sets, because the support of any non-empty set consists of only ∅.

1.4.2. Proposition. The full subcategory, 𝔖𝔭𝔢𝔠^{0}(ℌ)_{0}, generated by initial objects of 𝔖𝔭𝔢𝔠^{0}(ℌ) coincides with be the full subcategory ℌ_{1} = (ℌ^{1})_{0} of the category ℌ generated by initial objects of the subcategory ℌ^{1}.

Proof. By definition, the subcategory 𝔖𝔲𝔭𝔭_{ℌ}(x) (cf. 1.4) is not empty for every object x of 𝔖𝔭𝔢𝔠^{0}(ℌ). Therefore initial objects of the category ℌ do not belong to 𝔖𝔭𝔢𝔠^{0}(ℌ), i.e. 𝔖𝔭𝔢𝔠^{0}(ℌ) is contained in the subcategory ℌ^{1}. In particular, the subcategory 𝔖𝔭𝔢𝔠^{0}(ℌ)_{0} of initial objects of 𝔖𝔭𝔢𝔠^{0}(ℌ) is contained in ℌ_{1} = (ℌ^{1})_{0}. The converse inclusion is a consequence of 1.4.1.

1.4.3. Corollary. Let |𝔖𝔭𝔢𝔠^{1}(ℌ)| denote the set of isomorphism classes of objects of 𝔖𝔭𝔢𝔠^{1}(ℌ). Then

Ob𝔖𝔭𝔢𝔠^{1}(ℌ) = ⋃_{x∈|𝔖𝔭𝔢𝔠^{1}(ℌ)|} {y | (y, x → y) = ẑ, z ∈ Ob𝔖𝔭𝔢𝔠^{0}(x\ℌ)_{0}}.  (1)

In particular,

Ob𝔖𝔭𝔢𝔠^{1}(ℌ) ⊆ ⋃_{x∈|𝔖𝔭𝔢𝔠^{1}(ℌ)|} {y | (y, x → y) = ẑ, z ∈ Ob𝔖𝔭𝔢𝔠^{0}(x\ℌ)}.  (2)

Here ẑ is a final object of the category 𝔖𝔲𝔭𝔭_{x\ℌ}(z) (cf. 1.4).

Proof. The formula (1) follows from 1.4.2 applied to the category x\ℌ.

1.4.4. Lemma. A choice for every x ∈ Ob𝔖𝔭𝔢𝔠^{0}(ℌ) of a final object, x̂, of the category 𝔖𝔲𝔭𝔭_{ℌ}(x) extends to a functor 𝔖𝔭𝔢𝔠^{0}(ℌ) ⟶^{ϑ_{ℌ}} ℌ.

Proof. In fact, if x, y ∈ Ob𝔖𝔭𝔢𝔠^{0}(ℌ), and there is a morphism x ⟶ y, then 𝔖𝔲𝔭𝔭_{ℌ}(x) ⊆ 𝔖𝔲𝔭𝔭_{ℌ}(y). Therefore there exists a unique morphism x̂ ⟶ ŷ.

1.4.5. Remark. Notice that the functor 𝔖𝔭𝔢𝔠^{0}(ℌ) ⟶^{ϑ_{ℌ}} ℌ is faithful iff ℌ is a preorder, i.e. for any pair of objects, x, y, of ℌ, there is at most one morphism x ⟶ y.

1.4.6. Proposition. Suppose the category ℌ is a preorder with finite coproducts (i.e. supremums of pairs of objects). Then the functor 𝔖𝔭𝔢𝔠^{0}(ℌ) ⟶^{ϑ_{ℌ}} ℌ takes values in

𝔖𝔭𝔢𝔠^{1}(ℌ), i.e. it induces a functor 𝔖𝔭𝔢𝔠^{0}(ℌ) ⟶^{θ_{ℌ}} 𝔖𝔭𝔢𝔠^{1}(ℌ).

Proof. For any x ∈ Ob𝔖𝔭𝔢𝔠^{0}(ℌ), the final object, x̂, of the category 𝔖𝔲𝔭𝔭_{ℌ}(x) belongs to 𝔖𝔭𝔢𝔠^{1}(ℌ). More expicitly, we claim that the canonical coprojection, x̂ ⟶ x ⊔ x̂, is an initial object of the category (x̂\ℌ)^{1}.

In fact, let x̂ ⟶^{g} y be a morphism. Then one of two things happens: either y ∈ Ob𝔖𝔲𝔭𝔭_{ℌ}(x), or not. If y ∈ Ob𝔖𝔲𝔭𝔭_{ℌ}(x), then, since x̂ is a final object of the category 𝔖𝔲𝔭𝔭_{ℌ}(x), there is a unique morphism y ⟶^{h} x̂. It follows from the universal property of x̂ that h ∘ g = id_{x}. By hypothesis, ℌ is a preorder, in particular, h is a monomorphism. Therefore, h is an isomorphism inverse to g.

If y ∉ Ob𝔖𝔲𝔭𝔭_{ℌ}(x), then there exists an arrow x ⟶ y which, together with x̂ ⟶^{g} y, determines (and is determined by) a morphism (x ⊔ x̂, x̂ → x ⊔ x̂) ⟶ (y, x̂ →^{g} y). Since ℌ is a preorder, this is all we need.

1.4.6.1. Example. Let ℌ be a category with initial objects and such that ℌ(x, y) ≠ ∅ for every x ∈ Obℌ and ℌ(y, z) = ∅ if y is not an initial object and z is an initial object. Then 𝔖𝔲𝔭𝔭_{ℌ}(x) = ℌ_{0} for any x ∈ Obℌ^{1}. Therefore, 𝔖𝔭𝔢𝔠^{0}(ℌ) coincides with ℌ^{1}. If ℌ is a preorder, then, under conditions, ℌ^{1} is a connected groupoid, hence 𝔖𝔭𝔢𝔠^{1}(ℌ) = ℌ_{0} and

the functor 𝔖𝔭𝔢𝔠^{0}(ℌ) ⟶^{θ_{ℌ}} 𝔖𝔭𝔢𝔠^{1}(ℌ) is a category equivalence.

What might happen if ℌ is not a preorder is illustrated by the following.

1.4.6.2. Example. If ℌ = Sets, then, by 1.2.1.1, 𝔖𝔭𝔢𝔠^{1}(ℌ) = {∅, id_{∅}} and by 14.2.1, 𝔖𝔭𝔢𝔠^{0}(ℌ) = Sets^{1} – the category of all non-empty sets. There is only one functor 𝔖𝔭𝔢𝔠^{0}(ℌ) ⟶ 𝔖𝔭𝔢𝔠^{1}(ℌ).

1.4.7. Proposition. Suppose that the category ℌ is a preorder with coproducts. Then

Ob𝔖𝔭𝔢𝔠^{1}(ℌ) = ⋃_{x∈|𝔖𝔭𝔢𝔠^{1}(ℌ)|} {y | (y, x → y) = ẑ, z ∈ Ob𝔖𝔭𝔢𝔠^{0}(x\ℌ)}.  (3)

Proof. The assertion follows from 1.4.6 and 1.4.3(2).

1.4.8. Support in 𝔖𝔭𝔢𝔠^{0}(ℌ). For any object x of the category ℌ, we denote by 𝔖𝔲𝔭𝔭_{ℌ}^{0}(x) the preimage of 𝔖𝔲𝔭𝔭_{ℌ}(x) by the functor 𝔖𝔭𝔢𝔠^{0}(ℌ) ⟶ ℌ, z ⟼ ẑ, (cf. 1.4.4) and call it the support of x in 𝔖𝔭𝔢𝔠^{0}(ℌ). This means that 𝔖𝔲𝔭𝔭_{ℌ}^{0}(x) is a full subcategory of ℌ whose objects are all objects z of 𝔖𝔭𝔢𝔠^{0}(ℌ) such that ℌ(x, ẑ) = ∅, or, equivalently, 𝔖𝔲𝔭𝔭_{ℌ}(z) ⊆ 𝔖𝔲𝔭𝔭_{ℌ}(x). By 1.3.1, the latter means precisely that there exists a morphism z → x. Thus, 𝔖𝔲𝔭𝔭_{ℌ}^{0}(x) is a full subcategory of ℌ generated by all objects z of 𝔖𝔭𝔢𝔠^{0}(ℌ) such that z → x.

1.4.9. Proposition. (a) The map x ⟼ 𝔖𝔲𝔭𝔭_{ℌ}^{0}(x) is functorial: if there exists an arrow x → y, then 𝔖𝔲𝔭𝔭_{ℌ}^{0}(x) ⊆ 𝔖𝔲𝔭𝔭_{ℌ}^{0}(y).

(b) Let {x_{i} | i ∈ J} be a set of objects of ℌ such that there exists a coproduct, ∐_{i∈J} x_{i}.

Then

𝔖𝔲𝔭𝔭_{ℌ}^{0}( ∐_{i∈J} x_{i}) = ⋃_{i∈J} 𝔖𝔲𝔭𝔭_{ℌ}^{0}(x_{i}).  (1)

Proof. (a) The assertion follows from the fact that

Ob𝔖𝔲𝔭𝔭_{ℌ}^{0}(x) = {z ∈ Ob𝔖𝔭𝔢𝔠^{0}(ℌ) | ℌ(z, x) ≠ ∅}

(see the discussion in 1.4.8).

(b) An object z of 𝔖𝔭𝔢𝔠^{0}(ℌ) belongs to 𝔖𝔲𝔭𝔭_{ℌ}^{0}( ∐_{i∈J} x_{i}) iff ℌ( ∐_{i∈J} x_{i}, ẑ) = ∅. Since

ℌ( ∐_{i∈J} x_{i}, ẑ) = ∏_{i∈J} ℌ(x_{i}, ẑ), this occurs iff ℌ(x_{i}, ẑ) = ∅ for some i ∈ J.

1.5. Topologies and spectra.

1.5.1. Generalities on topologies. Let τ be a topology on ℌ^{op}, i.e. τ is a function which assigns to every object x of ℌ a set, τ(x), of subfunctors of the functor ℌ(x, −) (called the refinements of x) satisfying the following conditions:

(a) for every arrow x →^{f} y of ℌ and every R ∈ τ(x), the fibre product, R^{f}, of

R ⟶ ℌ(x, −) ⟵^{ℌ(f,−)} ℌ(y, −) belongs to τ(y);

(b) If R ∈ τ(x) and E is a subfunctor of ℌ(x, −) such that E^{f} ∈ τ(y) for any f ∈ R(y) and any y, then E ∈ τ(x).

1.5.1.1. Cocovers. A family of arrows x̃ = {x →^{u_{i}} x_{i} | i ∈ J} generates a subfunctor, R_{x}, of ℌ(x, −) defined as follows: R_{x}(y) consists of all arrows x ⟶ y which factor through x →^{u_{i}} x_{i} for some i ∈ J. The family x̃ = {x →^{u_{i}} x_{i} | i ∈ J} is called a cocover (or a cover in ℌ^{op}) for the topology τ if R_{x} ∈ τ(x).

1.5.1.2. Sheaves. Subcanonical topologies. A functor ℌ ⟶^{F} Sets (viewed as a presheaf of sets on ℌ^{op}) is called a sheaf (on (ℌ^{op}, τ)) if for every x ∈ Obℌ and any refinement R of x, the map F (x) ⟶ Hom(R, F) induced by the embedding R ↪ ℌ(x, −) and the Yoneda isomorphism F (x) ≃ Hom(ℌ(x, −), F), is a bijection.

The topology τ on ℌ^{op} is called subcanonical if every representable presheaf, i.e. a functor of the form ℌ(x, −), x ∈ Obℌ, is a sheaf.

1.5.1.3. Cosieves. It is convenient sometimes to describe topologies on ℌ^{op} in terms of cosieves. Recall that a cosieve in a category A is a full subcategory, B, of A such that for every x ∈ ObB, all arrows x ⟶ y belong to B.

Let x be an object of the category ℌ. There is a one-to-one correspondence between cosieves of the category x\ℌ and subfunctors of the functor ℌ(x, −). Namely, an object (y, x →^{ξ} y) of x\ℌ belongs to the cosieve R̃ corresoponding to a subfunctor R of ℌ(x, −) iff the morphism x →^{ξ} y is an element of R(y).

Thus, a topology, τ, on ℌ^{op} can be described as a function which assigns to each object of ℌ a non-empty family of cosieves of the category x\ℌ, which are also called refinements of x, satisfying the conditions reflecting properties (a) and (b) of 1.5.1.

One can see that a topology τ on ℌ^{op} is subcanonical iff for every x ∈ Obℌ each refinement of x is a terminal cone. In other words, for every refinement R̃ of x, the limit of the canonical functor R ⟶ ℌ, (y, x →^{ξ} y) ⟼ y, is isomorphic to x.

1.5.1.4. Pretopologies on ℌ^{op}. A pretopology on ℌ^{op} is a function, τ, which assigns to each object x of ℌ a family, τ_{x}, of sets of arrows {x → x_{i} | i ∈ J} (in ℌ) having the following properties:

(a) for every x ∈ Obℌ, {id_{x}} ∈ τ_{x};

(b) if {x → x_{i} | i ∈ J} ∈ τ_{x} and {x_{i} → x_{ij} | j ∈ J_{i}} ∈ τ_{x_{i}} for every i ∈ J, then {x → x_{ij} | i ∈ J, j ∈ J_{i}} ∈ τ_{x};

(c) for any x̃ = {x → x_{i} | i ∈ J} ∈ τ_{x} and any morphism x ⟶^{ϕ} y, there exists

ỹ = \{y →^{v_{j}} y_{j} | j ∈ I\} ∈ τ_{y} such that the morphism ϕ can be lifted to a morphism x̃ \underrightarrow{_{ϕ}} ỹ.

The latter means that for every j ∈ I, the composition of x ⟶^{ϕ} y and y ⟶ y_{j} factors through x ⟶ x_{i} for some i ∈ J.

Elements of τ_{x} are called cocovers of x. Arrows which belong to cocovers are interpreted as closed subsets. The corresponding arrows in ℌ^{op} are viewed as open subsets.

Every pretopology determines a topology obtained by taking cosieves (or subfunctors of representable functors) associated with cocovers.

1.5.2. Proposition. Suppose that τ is a subcanonical topology on ℌ^{op}; and let x̃ = {x →^{u_{i}} x_{i} | i ∈ J} be a cocover for τ. Then

𝔖𝔭𝔢𝔠^{1}(x\ℌ) = ⋃_{i∈J} 𝔖𝔭𝔢𝔠^{1}(x_{i}\ℌ).

Here 𝔖𝔭𝔢𝔠^{1}(x_{i}\ℌ) is identified with its image in x\ℌ via the morphism x →^{u_{i}} x_{i}.

Proof. Let an object z̃ = (z, x →^{𝔣} z) belong to 𝔖𝔭𝔢𝔠^{1}(x\ℌ). By definition, this means that the subcategory (z̃\(x\ℌ))^{1} has an initial object. Notice that the category z̃\(x\ℌ) is isomorphic to the category z\ℌ. Thus, the category (z\ℌ)^{1} has an initial object, (z^{⋆}, z →^{ξ} z^{⋆}). By condition, ξ is not an isomorphism.

Let R_{x} be a refinement of x associated with the cocover x̃ = {x →^{u_{i}} x_{i} | i ∈ J}; and let R_{x}^{ξ} be the corresponding refinement of the object z. Notice that there exists y ∈ Obℌ

and an element z →^{g} y of R_{x}^{ξ}(y) such that (y, z →^{g} y) ∉ Ob(z\ℌ)^{1}.

In fact, if such element would not exist, then, since (z^{⋆}, z →^{ξ} z^{⋆}) is an initial object of (z\ℌ)^{1}, every element z →^{g} y of R_{x}^{ξ}(y) factors through z →^{ξ} z^{⋆}, and this factorization is

unique. Since the topology τ is subcanonical, (z, id_{z}) is an initial object of the sieve R̂_{x}^{ξ}

associated with R_{x}^{ξ}. Therefore, there exists a unique morphism (z^{⋆}, ξ) ⟶ (z, id_{z}). But, this cannot happen (see the argument of 1.4.6).

Thus, there exists an element z →^{g} y of R_{x}^{ξ}(y) such that (y, z →^{g} y) ∉ Ob(z\ℌ)^{1}, or,

what is the same, the arrow z →^{g} y is an isomorhism. By the definition of R̂_{x}^{ξ}, there exists a commutative diagram

\begin{matrix}x & ⟶^{𝔣} & z \\ u_{i} ↓ & & ↓ g \\ x_{i} & ⟶^{𝔣_{i}} & y \end{matrix}  (4)

for some i ∈ J. Since the arrow g in (4) is an isomorphism, it follows from (4) that x →^{𝔣} z factors through the element x →^{u_{i}} x_{i} of the cocover x̃. Therefore, the object (z, 𝔣) of 𝔖𝔭𝔢𝔠^{1}(x\ℌ) is the image of an object (z, g𝔣_{i}) of 𝔖𝔭𝔢𝔠^{1}(x_{i}\ℌ); hence the assertion.

For any x ∈ Obℌ, let U_{ℌ}(x) denote the full subcategory of ℌ generated by all y ∈ Obℌ such that ℌ(y, x) = ∅. Thus, U_{ℌ}(x) coincides with (𝔖𝔲𝔭𝔭_{ℌ^{op}}(x))^{op}.

1.5.3. Proposition. Let x ∈ Obℌ be such that for any z ∈ Obℌ, there exists a coproduct, x ⊔ z. Then the map z ⟼ (x ⊔ z, x → x ⊔ z) defines a functor

𝔖𝔭𝔢𝔠^{0}(ℌ) ⋂ U_{ℌ}(x) ⟶ 𝔖𝔭𝔢𝔠^{0}(x\ℌ).

Proof. For any y ∈ Obℌ, we set f^{∗}(y) = (y⊔x, x → y⊔x). The map y ↦ f^{∗}(y) extends

to a functor, ℌ ⟶^{f^{∗}} x\ℌ, which is left adjoint to the functor x\ℌ ⟶^{f_{∗}} ℌ, (v, x → v) ↦ v.

Let z be an object of the subcategory 𝔖𝔭𝔢𝔠^{0}(ℌ) ⋂ U_{ℌ}(x), and let ẑ be a final object of the category 𝔖𝔲𝔭𝔭_{ℌ}(z). Since z ∈ ObU_{ℌ}(x), i.e. ℌ(z, x) = ∅, there exists a unique morphism x → ẑ. We claim that the (ẑ, x → ẑ) is a final object of the category 𝔖𝔲𝔭𝔭_{x\ℌ}(f^{∗}(z)).

In fact, x\ℌ(f^{∗}(z), (y, x → y)) ≃ ℌ(z, y) which shows that (y, x → y) is an object of 𝔖𝔲𝔭𝔭_{x\ℌ}(f^{∗}(z)) iff y is an object of 𝔖𝔲𝔭𝔭_{ℌ}(z). Therefore, (ẑ, x → ẑ) belongs to 𝔖𝔲𝔭𝔭_{x\ℌ}(f^{∗}(z)) and, moreover, is a final object of this category.

1.5.4. Corollary. Let x →^{u} y be a morphism of ℌ such that for any other morphism, x →^{v} z, there exists a fibred coproduct, y ⊔_{x} z. Then the functor

y\ℌ ⟶^{u_{∗}} x\ℌ, (z, y →^{g} z) ⟼ (z, x →^{gu} z),

has a left adjoint, u^{∗}; and u^{∗} induces a functor

𝔖𝔭𝔢𝔠^{0}(x\ℌ) ⋂ U_{x\ℌ}(y, u) ⟶ 𝔖𝔭𝔢𝔠^{0}(y\ℌ).

Proof. The fact is a consequence of 1.5.3 applied to the category x\ℌ.

1.5.5. Proposition. Let x = {x →^{u_{i}} x_{i} | i ∈ J} be a set of arrows such that the cone x ⟶ R̃_{x} is terminal (i.e. x = lim R̃_{x}) and for any arrow x ⟶ y, there exist fibred coproducts x_{i} ⊔_{x} y. If ℌ is a preorder, then the image of the canonical map 𝔖𝔭𝔢𝔠^{0}(x\ℌ) ⟶ 𝔖𝔭𝔢𝔠^{1}(x\ℌ) is contained in the union of images of 𝔖𝔭𝔢𝔠^{0}(x_{i}\ℌ), i ∈ J, in 𝔖𝔭𝔢𝔠^{1}(x\ℌ).

Proof. By 1.5.4, there are natural functors

𝔖𝔭𝔢𝔠^{0}(x\ℌ) ⋂ U_{x\ℌ}(x_{i}, u_{i}) ⟶ 𝔖𝔭𝔢𝔠^{0}(x_{i}\ℌ)

Therefore, it suffices to show that for any object (z, ξ) of 𝔖𝔭𝔢𝔠^{0}(x\ℌ), there exists i ∈ J such that there are no morphisms from (z, ξ) to (x_{i}, u_{i}).

Suppose that for each i ∈ J, there is an arrow (z, ξ) ⟶ (x_{i}, u_{i}). Since ℌ is a preorder, these arrows determine a cone z ⟶ R̃_{x}. By hypothesis, x = lim R̃_{x}, hence there exists a morphism (z, ξ) ⟶ (x, id_{x}). Since (x, id_{x}) is an initial object of the category x\ℌ, this means that for any object of x\ℌ there is an arrow from (z, ξ) to this object, which cannot happen, because (z, ξ) belongs to 𝔖𝔭𝔢𝔠^{0}(x\ℌ). Thus, there exists i ∈ J such that there are no morphisms from (z, ξ) to (x_{i}, u_{i}).

1.5.6. The spectrum of a precosite. Let τ be a pretopology on ℌ^{op}. We call the pair (ℌ, τ) a precosite. Let ℌ_{τ} denote the subcategory of ℌ formed by arrows which belong to some cocovers. For every x ∈ Obℌ, we denote by τ_{x} the induced pretopology on ℌ^{op}/x = (x\ℌ)^{op}. We denote by 𝔖𝔭𝔢𝔠^{0}(x\ℌ, τ_{x}) the full subcategory of x\ℌ generated by the images of 𝔖𝔭𝔢𝔠^{0}(y\ℌ) for all arrows x → y of the subcategory ℌ_{τ}. We call 𝔖𝔭𝔢𝔠^{0}(x\ℌ, τ_{x}) the spectrum of the precosite (x\ℌ, τ_{x}).

Thus, if the category ℌ has an initial object, we obtain the spectrum, 𝔖𝔭𝔢𝔠^{0}(ℌ, τ), of the precosite (ℌ, τ). If, in addition, all arrows of the subcategory ℌ_{τ} are isomorphisms (for instance, the pretopology τ is discrete), then 𝔖𝔭𝔢𝔠^{0}(ℌ, τ) coincides with 𝔖𝔭𝔢𝔠^{0}(ℌ).

1.5.6.1. Proposition. Suppose that ℌ_{τ} = ℌ and ℌ is a preorder with finite coproducts and an initial object. Then 𝔖𝔭𝔢𝔠^{0}(ℌ, τ) is isomorphic to 𝔖𝔭𝔢𝔠^{1}(ℌ).

Proof. The assertion is a consequence of 1.4.6.

1.5.6.2. Proposition. To any morphism, x ⟶ y, of the subcategory ℌ_{τ}, there corresponds an inclusion 𝔖𝔭𝔢𝔠^{0}(y\ℌ, τ_{y}) ⊆ 𝔖𝔭𝔢𝔠^{0}(x\ℌ, τ_{x}), i.e. the map x ⟼ 𝔖𝔭𝔢𝔠^{0}(x\ℌ, τ_{x}) extends to a functor ℌ_{τ}^{op} ⟶ Cat.

Proof. The fact follows from definitions.

1.6. Relative spectra. Let 𝔊 ⟶^{F} ℌ be a functor. We define the relative spectra, 𝔖𝔭𝔢𝔠^{1}(𝔊, F) and 𝔖𝔭𝔢𝔠^{0}(𝔊, F), via cartesian squares

\begin{matrix}𝔖𝔭𝔢𝔠^{1}(𝔊, F) & ⟶^{θ_{F}^{1}} & 𝔊 & & 𝔖𝔭𝔢𝔠^{0}(𝔊, F) & ⟶^{ϑ_{F}} & 𝔊 \\ π_{1}^{F} ↓ & & ↓ F & and & π_{0}^{F} ↓ & & ↓ F \\ 𝔖𝔭𝔢𝔠^{1}(ℌ) & ⟶^{θ_{ℌ}^{1}} & ℌ & & 𝔖𝔭𝔢𝔠^{0}(ℌ) & ⟶^{ϑ_{ℌ}} & ℌ \end{matrix}  (1)

(in the bicategorical sense, i.e. the squares quasi-commute), where 𝔖𝔭𝔢𝔠^{0}(ℌ) ⟶^{ϑ_{ℌ}} ℌ is the canonical functor of 1.4.4.

Explicitly, objects of the category 𝔖𝔭𝔢𝔠^{1}(𝔊, F) are triples (z, x; ϕ), where z is an object of 𝔖𝔭𝔢𝔠^{1}(ℌ), x ∈ Ob𝔊, and ϕ is an isomorphism z ⟶^{∼} F (x). Morphisms from (z, x; ϕ) to (z′, x′; ϕ′) are given by pairs of arrows, z ⟶^{g} z′ and x ⟶^{h} x′ such that the diagram

z ⟶^{g} z′
ϕ ↓≀  ≀↓ ϕ′
F (x) ⟶^{F (h)} F (x′)

commutes. The projections 𝔖𝔭𝔢𝔠^{1}(ℌ) ⟵^{π_{1}^{F}} 𝔖𝔭𝔢𝔠^{1}(𝔊, F) ⟶^{θ_{F}^{1}} 𝔊 in the left diagram (1) are defined by π_{1}^{F} (z, x; ϕ) = z and θ_{F}^{1} (z, x; ϕ) = x.

Similarly, objects of the category 𝔖𝔭𝔢𝔠^{0}(𝔊, F) are triples (z, x; ψ), where z is an object of 𝔖𝔭𝔢𝔠^{0}(ℌ), x ∈ Ob𝔊, and ψ is an isomorphism ϑ_{ℌ}(z) ⟶^{∼} F (x).

1.6.1. Proposition. Let i be 0 or 1. The map (𝔊, F) ⟼ 𝔖𝔭𝔢𝔠^{i}(𝔊, F) extends to a pseudo-functor 𝔖𝔭𝔢𝔠^{i} : Cat/ℌ ⟶ Cat.

Proof. The assertion follows from the universal property of cartesian squares.

1.6.2. Proposition. Suppose ℌ is a preorder with finite coproducts. Then for every functor 𝔊 ⟶^{F} ℌ, there is a canonical functor

𝔖𝔭𝔢𝔠^{0}(𝔊, F) ⟶^{ϑ_{(𝔊,F)}} 𝔖𝔭𝔢𝔠^{1}(𝔊, F).  (2)

The family {ϑ_{(𝔊,F)} | (𝔊, F) ∈ ObCat/ℌ} is a morphism of pseudo-functors,

𝔖𝔭𝔢𝔠^{0} ⟶^{ϑ} 𝔖𝔭𝔢𝔠^{1}.  (3)

Proof. Since ℌ is a preorder with finite coproducts, the functor 𝔖𝔭𝔢𝔠^{0}(ℌ) ⟶^{ϑ_{ℌ}} ℌ takes values in 𝔖𝔭𝔢𝔠^{1}(ℌ), hence it factors through the embedding 𝔖𝔭𝔢𝔠^{1}(ℌ) ⟶ ℌ (see 1.4.6). By the universal property of cartesian squares, there exists a unique functor (2) such that θ_{F}^{1} ∘ ϑ_{(𝔊,F)} = ϑ_{F} and π_{1}^{F} ∘ ϑ_{(𝔊,F)} = π_{0}^{F} (see the diagram (2)).

It is useful to have an explicit description of the functor (2) in terms of the descripitons of 𝔖𝔭𝔢𝔠^{0}(𝔊, F) and 𝔖𝔭𝔢𝔠^{1}(𝔊, F) given above. The functor ϑ_{(𝔊,F)} maps an object (z, x; ψ) of 𝔖𝔭𝔢𝔠^{0}(𝔊, F) to the object (ϑ_{ℌ}(z), x; ψ) of 𝔖𝔭𝔢𝔠^{1}(𝔊, F).

It follows from this description that ϑ = {ϑ_{(𝔊,F)} | (𝔊, F) ∈ ObCat/ℌ} is a morphism of pseudo-functors.

2. Applications: spectra of ’spaces’.

2.0. Preliminaries on ’spaces’ and localizations.

’Spaces’ here are spaces of noncommutative algebraic geometry. In the simplest (or the most abstract) setting, they are represented by categories. Morphisms of ’spaces’ are functors regarded as inverse image functors. We denote by C_{X} the category representing a ’space’ X and by f^{∗} a functor C_{Y} ⟶ C_{X} representing a morphism X ⟶^{f} Y. Formally, ’spaces’ are objects of the category Cat^{op} opposite to the category Cat. The bicategory of ’spaces’ is the bicategory Cat^{op}. The category of ’spaces’ is the category |Cat|^{o} having same objects as Cat^{op}. Morphisms from X to Y are isomorphism classes of (inverse image) functors C_{Y} ⟶ C_{X}.

2.0.1. Localizations and conservative morphisms. Let X be a ’space’ and Σ a family of arrows of the category C_{X}. We denote by Σ^{−1}X the object of |Cat|^{o} such that the corresponding category coincides with (the standard realization of) the category of fractions ot C_{X} for Σ (cf. [GZ], 1.1): C_{Σ^{−1}X} = Σ^{−1}C_{X}. We call Σ^{−1}X the ’space’ of

fractions of X for Σ. The canonical localization functor C_{X} ⟶^{q_{Σ}^{∗}} Σ^{−1}C_{X} is regarded as an inverse image functor of a morphism, Σ^{−1}X ⟶^{q_{Σ}} X.

For any morphism f : X ⟶ Y in |Cat|^{o}, we denote by Σ_{f} the family of all morphisms s of the category C_{Y} such that f^{∗}(s) is invertible (notice that Σ_{f} does not depend on the choice of an inverse image functor f^{∗}). Thanks to the universal property of localizations, f^{∗} is represented as the composition of the localization functor p_{f}^{∗} = p∗_{Σ_{f}} : C_{Y} ⟶ Σ_{f}^{−1}C_{Y}

and a uniquely determined functor f_{𝔠}^{∗} : Σ_{f}^{−1}C_{Y} ⟶ C_{X}. In other words, f = p_{f} ∘ f_{𝔠} for a uniquely determined morphism f_{𝔠} : X ⟶ Σ_{f}^{−1}Y.

A morphism f : X ⟶ Y is called conservative if Σ_{f} consists of isomorphisms only, or, equivalently, p_{f} is an isomorphism.

A morphism f : X ⟶ Y is called a localization if f_{𝔠} is an isomorphism, i.e. the functor f_{𝔠}^{∗} is an equivalence of categories.

Thus, f = p_{f} ∘ f_{𝔠} is a decomposition of a morphism f into a localization and a conservative morphism.

2.0.2. Multiplicative systems. A family of arrows Σ of a category C_{X} is called a left multiplicative system if it has the following properties:

(S1) Σ is closed under composition and contains all identical arrows of C_{X}.

(SL2) Every diagram M′ ⟵^{s} M ⟶^{f} L, where s ∈ Σ, can be completed to a commutative square

M ⟶^{f} L
s ↓  ↓ s′
M′ ⟶^{f′} L′

where s′ ∈ Σ.

(SL3) If M \overrightarrow{⟶}^{f}_{g} N is a pair of arrows such that f ∘ s = g ∘ s for some s ∈ Σ, then

there exists a morphism N ⟶^{t} N′ of Σ such that t ∘ f = t ∘ g.

A family Σ ⊆ HomC_{X} is a right multiplicative system if it has dual properties. Finally, Σ is called multiplicative system if it is both right and left multiplicative.

We denote by SM_{ℓ}(X) (resp. by SM_{𝔯}(X)) the family of all left (resp. right) multiplicative systems in C_{X}. We denote by SM(X) the family SM_{ℓ}(X) ⋂ SM_{𝔯}(X) of all multiplicative systems in C_{X}.

We regard SM_{ℓ}(X), SM_{𝔯}(X), and SM(X) as preorders with respect to ⊆.

2.0.3. Saturated multiplicative systems. Let Σ be a family of morphisms of the

category C_{X}. Let q_{Σ} be the localization morphism Σ^{−1}X ⟶ X and C_{X} ⟶^{q_{Σ}^{∗}} C_{Σ^{−1}X} = Σ^{−1}C_{X} its canonical inverse image functor.

The family Σ^{𝔰} = Σ_{q_{Σ}} of all arrows of C_{X} which q∗_{Σ} transfers into isomorphisms (cf. 1.2) is called the saturation of Σ. A family of morphisms Σ is called saturated if Σ coincides with its saturation.

We denote by S^{𝔰}M_{ℓ}(X) (resp. by S^{𝔰}M_{𝔯}(X)) the family of all saturated left (resp. right) multiplicative systems in C_{X}. We denote by S^{𝔰}M(X) the family of all saturated (left and right) multiplicative systems in C_{X}; that is S^{𝔰}M(X) = S^{𝔰}M_{ℓ}(X) ⋂ S^{𝔰}M_{𝔯}(X).

We regard S^{𝔰}M_{ℓ}(X), S^{𝔰}M_{𝔯}(X), and S^{𝔰}M(X) as preorders with respect to ⊆.

2.0.3.1. Saturated multiplicative systems and thick subcategories. If C_{X} is an abelian category, then there is an isomorphism between the preorder S^{𝔰}M(X) of saturated multiplicative systems of C_{X} and the preorder 𝔗𝔥(X) of thick subcategories of C_{X}. Recall that a full subcategory 𝕋 of C_{X} is thick if it is closed under taking subquotients and extensions. This isomorphism is given by the map which assigns to each thick subcategory 𝕋 of C_{X} the family Σ_{𝕋} of all arrows s of C_{X} such that both kernel and cokernel of s belong to 𝕋. Inverse isomorphism assigns to a multiplicative system Σ the kernel of the localization at Σ.

2.0.4. Right exact, left exact, and exact morphisms. A morphism X ⟶^{f} Y is called right exact (resp. left exact, resp exact), if its inverse image functor preserves colimits (resp. limits, resp. both limits and colimits) of arbitrary finite diagrams.

The following assertion is a reformulation of Propositions 1.3.1 and 1.3.4 in [GZ].

2.0.4.1. Proposition. (a) Let Σ be a left multiplicative system in C_{X}. Then the canonical morphism Σ^{−1}X ⟶^{q_{Σ}} X is right exact.

(b) Let f = p_{f} ∘ f_{𝔠} be the canonical decomposition of a morphism f : X ⟶ Y into

a conservative morphism X ⟶^{f_{𝔠}} Σ_{f}^{−1}Y and a localization Σ_{f}^{−1}Y ⟶^{p_{f}} Y. Suppose C_{Y} has finite limits (resp. finite colimits). Then f is left exact (resp. right exact) iff the family of arrows Σ_{f} is a left (resp. right) multiplicative system. In this case both the localization p_{f} and the conservative morphism f_{𝔠} are left (resp. right) exact.

In particular, if the category C_{Y} has limits and colimits of finite diagrams, then f is exact iff both the localization p_{f} and the conservative component f_{𝔠} are exact. The exactness of p_{f} is equivalent to that Σ_{f} ∈ S^{𝔰}M(X).

Thus, if a category C_{X} admits finite colimits, then saturated left multiplicative systems of C_{X} classify right exact localizations of C_{X}.

2.0.5. Closed multiplicative systems. Let Σ ⊆ HomC_{X}. We say that an object, M, of the category C_{X} is Σ-torsion free if every morphism M ⟶^{t} N which belongs to Σ is a strict monomorphism.

We say that Σ is closed, or right closed, if for every M ∈ ObC_{X}, there exist a Σ-torsion free object M̃ and an arrow M ⟶ M̃ of Σ which belongs to Σ.

We denote by ℭS^{𝔰}M(X) the preorder of all closed saturated multiplicative systems of the category C_{X}.

2.0.6. Flat multiplicative systems. We call a saturated multiplicative system Σ

flat if the localization functor C_{X} ⟶^{q_{Σ}^{∗}} Σ^{−1}C_{X} has a right adjoint.

We denote by 𝔏𝔠(X) the preorder of all flat multiplicative systems of C_{X}. By [R4, 5.2.1], every flat multiplicative system is closed, i.e. 𝔏𝔠(X) ⊆ ℭS^{𝔰}M(X).

2.1. The spectra of exact localizations.

For a ’space’ X, we take as ℌ the preorder S^{𝔰}M(X) of saturated multiplicative systems of C_{X} and set

Spec_{𝔏}^{1}(X) = 𝔖𝔭𝔢𝔠^{1}(S^{𝔰}M(X)) and Spec_{𝔏}^{0}(X) = 𝔖𝔭𝔢𝔠^{0}(S^{𝔰}M(X)).

Since S^{𝔰}M(X) is a preorder, there exists a canonical injective morphism

Spec_{𝔏}^{0}(X) ⟶ Spec_{𝔏}^{1}(X), Σ ⟼ Σ̂

(cf. 1.4.6). Notice that the support, 𝔖𝔲𝔭𝔭_{S^{𝔰}M(X)}(Σ), of Σ consists of all saturated multiplicative systems of C_{X} which do not contain Σ, and its final object, Σ̂, is the union of all multiplicative systems which belong to the support of Σ.

2.2. Closed spectra and flat spectra.

The closed spectra of a ’space’ X are relative spectra corresponding to the inclu-

sion functor, ℭS^{𝔰}M(X) ⟶^{𝔍_{ℭ}(X)} S^{𝔰}M(X), where ℭS^{𝔰}M(X) is the preorder of all closed saturated multiplicative systems of C_{X} (cf. 2.0.5). Thus,

Spec_{ℭ}^{i}(X) = 𝔖𝔭𝔢𝔠^{i}(ℭS^{𝔰}M(X), 𝔍_{ℭ}(X)), i = 0, 1.

Similarly, the flat spectra of X are relative spectra of (𝔏𝔠(X), 𝔍_{𝔏}(X)), where 𝔏𝔠(X) is the preorder of all flat saturated multiplicative systems of C_{X} (cf. 2.0.6) and 𝔍_{𝔏}(X) is the inclusion functor 𝔏𝔠(X) ↪ S^{𝔰}M(X). Thus,

Spec_{𝔣𝔏}^{i}(X) = 𝔖𝔭𝔢𝔠^{i}(𝔏𝔠(X), 𝔍_{𝔏}(X)), i = 0, 1.

2.3. Spectra of ’spaces’ represented by abelian categories.

Fix a ’space’ X such that C_{X} is an abelian category.

2.3.1. Thick spectra. We take as ℌ the preorder 𝔗𝔥(X) of thick subcategories of C_{X} and set

Spec_{𝔗𝔥}^{i}(X) = 𝔖𝔭𝔢𝔠^{i}(𝔗𝔥(X)), i = 0, 1.

Since the preorder 𝔗𝔥(X) is naturally isomorphic to the preorder S^{𝔰}M(X) of saturated multiplicative systems, the isomorphism 𝔗𝔥(X) ⟶^{∼} S^{𝔰}M(X) induces isomorphisms

Spec_{𝔗𝔥}^{i}(X) ⟶^{∼} Spec_{𝔏}^{i}(X), i = 0, 1

such that the diagram

Spec_{𝔗𝔥}^{0}(X) ⟶^{∼} Spec_{𝔏}^{0}(X)
↓  ↓
Spec_{𝔗𝔥}^{1}(X) ⟶^{∼} Spec_{𝔏}^{1}(X)

commutes. Here the vertical arrows are canonical embeddings of 1.4.6.

2.3.2. Representatives of Spec_{𝔗𝔥}^{0}(X). Let C_{𝔐(X)} denote the subcategory of C_{X} formed by all monomorphisms of C_{X}. The map which assigns to each object, M, of the category C_{X} the smallest thick subcategory, [M]_{•}, containing M defines a functor C_{𝔐(X)} ⟶^{𝔉_{X}} 𝔗𝔥(X). We denote by Spec_{𝔗𝔥}^{0}(X) the preimage, 𝔉_{X}^{−1}(Spec_{𝔗𝔥}^{0}(X)), of the spectrum Spec_{𝔗𝔥}^{0}(X). An object M of Spec_{𝔗𝔥}^{0}(X) is regarded as a representative of the object [M]_{•} of Spec_{𝔗𝔥}^{0}(X).

It follows from [R4, 7.1.1] that the functor

Spec_{𝔗𝔥}^{0}(X) ⟶ Spec_{𝔗𝔥}^{0}(X), M ⟼ [M]_{•},  (1)

is surjective. Namely, if P is an object of Spec_{𝔗𝔥}^{0}(X), then P = [M]_{•} for any M ∈ ObP − ObP̂. Here P̂ is the union of all thick subcategories of C_{X} which do not contain P.

2.3.3. Closed and flat spectra. Let ℭ𝔗𝔥(X) be the preorder of coreflective thick subcategories of the category C_{X}, and 𝔍_{ℭ𝔗}(X) the inclusion functor ℭ𝔗𝔥(X) ↪ 𝔗𝔥(X). Recall that a full subcategory 𝕋 of C_{X} is coreflective if the inclusion functor 𝕋 ⟶ C_{X} has a right adjoint. In other words, every object of C_{X} has the biggest subobject which belongs to 𝕋.

The coreflective spectra of X are defined by

Spec_{ℭ𝔗𝔥}^{i}(X) = 𝔖𝔭𝔢𝔠^{i}(ℭ𝔗𝔥(X), 𝔍_{ℭ𝔗}(X)), i = 0, 1.

By [R4, 7.2.1], the isomorphism 𝔗𝔥(X) ⟶^{∼} S^{𝔰}M(X) induces an isomorphism of the preorder ℭ𝔗𝔥(X) and the preorder ℭS^{𝔰}M(X) of closed saturated multiplicative systems. Therefore, Spec_{ℭ𝔗𝔥}^{i}(X) is isomorphic to the closed spectrum, Spec_{ℭ}^{i}(X) defined in 2.2.

Let 𝔗𝔥_{𝔠}(X) denote the preorder of all thick subcategories 𝕋 such that the localization functor C_{X} ⟶ C_{X}/𝕋 has a right adjoint. And let 𝔍_{𝔠}(X) denote the inclusion functor 𝔗𝔥_{𝔠}(X) ↪ 𝔗𝔥(X). Since the preorder 𝔗𝔥_{𝔠}(X) is isomorphic to the preorder 𝔏𝔠(X) of flat saturated systems of C_{X} (cf. 2.2), the spectrum

Spec_{𝔗𝔥_{𝔠}}^{i}(X) = 𝔖𝔭𝔢𝔠^{i}(𝔗𝔥_{𝔠}(X), 𝔍_{𝔠}(X))

is isomorphic to the corresponding flat spectrum Spec_{𝔣𝔏}^{i}(X) defined in 2.2. Here i = 0, 1.

2.3.4. Relative Serre spectrum. Fix an abelian category C_{X}. Let 𝕋 be a subcategory of C_{X}. We denote by 𝕋^{−} the full subcategory of C_{X} generated by all objects L of C_{X} such that any nonzero subquotient of L has a nonzero subobject which belongs to 𝕋. By [R, III.2.3.2], for any subcategory 𝕋, the category 𝕋^{−} is thick, and (𝕋^{−})^{−} = 𝕋^{−}.

A subcategory 𝕋 of C_{X} is called a Serre subcategory if 𝕋 = 𝕋^{−}.

Recall that X (or the category C_{X}) has the property (sup) if for any ascending chain, Ω, of subobjects of an object M, the supremum of Ω exists, and for any subobject L of M, the natural morphism

sup(N ∩ L | N ∈ Ω) ⟶ (supΩ) ∩ L

is an isomorphism.

2.3.4.1. Lemma. (a) Any coreflective thick subcategory of an abelian category C_{X} is a Serre subcategory.

(b) If X has the property (sup), then any Serre subcategory of C_{X} is coreflective.

Proof. See [R, III.2.4.4].

Let 𝔖𝔢(X) be the preorder of Serre subcategories of C_{X} and 𝔍_{𝔖𝔢}(X) the inclusion functor 𝔖𝔢(X) ↪ 𝔗𝔥(X). The Serre spectra, or, shortly, S-spectra of X are defined by

Spec_{𝔖𝔢}^{i}(X) = 𝔖𝔭𝔢𝔠^{i}(𝔖𝔢(X), 𝔍_{𝔖𝔢}(X)), i = 0, 1.

Since 𝔗𝔥_{𝔠}(X) ⊆ ℭ𝔗𝔥(X) ⊆ 𝔖𝔢(X), there are inclusions of spectra,

Spec_{𝔗𝔥_{𝔠}}^{i}(X) ⊆ Spec_{ℭ𝔗𝔥}^{i}(X) ⊆ Spec_{𝔖𝔢}^{i}(X), i = 0, 1.

By 2.3.4.1, if C_{X} is a category with the property (sup), then ℭ𝔗𝔥(X) = 𝔖𝔢(X), in particular the spectra Spec_{𝔖𝔢}^{i}(X) and Spec_{ℭ𝔗𝔥}^{i}(X) coincide. If C_{X} is a Grothendieck category, then 𝔗𝔥_{𝔠}(X) = ℭ𝔗𝔥(X) = 𝔖𝔢(X), hence in this case,

Spec_{𝔗𝔥_{𝔠}}^{i}(X) = Spec_{ℭ𝔗𝔥}^{i}(X) = Spec_{𝔖𝔢}^{i}(X), i = 0, 1.

2.3.4.3. The category Spec_{𝔰}^{0}(X). We denote by Spec_{𝔰}^{0}(X) the full subcategory of the category C_{𝔐(X)} generated by nonzero objects M such that M ∈ Ob[N]_{•} if there exists a nonzero arrow N ⟶ M (see [R4, 7.3.4]). By [R4, 7.3.5], Spec_{𝔰}^{0}(X) ⊆ Spec_{𝔖𝔢}^{0}(X).

It is easy to see that a nonzero object M belongs to Spec_{𝔰}^{0}(X) iff [L]_{•} = [M]_{•} for any nonzero subobject L of M.

2.3.4.4. Proposition. The image, Spec_{𝔰}^{0}(X), of the map

Spec_{𝔰}^{0}(X) ⟶ Spec_{𝔗𝔥}^{0}(X), M ⟼ [M]_{•},  (2)

contains Spec_{ℭ𝔗𝔥}^{0}(X). If the category C_{X} has the property (sup), then the image of (2) coincides with Spec_{ℭ𝔗𝔥}^{0}(X).

Proof. Let P is an object of Spec_{ℭ𝔗𝔥}^{0}(X); i.e. P is an object of Spec_{𝔗𝔥}^{0}(X) such that the thick subcategory P̂ is coreflective. Let M ∈ ObP − ObP̂. Since P̂ is coreflective, M has a P̂-torsion, 𝔱_{P}M. Replacing M by the quotient M/𝔱_{P}M, we can assume that M is P̂-torsion free. Since P̂ = ⟨M⟩_{•}, it follows from [R4, 7.3.6] that M is an object of Spec_{𝔰}^{0}(X) such that [M]_{•} = P.

By [R4, 7.3.5], the subcategory ⟨M⟩_{•} is a Serre subcategory for every object M of Spec_{𝔰}^{0}(X). If the category C_{X} has the property (sup), then, by [R4, 7.3.8], every Serre subcategory of C_{X} is coreflective.

2.3.5. Spectra related to topologizing subcategories. Let 𝔗(X) denote the preorder of all topologizing subcategories of the category C_{X}. Let 𝔍_{X}^{𝔱} denote the inclusion functor 𝔗𝔥(X) ⟶ 𝔗(X).

Thus, we have two spectra associated to this functor,

Spec_{𝔱}^{i}(X) = 𝔖𝔭𝔢𝔠^{i}(𝔗𝔥(X), 𝔍_{X}^{𝔱}), i = 0, 1.

and a canonical morphism from one to another, Spec_{𝔱}^{0}(X) ⟶ Spec_{𝔱}^{1}(X).

2.3.5.1. Proposition. (a) There is a natural map Spec_{𝔱}^{1}(X) ⟶ Spec_{𝔗𝔥}^{1}(X).

(b) The functor 𝔗(X) ⟶ 𝔗𝔥(X) which assigns to every topologizing subcategory 𝕋 the smallest thick subcategory, 𝕋_{•}, containing 𝕋, induces a functor Spec_{𝔱}^{0}(X) ⟶ Spec_{𝔗𝔥}^{0}(X).

Proof. (a) The category Spec_{𝔱}^{1}(X) is defined by the cartesian square

Spec_{𝔱}^{1}(X) ⟶ 𝔗𝔥(X)
↓  ↓
𝔖𝔭𝔢𝔠^{1}(𝔗(X)) ⟶ 𝔗(X)

in which the right vertical arrow and the lower horizontal arrow are inclusions (see 1.6). It follows from this description (or from the explicit description of relative spectra in 1.6) that objects of Spec_{𝔱}^{1}(X) are naturally identified with thick subcategories, P, which are objects of 𝔖𝔭𝔢𝔠^{1}(𝔗(X)). The latter means that that there exists the smallest topologizing subcategory, P^{𝔱}, properly containing P. Therefore, the smallest thick subcategory, [P^{𝔱}]_{•}, containing P^{𝔱} is the smallest thick subcategory properly containing P, hence P belongs to Spec_{𝔗𝔥}^{1}(X).

(b) The spectrum Spec_{𝔱}^{0}(X) is defined by the cartesian square

\begin{matrix}Spec_{𝔱}^{0}(X) & ⟶ & 𝔗𝔥(X) \\ ↓ & & ↓ \\ 𝔖𝔭𝔢𝔠^{0}(𝔗(X)) & ⟶ & 𝔗(X) \end{matrix}  (1)

(see 1.6). By definition, objects of 𝔖𝔭𝔢𝔠^{0}(𝔗(X)) are topologizing subcategories, P, such that 𝔖𝔲𝔭𝔭_{𝔗(X)}(P) has a final object. This means, precisely, that the union, P̂^{𝔱}, of all topologizing subcategories which do not contain P is a topologizing subcategory. The lower horizontal arrow of the diagram (1) maps an element P to P̂^{𝔱}.

It follows that objects of Spec_{𝔱}^{0}(X) can be identified with topologizing subcategories, P, such that P̂^{𝔱} is a thick subcategory. Therefore, P̂^{𝔱} is a final object of the support 𝔖𝔲𝔭𝔭_{𝔗𝔥(X)}(P_{•}), where P_{•} is the smallest thick subcategory containing P. This implies that the map

Spec_{𝔱}^{0}(X) ⟶ 𝔗𝔥(X), P ⟼ P_{•},

takes values in Spec_{𝔗𝔥}^{0}(X), hence the assertion.

2.3.5.2. Representatives of Spec_{𝔱}^{0}(X). Let C_{𝔐(X)} denote the subcategory of C_{X} formed by all monomorphisms of C_{X}. The map which assigns to each object, M, of the category C_{X} the smallest topologizing subcategory, [M], containing M defines a functor C_{𝔐(X)} ⟶^{𝔉_{X}} 𝔗(X). We denote by Spec_{𝔱}^{0}(X) the preimage, 𝔉_{X}^{−1}(Spec_{𝔱}^{0}(X)), of the spectrum Spec_{𝔱}^{0}(X). An object M of Spec_{𝔱}^{0}(X) is regarded as a representative of the object [M] of Spec_{𝔱}^{0}(X).

2.3.5.3. Proposition. The functor

Spec_{𝔱}^{0}(X) ⟶ Spec_{𝔱}^{0}(X), M ⟼ [M],

is surjective.

Proof. Let P be an object of Spec_{𝔱}^{0}(X). For any M ∈ ObP − ObP̂^{𝔱}, the union, ⟨M⟩, of all topologizing subcategories of C_{X} which do not contain the object M coincides with P̂^{𝔱}. In fact, ⟨M⟩ ⊆ P̂^{𝔱}, because M ∈ ObP; and P̂^{𝔱} ⊆ ⟨M⟩, because M ∉ ObP̂^{𝔱}.

It remains to notice that [M] = P. Clearly [M] ⊆ P. The inverse inclusion, P ⊆ [M], holds because if P ⊈ [M], then [M] ⊆ ⟨M⟩ which is impossible by the definition of ⟨M⟩.

2.3.6. Closed spectra defined by topologizing subcategories. Let 𝔍_{X}^{𝔰} be the inclusion functor ℭ𝔗𝔥(X) ⟶ 𝔗(X). This functor creates two spectra,

Spec_{ℭ𝔱}^{i}(X) = 𝔖𝔭𝔢𝔠^{i}(ℭ𝔗𝔥(X), 𝔍_{X}^{𝔰}), i = 0, 1.

and a canonical morphism from one to another, Spec_{ℭ𝔱}^{0}(X) ⟶ Spec_{ℭ𝔱}^{1}(X).

2.3.6.1. The spectrum Spec(X). We denote by Spec(X) the full subcategory of the category C_{𝔐(X)} generated by nonzero objects M such that M ∈ Ob[N] if there exists a nonzero morphism N ⟶ M.

Since [N] ⊆ [N]_{•} for every object N, it follows that Spec(X) ⊆ Spec_{𝔰}^{0}(X). In particular, Spec(X) ⊆ Spec_{𝔖𝔢}^{0}(X).

It is easy to see that a nonzero object M belongs to Spec(X) iff [L] = [M] for every nonzero subobject L of M. In other words, the functor

C_{𝔐(X)} ⟶ 𝔗(X), M ⟼ [M],

maps every nonzero (mono)morphism L ⟶ M to the (identical) isomorphism.

2.3.6.2. Proposition. The image Spec(X) of the map

Spec(X) ⟶ Spec_{𝔱}^{0}(X), M ⟼ [M],  (1)

contains Spec_{ℭ𝔱}^{0}(X). If the category C_{X} has the property (sup), then the image of (1) coincides with Spec_{ℭ𝔱}^{0}(X).

Proof. Let P is an object of Spec_{ℭ𝔱}^{0}(X); i.e. P is an object of Spec_{𝔱}^{0}(X) such that P̂^{𝔱} is a coreflective thick subcategory. Let M ∈ ObP − ObP̂^{𝔱}. Since P̂^{𝔱} is coreflective, M has a P̂^{𝔱}-torsion, 𝔱_{P^{𝔱}}M. Replacing M by the quotient M/𝔱_{P^{𝔱}}M, we assume that M is P̂^{𝔱}-torsion free. Since P̂^{𝔱} is the union, ⟨M⟩, of all topologizing subcategories of C_{X} which do not contain M, it follows that M is an object of Spec(X) such that [M] = P.

Since ⟨M⟩ is thick, it coincides with ⟨M⟩_{•}. In particular, it is (by [R4, 7.3.5]) a Serre subcategory. If the category C_{X} has the property (sup), then, by [R4, 7.3.8], every Serre subcategory of C_{X} is coreflective.

2.4. Spectra defined by Serre subcategories.

Let ℌ be the preorder 𝔖𝔢(X) of all Serre subcategories of the category C_{X}. Thus, we have two spectra and an embedding:

𝔖𝔭𝔢𝔠^{0}(𝔖𝔢(X)) ⟶ 𝔖𝔭𝔢𝔠^{1}(𝔖𝔢(X)).

2.4.1. Proposition. There are natural functors

Spec_{𝔖𝔢}^{i}(X) ⟶ 𝔖𝔭𝔢𝔠^{i}(𝔖𝔢(X)), i = 0, 1,

such that the diagram

Spec_{𝔖𝔢}^{0}(X) ⟶ 𝔖𝔭𝔢𝔠^{0}(𝔖𝔢(X))
↓  ↓
Spec_{𝔖𝔢}^{1}(X) ⟶ 𝔖𝔭𝔢𝔠^{1}(𝔖𝔢(X))

commutes.

Proof. The functor Spec_{𝔖𝔢}^{1}(X) ⟶ 𝔖𝔭𝔢𝔠^{1}(𝔖𝔢(X)) is the inclusion. The functor Spec_{𝔖𝔢}^{0}(X) ⟶ 𝔖𝔭𝔢𝔠^{0}(𝔖𝔢(X)) assigns to each object P of Spec_{𝔖𝔢}^{0}(X) the Serre subcategory P^{−}.

2.5. Spectra defined by closed cosubspaces.

Let ℭ𝔗(X) denote the preorder of all coreflective topologizing subcategories of the category C_{X}. Let 𝔍_{X}^{𝔠𝔱} denote the inclusion functor ℭ𝔗𝔥(X) ⟶ ℭ𝔗(X).

Thus, we have two spectra associated to this functor,

Spec_{𝔠}^{i}(X) = 𝔖𝔭𝔢𝔠^{i}(𝔗𝔥(X), 𝔍_{X}^{𝔠𝔱}), i = 0, 1.

and a canonical morphism from one to another, Spec_{𝔠}^{0}(X) ⟶ Spec_{𝔠}^{1}(X).

If the category C_{X} has the property (sup), then the preorder ℭ𝔗𝔥(X) of coreflective thick subcategories coincides with the preorder 𝔖𝔢(X) of Serre subcategories.

2.5.1. Proposition. Suppose that the category C_{X} has the property (sup).

(a) There is a natural map Spec_{𝔠}^{1}(X) ⟶ 𝔖𝔭𝔢𝔠^{1}(𝔖𝔢(X).

(b) The functor

ℭ𝔗(X) ⟶ 𝔖𝔢(X), 𝕋 ⟼ 𝕋^{−},

(see 2.4) induces a functor Spec_{𝔠}^{0}(X) ⟶ 𝔖𝔭𝔢𝔠^{0}(𝔖𝔢(X).

Proof. The argument is similar to that of 2.3.5.1. Details are left to the reader.

2.5.2. The spectrum Spec_{𝔠}^{0}(X). We denote by Spec_{𝔠}^{0}(X) the full subcategory of the category C_{𝔐(X)} generated by nonzero objects M such that M ∈ Ob[N]_{𝔠} if there exists a nonzero morphism N ⟶ M. Here [N]_{𝔠} denotes the smallest coreflective topologizing subcategory of C_{X} containing the object N.

Since [N]_{𝔠} ⊆ [N]^{−} for every object N, it follows that Spec_{𝔠}^{0}(X) ⊆ Spec_{𝔰}^{0}(X). In particular, Spec(X) ⊆ Spec_{𝔖𝔢}^{0}(X).

2.5.2.1. Remarks. (a) It is easy to show that a nonzero object M belongs to Spec_{𝔠}^{0}(X) iff [L]_{𝔠} = [M]_{𝔠} for every nonzero subobject L of M. In other words, the functor

C_{𝔐(X)} ⟶ ℭ𝔗(X), M ⟼ [M]_{𝔠},

maps every nonzero (mono)morphism L ⟶ M to the (identical) isomorphism.

(b) Suppose the category C_{X} has infinite coproducts. Then one can show that, for any object M ∈ ObC_{X}, objects of the subcategory [M]_{𝔠} are subquotients of a coproduct of a set of copies of M, while objects of the subcategory [M] are subquotients of a coproduct of a finite set of copies of M.

Thus, a nonzero object M belongs to Spec_{𝔠}^{0}(X) iff M is a subquotient of a coproduct of a set of copies of any of its nonzero subobjects. And a nonzero object M′ belongs to Spec(X) iff M′ is a subquotient of a coproduct of a finite set of copies of any of its nonzero subobjects.

2.5.3. Proposition. Suppose that the category C_{X} has the property (sup). Then the map

Spec_{𝔠}^{0}(X) ⟶ Spec_{𝔠}^{0}(X), M ⟼ [M]_{𝔠},  (1)

is surjective.

Proof. Let P is an object of Spec_{𝔠}^{0}(X); i.e. P is a coreflective topologizing subcategory such that the union, P̂^{𝔠𝔱}, of all coreflective topologizing subcategories of C_{X} which do not contain P is a Serre subcategory. Let M ∈ ObP − ObP̂^{𝔠𝔱}. Since P̂^{𝔠𝔱} is coreflective, we can and will assume M that M is P̂^{𝔠𝔱}-torsion free. Since P̂^{𝔠𝔱} is the union, ⟨M⟩_{𝔠}, of all topologizing subcategories of C_{X} which do not contain M, it follows that M is an object of Spec_{𝔠}^{0}(X) such that [M]_{𝔠} = P.

Recall that an object M of a category C_{X} is of finite type if the functor C_{X}(M, −) preserves colimits of filtered systems of monomorphisms. If the category C_{X} has the property (sup), then M is of finite type iff the following condition holds: if M is the supremum of a family, 𝔉, of its subobjects, then M is the supremum of a finite subfamily of 𝔉. If C_{X} is the category of modules over some associative ring, then its objects of finite type are finitely generated modules.

2.5.4. Proposition. Suppose that the category C_{X} has the property (sup) and every nonzero object of C_{X} has a nonzero subobject of finite type. Then Spec_{𝔠}^{0}(X) = Spec(X).

Proof. The inclusion Spec(X) ⊆ Spec_{𝔠}^{0}(X) holds without any additional hypothesis. The inverse inclusion is a consequence of the following observations.

(a) Thanks to the property (sup), the smallest coreflective subcategory spanned by a topologizing subcategory, 𝕋, is generated by objects which are supremums of objects of 𝕋. In particular, for any object N of the category C_{X}, objects of the subcategory [N]_{𝔠} are supremums (of a filtered family) of their subobjects which belong to [N]. This implies that every object of finite type of the category [N]_{𝔠} belongs to [N].

(b) Let P be an object of Spec_{𝔠}^{0}(X). By 2.5.3, P = [M]_{𝔠} for some object M of Spec_{𝔠}^{0}(X). Suppose that M is of finite type. Then M belongs to Spec(X).

In fact, M belongs to the subcategory [N]_{𝔠} for any nonzero subobject N of M. By (a), since M is of finite type, it belongs to [N]. This means that M is an object of Spec(X).

(c) Since [M]_{𝔠} = [L]_{𝔠} for any nonzero subobject, L, of M, and, by hypothesis, M has a nonzero subobject of finite type, we can choose M to be of finite type.

2.5.4.1. Corollary. If C_{X} is the category of left (or right) modules over an associative unital ring, then Spec_{𝔠}^{0}(X) = Spec(X).

2.6. Spectra of ’spaces’ represented by triangulated categories.

The reader is referred to [Ve1] and [Ve2] for background on triangulated categories. Here we only fix notations. Recall that a ℤ-category is a pair (C_{X}, γ), where C_{X} is a category and γ is an action of ℤ on C_{X}; i.e. γ is a monoidal functor from ℤ to the monoidal category 𝔈𝔫𝔡(C_{X}) of functors C_{X} ⟶ C_{X}. Here ℤ is regarded as a discrete category with the monoidal structure given by m ⊙ n = m + n.

A triangulated category is a triple (C_{X}, γ; 𝔇), where (C_{X}, γ) is an additive ℤ-category, and 𝔇 a class of ’distinguished’ triangles having certain properties (Tr1) – (Tr4) formulated by Verdier (cf. [Ve1] or [Ve2, Ch.2]). We shall denote a triangulated category (C_{X}, γ; 𝔇) by CT_{𝔛} and regard it as a triangulated category representing a triangulated ’space’ 𝔛.

2.6.1. The spectra of exact localizations. Let CT_{𝔛} = (C_{X}, γ; 𝔇) be a triangulated category. Let 𝔗𝔥𝔱(𝔛) denote the preorder of all thick triangulated subcategories of CT_{𝔛}. Following the standard procedure, we associate with the preorder 𝔗𝔥𝔱(𝔛) two spectra of the ’space’ 𝔛 represented by the triangulated category CT_{𝔛}

Spec_{𝔗𝔥𝔱}^{i}(𝔛) = 𝔖𝔭𝔢𝔠^{i}(𝔗𝔥𝔱(𝔛)), i = 0, 1.

and a canonical morphism from one to another,

Spec_{𝔗𝔥𝔱}^{0}(𝔛) ⟶ Spec_{𝔗𝔥𝔱}^{1}(𝔛)  (1)

which assigns to every object P of Spec_{𝔗𝔥𝔱}^{0}(𝔛) the union, P̂^{𝔱𝔯}, of all thick triangulated subcategories of CT_{𝔛} which do not contain the subcategory P.

2.6.2. Flat spectra. Let 𝔖𝔢(𝔛) denote the family of all thick triangulated categories

𝕋 of C_{X} such that the localization functor C_{X} ⟶^{q_{𝕋}^{∗}} C_{X}/𝕋 has a right adjoint, q_{_{𝕋}∗}. The flat spectra of 𝔛 are relative spectra

Spec_{𝔣𝔏}^{i}(𝔛) = 𝔖𝔭𝔢𝔠^{i}(𝔖𝔢(𝔛), 𝔍_{𝔖}), i = 0, 1,

corresponding to the inclusion functor 𝔖𝔢(𝔛) ⟶^{𝔍_{𝔖}} 𝔗𝔥𝔱(𝔛). The morphism (1) induces a canonical morphism

Spec_{𝔣𝔏}^{i}(𝔛) ⟶ Spec_{𝔣𝔏}^{i}(𝔛).  (2)

Let Spec_{𝔣𝔏}^{1/2}(𝔛) denote the full subpreorder of 𝔗𝔥𝔱(𝔛) whose objects are thick triangulated subcategories Q such that ^{⊥}Q belongs to Spec_{𝔣𝔏}^{1}(𝔛) and every thick triangulated subcategory of CT_{𝔛} properly containing ^{⊥}Q contains Q; i.e. ^{⊥}Q ∨ Q is the smallest thick triangulated subcategory of CT_{𝔛} properly containing ^{⊥}Q. Here ^{⊥}Q is the left orthogonal to Q, i.e. the full subcategory of CT_{𝔛} generated by all objects L such that CT_{𝔛}(L, M) = 0 for every M ∈ ObQ.

2.6.2.1. Proposition. (a) The map

𝔗𝔥𝔱(𝔛) ⟶ 𝔗𝔥𝔱(𝔛), Q ⟼ ^{⊥}Q,

induces an isomorphism

Spec_{𝔣𝔏}^{1/2}(𝔛) ⟶^{∼} Spec_{𝔣𝔏}^{1}(𝔛).  (3)

(b) Spec_{𝔣𝔏}^{0}(𝔛) = Spec_{𝔗𝔥𝔱}^{0}(𝔛) ⋂ Spec_{𝔣𝔏}^{1/2}(𝔛). The canonical morphism (2) is the com-

position of the inclusion Spec_{𝔣𝔏}^{0}(𝔛) ↪ Spec_{𝔣𝔏}^{1/2}(𝔛) and the isomorphism (3).

Proof. See [R4, 12.7.1].

For any object M of the category CT_{𝔛}, let [M]_{𝔱} denote the smallest thick triangulated subcategory of CT_{𝔛} containing M, and let ^{⊥}M be the left orthogonal to [M]_{𝔱}.

2.6.2.2. Proposition. (a) If Q is an element of Spec_{𝔣𝔏}^{1/2}(𝔛), then Q = [M]_{𝔱} (hence ⊥Q = ^{⊥}M) for any nonzero object M of Q.

(b) The following properties of a nonzero object M of the triangulated category CT_{𝔛} are equivalent:

(i) The thick envelope, [M]_{𝔱}, of M belongs to Spec_{𝔣𝔏}^{1/2}(𝔛).

(ii) The inclusion functor ^{⊥}M ⟶ CT_{𝔛} has a right adjoint and M belongs to any thick triangulated subcategory of CT_{𝔛} properly containing ^{⊥}M.

(iii) [M]_{𝔱} is a minimal nonzero thick subcategory, and ^{⊥}M belongs to Spec_{𝔣𝔏}^{1}(𝔛). (iv) M belongs to every nonzero thick triangulated subcategory of (^{⊥}M)^{⊥} and such

that the inclusion functor ^{⊥}M ⟶ CT_{𝔛} has a right adjoint.

Proof. See [R4, 12.7.2].

3. The left spectra.

There are ’spaces’ with only trivial saturated multiplicative systems. They might be called simple in the same sense as a ring with only trivial two-sided ideals is called simple. If X is such a ’space’, then Spec_{𝔏}^{1}(X) = {Iso(C_{X})} and Spec_{𝔏}^{0}(X) = {HomC_{X}}. It follows that all other spectra introduced above (closed, flat, etc.) are one-element sets too. Some of simple ’spaces’ have quite meaningful left spectra. The latter are associated with the preorder of saturated left multiplicative systems.

A fundamental example of a simple ’space’ is the ’space’ represented by the category Sets = Sets_{𝔘} of sets which belong to a given universe 𝔘.

3.1. Basic left spectra. Let X be a ’space’. We take as ℌ the preorder S^{𝔰}M_{ℓ}(X) of saturated left multiplicative systems on C_{X} and set

Spec_{𝔏,ℓ}^{i}(X) = 𝔖𝔭𝔢𝔠^{i}(S^{𝔰}M_{ℓ}(X)), i = 0, 1.

By 1.4.6, there exists a canonical injective map

Spec_{𝔏,ℓ}^{0}(X) ⟶ Spec_{𝔏,ℓ}^{1}(X), Σ ⟼ Σ̂.

3.1.1. The spectrum Spec_{𝔏,ℓ}^{1}(X) and left local quotient ’spaces’. We call a ’space’ Y left local if the preorder (S^{𝔰}M_{ℓ}(Y), ⊆) is local, i.e. there is the smallest nontrivial saturated left multiplicative system on C_{Y}. It follows that Spec_{𝔏,ℓ}^{1}(X) is formed by all Σ ∈ S^{𝔰}M_{ℓ}(X) such that the quotient ’space’ Σ^{−1}X is left local.

3.2. Closed left spectra. Let ℭS^{𝔰}M_{ℓ}(X) denote the preorder of all closed saturated left multiplicative systems on C_{X} (cf. 2.0.5). They give rise to the spectra

Spec_{𝔣,ℓ}^{i}(X) = 𝔖𝔭𝔢𝔠^{i}(ℭS^{𝔰}M_{ℓ}(X)), i = 0, 1.

and the relative spectra

Spec_{ℭ,ℓ}^{i}(X) = 𝔖𝔭𝔢𝔠^{i}(ℭS^{𝔰}M_{ℓ}(X), 𝔍_{ℭ,ℓ}), i = 0, 1,

where 𝔍_{ℭ,ℓ} denotes the embedding ℭS^{𝔰}M_{ℓ}(X) ↪ S^{𝔰}M_{ℓ}(X).

They are related by canonical injective maps

Spec_{𝔣,ℓ}^{0}(X) ⟶ Spec_{𝔣,ℓ}^{1}(X),
Spec_{ℭ,ℓ}^{0}(X) ⟶ Spec_{ℭ,ℓ}^{1}(X)

(see 1.4.6 and 1.6.2).

3.3. Continuous left multiplicative systems and continuous left spectra. A left multiplicative system Σ in C_{X} is called continuous if the corresponding localization functor C_{X} ⟶ Σ^{−1}C_{X} has a right adjoint.

Let 𝔏𝔠_{ℓ}(X) denote the preorder of continuous saturated left multiplicative systems. By [R4, 5.2.1, 5.2.2], every continuous saturated left multiplicative system is closed, i.e. 𝔏𝔠_{ℓ}(X) ⊆ ℭS^{𝔰}M_{ℓ}(X).

Let 𝔍_{𝔠,ℓ} denote the embedding 𝔏𝔠_{ℓ}(X) ↪ S^{𝔰}M_{ℓ}(X). This data provides us with the continuous left spectra

Spec_{𝔉𝔩,ℓ}^{i}(X) = 𝔖𝔭𝔢𝔠^{i}(𝔏𝔠_{ℓ}(X)), i = 0, 1.

and the relative continuous left spectra

Spec_{𝔣𝔏,ℓ}^{i}(X) = 𝔖𝔭𝔢𝔠^{i}(𝔏𝔠_{ℓ}(X), 𝔍_{𝔠,ℓ}), i = 0, 1,

together with the canonical injective maps

Spec_{𝔉𝔩,ℓ}^{0}(X) ⟶ Spec_{𝔉𝔩,ℓ}^{1}(X),
Spec_{𝔣𝔏,ℓ}^{0}(X) ⟶ Spec_{𝔣𝔏,ℓ}^{1}(X).

3.3.1. Another realization of continuous localizations and continuous left spectra. Fix a ’space’ X. Consider the preorder 𝔣𝔏_{ℓ}(X) of all strictly full subcategories

C_{Y} of C_{X} such that the inclusion functor C_{Y} ⟶^{ι_{_{Y} ∗}} C_{X} has a left adjoint C_{X} ⟶^{ι_{Y}^{∗}} C_{Y}. These functors are regarded as resp. direct and inverse image functors of a continuous strictly full embedding Y ↪^{ι_{Y}} X. The map which assigns to every such subcategory the family of arrows Σ_{ι_{Y}^{∗}} = ι_{Y}^{∗^{−1}}(Iso(C_{Y})) is an isomorphism of the preorder (𝔣𝔏_{ℓ}(X), ⊇) onto the preorder (𝔏𝔠_{ℓ}(X), ⊆) of continuous saturated left multiplicative systems.

Thus, the continuous left spectrum Spec_{𝔣𝔏}^{1}(X) can be identified with the preorder of

all continuous strictly full embeddings Y ↪^{ι_{Y}} X such that Y is a left local ’space’.

Let 𝔣𝔏_{ℓ}^{⋆}(X) denote the set 𝔣𝔏(X) − {id_{X}} of all proper continuous strictly full embeddings. Thanks to the isomorphism (𝔏𝔠(X), ⊆) ⟶^{∼} (𝔣𝔏(X), ⊇) elements of Spec_{𝔉𝔩,ℓ}^{1}(X)

can be identified with continuous strictly full embeddings Y ↪^{ι_{Y}} X such that the preorder (𝔣𝔏^{⋆}(Y), ⊆) of proper strictly full continuous embeddings into Y has the biggest element.

3.4. The left spectra of Sets. Let E denote the ’space’ represented by the category Sets = Sets_{𝔘} of sets which belong to a fixed universe 𝔘, i.e. C_{E} = Sets.

The preorder S^{𝔰}M_{𝔯}(E) of right saturated multiplicative systems on Sets is trivial: it consists only of Iso(C_{E}) and HomC_{E}. In particular, the preorder S^{𝔰}M(E) of saturated multiplicative systems on Sets is trivial.

For an infinite cardinal number α, let Σ_{α} denote the family of all maps M ⟶^{f} N (– morphisms of C_{E}) such that

(a) M ≠ ∅ if N ≠ ∅,

(b) Card(N − f(M)) < α,

(c) There exists a subset M′ of M such that Card(M − M′) < α and the restriction of the map f to M′ is injective.

Let Σ_{α∗} denote the family of all maps M ⟶^{f} N satisfying (b) and (c) only. So that Σ_{α} ⊂ Σ_{α∗}. Explicitly, Σ_{α∗} = Σ_{α} ⋃{∅ ⟶ N | Card(N) < α}.

Both Σ_{α} and Σ_{α∗} are saturated left multiplicative systems on Sets. Moreover, every saturated left multiplicative system on Sets is either Σ_{α} or Σ_{α∗} for a suitable infinite cardinal α (see [GZ], I.2.5f) and I.3.5).

Let 𝔖𝔭(𝔘) denote the order of non-limit cardinals which belong to the universe 𝔘.

3.4.1. Proposition. (a) Let α be an infinite cardinal number. Then the following conditions are equivalent:

(i) α is a non-limit cardinal, (ii) Σ_{α} belongs to Spec_{𝔏,ℓ}^{0}(E).

(b) The map α ⟼ Σ_{α} defines an isomorphism of preorders 𝔖𝔭(𝔘) ⟶^{ϱ^{0}} Spec_{𝔏,ℓ}^{0}(E).

(c) If α is a non-limit infinite cardinal, then Σ_{α−1∗} belongs to Spec_{𝔏,ℓ}^{1}(E). The map

α ⟼ Σ_{α−1∗} defines an isomorphism of preorders 𝔖𝔭(𝔘) ⟶^{ϱ^{1}} Spec_{𝔏,ℓ}^{1}(E) such that the diagram

Spec_{𝔏,ℓ}^{0}(E) ⟶^{θ_{ℓ}(E)} Spec_{𝔏,ℓ}^{1}(E)

ϱ^{0} ↖ ↗ ϱ^{1}

𝔖𝔭(𝔘)

(1)

commutes. In particular, the canonical preorder morphism Spec_{𝔏,ℓ}^{0}(E) ⟶^{θ_{ℓ}(E)} Spec_{𝔏,ℓ}^{1}(E) is an isomorphism.

Proof. (a) Fix an infinite cardinal number α.

(i)⇒(ii). Let α be a non-limit cardinal. Then the union, Σ̂_{α} of all elements of the support of Σ_{α} (that is the union of all saturated left multiplicative system on C_{E} which do not contain Σ_{α}) coincides with Σ_{α−1∗}. This shows that Σ_{α} is an element of Spec_{𝔏,ℓ}^{0}(E).

(ii)⇒(i). If α is a limit cardinal, then Σ̂_{α} = ⋃_{β<α} Σ_{β∗} ⊇ ⋃_{β<α} Σ_{β} = Σ_{α}. This shows that

the support, 𝔖𝔲𝔭𝔭(Σ_{α}), of Σ_{α} does not have the final object, i.e. the left multiplicative system Σ_{α} does not belong to Spec_{𝔏,ℓ}^{0}(E).

(b) Let α be any cardinal of a set from 𝔘. The support of Σ_{α∗} consists of all Σ_{β} and all Σ_{γ∗} with γ < α. Thus Σ̂_{α∗} = HomC_{E^{1}} ⋃{∅ ⟶ N | Card(N) < γ, γ < α}, where C_{E^{1}} is the full subcategory of C_{E} formed by all non-empty sets. It follows from this description that Σ̂_{α∗} is not closed under composition, hence the support of Σ_{α∗} does not have the final object, i.e. the left multiplicative system Σ_{α∗} does not belong to Spec_{𝔏,ℓ}^{0}(E). Since every saturated left multiplicative system on Sets is either Σ_{α}, or Σ_{α∗}, this proves that the map α ⟼ Σ_{α} is an isomorphism 𝔖𝔭(𝔘) ⟶ Spec_{𝔏,ℓ}^{0}(E).

(c) Let α be an infinite non-limit cardinal number. That Σ_{α−1∗} is an element of Spec_{𝔏,ℓ}^{1}(E) follows, together with the commutativity of the diagram (1), from the fact that Σ_{α−1∗} is the final object of the support of Σ_{α} (see the argument (i)⇒(ii) above) and 1.4.6. Clearly the map

𝔖𝔭(𝔘) ⟶^{ϱ^{1}} Spec_{𝔏,ℓ}^{1}(E), α ⟼ Σ_{α−1∗},  (2)

is a morphism of preorders. It remains to show that this map is bijective.

In fact, for any pair of infinite cardinal numbers β and γ such that β < γ, the system Σ_{β} is contained properly in Σ_{γ} and in Σ_{β∗}. On the other hand, Σ_{β} = Σ_{γ} ⋂ Σ_{β∗}. Therefore Σ_{β} does not belong to Spec_{𝔏,ℓ}^{1}(E) for any infinite cardinal number β. Therefore elements of Spec_{𝔏,ℓ}^{1}(E) are systems Σ_{β∗} for some β. Suppose Σ_{β∗} belongs to Spec_{𝔏,ℓ}^{1}(E), i.e. there exists the smallest system Σ_{β∗}^{⋆} in S^{ℓ}M_{𝔯}(E) properly containing Σ_{β∗}. Since Σ_{β∗} is not contained in Σ_{σ} for any σ, the system Σ_{β∗}^{⋆} should coincide with Σ_{α∗} for some α. But, Σ_{β∗} ⊊ Σ_{γ∗} iff β ⪇ γ. Therefore, β ⪇ α and there are no intermediate cardinal numbers, i.e. α is a non-limit cardinal number and β = α − 1.

3.4.2. Other spectra. Let Σ ⊆ HomC_{E}. By definition, and object M of C_{E} is Σ-torsion free if every morphism M ⟶ M′ which belongs to Σ is a monomorphism. Suppose Σ is Σ_{α} or Σ_{α∗} for some infinite cardinal number α. Then a set M is Σ-torsion free iff Card(M) ≤ 1 (that is either M = ∅, or M is a one-element set). It follows from the definitions of Σ_{α} and Σ_{α∗} that objects of C_{E} having a morphism to a Σ-torsion free object are precisely sets N such that Card(N) < α. This shows that the only closed saturated left multiplicative systems on C_{E} are Iso(C_{E}) and HomC_{E}. Since, by [R4, 5.2.2], every continuous saturated left multiplicative system is closed, there are no non-trivial continuous saturated left multiplicative systems either. Therefore, Spec_{𝔣,ℓ}^{0} = Spec_{𝔉𝔩,ℓ}^{0} = {HomC_{E}} and Spec_{𝔣,ℓ}^{1} = Spec_{𝔉𝔩,ℓ}^{1} = {Iso(C_{E})} (see notations in 3.2 and 3.3).

The relative continuous and closed left spectra (cf. 3.2, 3.3) are empty.

3.4.3. Sets without emptyset. Let C_{E^{1}} = Sets_{𝔘}^{1}, where (in accordance with notations in Section 1), Sets_{𝔘}^{1} is the full subcategory of Sets_{𝔘} formed by non-empty sets which belong to the universe 𝔘. It follows that the preorder S^{𝔰}M_{𝔯}(E^{1}) of saturated right multiplicative systems on Sets_{𝔘}^{1} is trivial and the set S^{𝔰}M_{ℓ}(E^{1}) of saturated left multiplicative systems consists of all systems Σ_{α}^{1} = Σ_{α} ⋂ HomC_{E^{1}}, where α runs through infinite cardinal numbers (notice that Σ_{α∗} ⋂ HomC_{E^{1}} = Σ_{α}^{1} for any α). There is a following analogue of 3.4.1:

3.4.4. Proposition. (a) Let α be an infinite cardinal number. Then the following conditions are equivalent:

(i) α is a non-limit cardinal, (ii) Σ_{α}^{1} belongs to Spec_{𝔏,ℓ}^{0}(E^{1}).

(b) The map α ⟼ Σ_{α}^{1} defines an isomorphism of preorders 𝔖𝔭(𝔘) ⟶^{ν^{0}} Spec_{𝔏,ℓ}^{0}(E^{1}).

(c) If α is a non-limit infinite cardinal, then Σ_{α−1}^{1} belongs to Spec_{𝔏,ℓ}^{1}(E^{1}). The map

α ⟼ Σ_{α−1}^{1} defines an isomorphism of preorders 𝔖𝔭(𝔘) ⟶^{ν^{1}} Spec_{𝔏,ℓ}^{1}(E^{1}) such that the diagram

Spec_{𝔏,ℓ}^{0}(E^{1}) ⟶^{θ_{ℓ}(E^{1})} Spec_{𝔏,ℓ}^{1}(E^{1})

ν^{0} ↖ ↗ ν^{1}

𝔖𝔭(𝔘)

(1)

commutes. In particular, the canonical morphism Spec_{𝔏,ℓ}^{0}(E^{1}) ⟶^{θ_{ℓ}(E^{1})} Spec_{𝔏,ℓ}^{1}(E^{1}) is an isomorphism.

Proof. The assertion follows from 3.4.1. Details are left to the reader.

The canonical map Spec_{𝔏,ℓ}^{0}(E^{1}) ⟶^{θ_{ℓ}(E^{1})} Spec_{𝔏,ℓ}^{1}(E^{1}) (of 1.4.6) assigns to each element Σ_{α}^{1} of Spec_{𝔏,ℓ}^{0}(E^{1}) the system Σ_{α−1}^{1}. Notice that the inverse map assigns to each element Σ of Spec_{𝔏,ℓ}^{1}(E^{1}) the smallest saturated left multiplicative system properly containing Σ.

3.5. Example: finite sets. Let C_{E_{𝔣}} be the category Set_{𝔣} of finite sets. There are no non-trivial left or right multiplicative systems on Sets_{𝔣}; so that the ’space’ E_{𝔣} can be viewed as an analog of a ’point’ – the spectrum of a field.

4. Left exact multiplicative systems and injective spectra.

4.1. Left exact multiplicative systems. Fix a ’space’ X. We call a saturated

left multiplicative system Σ left exact if the localization functor C_{X} ⟶^{q_{Σ}^{∗}} Σ^{−1}C_{X} = C_{Σ^{−1}X} maps strict monomorphisms to strict monomorphisms. Let S_{ℓs}^{𝔰}M(X) denote the preorder of all left exact multiplicative systems.

If C_{X} is an abelian category, then every left exact multiplicative system is a right multiplicative system, i.e. S_{ℓs}^{𝔰}M(X) = S^{𝔰}M(X).

4.2. Proposition. Suppose that the category C_{X} has finite colimits and kernels of pairs of arrows. Let C_{X} ⟶^{f^{∗}} C_{Y} be a right exact functor (i.e. it preserves finite colimits) which maps strict monomorphisms to strict monomorphisms, and let Σ = Σ_{f^{∗}} = {s ∈ HomC_{X} | f^{∗}(s) ∈ Iso(C_{X})}. Then both functors, p_{f}^{∗} and q_{f}^{∗}, in the canonical decomposition f^{∗} = p_{f}^{∗}q_{f}^{∗} (here q_{f}^{∗} is the localization functor C_{X} ⟶ Σ^{−1}C_{X} and p_{f}^{∗} is a conservative functor) are right exact and map strict monomorphisms to strict monomorphisms. In particular, Σ ∈ S_{ℓs}^{𝔰}M(X).

Proof. (a) By [GZ, I.3.4], the functors q_{f}^{∗} and p_{f}^{∗} are right exact and Σ = Σ_{f^{∗}} belongs to S_{ℓ}^{𝔰}M(X). It remains to show that q_{f}^{∗} and p_{f}^{∗} map strict monomorphisms to strict monomorphisms.

(b) Let L ⟶^{𝔧} M \overrightarrow{⟶}^{p_{1}}_{p_{2}}M_{1} = M ∐_{L} M be an exact diagram. We claim that its image

by the localization functor q_{f}^{∗} is exact too.

In fact, let a morphism q_{f}^{∗}(N) ⟶^{g′} q_{f}^{∗}(M) equalize the pair q_{f}^{∗}(M \overrightarrow{⟶}^{p_{1}}_{p_{2}}M_{1}). Since Σ

is a left multiplicative system, there exist arrows N ⟶^{g} M′ ⟵^{s} M such that s ∈ Σ and g′ = q_{f}^{∗}(s)^{−1}q_{f}^{∗}(g), and there exist commutative diagrams

M ⟶^{p_{i}} M_{1}
s ↓  ↓ s_{i}
M′ ⟶^{p′_{i}} N_{i}  i = 1, 2,

with s_{1}, s_{2} ∈ Σ. By the same reason, there exists a commutative diagram

M_{1} ⟶^{s_{1}} N_{1}
s_{2}↓  ↓ s′_{2}
N_{2} ⟶^{s′_{1}} N′_{1}

with s′_{2} ∈ Σ. Since the system Σ is saturated and the arrows s_{1}, s_{2}, s′_{2} belong to Σ, the remaining arrow, s′_{1}, belongs to Σ too. Thus, we obtain a commutative diagram

\begin{aligned}L & ⟶^{𝔧} & M & \overrightarrow{⟶}^{p_{1}}_{p_{2}} & M_{1} \\ & & s ↓ & & ↓ t \\ N & ⟶^{g} & M′ & \overrightarrow{⟶}^{ϕ_{1}}_{ϕ_{2}} & N′_{1} \end{aligned}  (1)

where s and t = s′_{2}s_{1} are arrows from Σ and ϕ_{i} = s′_{i}p′_{i}. It follows from the definition of N ⟶^{g} M′ and the commutativity of the diagram (1) that q_{f}^{∗}(ϕ_{1}g) = q_{f}^{∗}(ϕ_{2}g). Since Σ is saturated, this means precisely that there exists an arrow N′_{1} ⟶^{u} M′′_{1} in Σ such that

(uϕ_{1})g = (uϕ_{2})g. Set ϕ′_{i} = uϕ_{i}, i = 1, 2, and let L′ ⟶^{𝔧′} M′ denote the kernel of the

pair of arrows M′ \overrightarrow{⟶}^{ϕ′_{1}}_{ϕ′_{2}}M′′_{1}. Then we have the diagram L′ ⟶^{𝔧′} M′ \overrightarrow{⟶}^{π_{1}}_{π_{2}}M′_{1} ⟶^{σ} M′′_{1}, where

M′_{1} = M′ ∐_{L′} M′, π_{1}, π_{2} are coprojections, and σ is a morphism uniquely determined by the equalities σπ_{i} = ϕ′_{i}, i = 1, 2. Combining these decompositions with (1), we obtain a commutative diagram

\begin{aligned}& & L & ⟶^{𝔧} & M & \overrightarrow{⟶}^{p_{1}}_{p_{2}} & M_{1} & ⟶^{t} & N′_{1} \\ & & s′↓ & & s ↓ & & & & ↓ u \\ N & ⟶^{g′} & L′ & ⟶^{𝔧′} & M′ & \overrightarrow{⟶}^{π_{1}}_{π_{2}} & M′_{1} & ⟶^{σ} & M′′_{1} \end{aligned}  (2)

Here N ⟶^{g′} L′ ⟶^{𝔧′} M′ is the unique decomposition of the morphism N ⟶^{g} M′ and the arrow L ⟶^{s′} L′ is uniquely determined by the commutativity of the diagram (2) and

the fact that L′ ⟶^{𝔧′} M′ is the kernel of the pair of arrows M′ \overrightarrow{⟶}^{σπ_{1}}_{σπ_{2}}M′′_{1}. Applying the

localization functor q_{f}^{∗} to (2) and using that arrows s, t, u, σ belong to Σ, we obtain a commutative diagram

\begin{aligned}& & q_{f}^{∗}(L) & ⟶^{q_{f}^{∗}(𝔧)} & q_{f}^{∗}(M) & \overrightarrow{⟶} & q_{f}^{∗}(M_{1}) \\ & & q_{f}^{∗}(s′) ↓ & & ≀↓ & & ↓≀ \\ q_{f}^{∗}(N) & ⟶^{q_{f}^{∗}(g′)} & q_{f}^{∗}(L′) & ⟶^{q_{f}^{∗}(𝔧′)} & q_{f}^{∗}(M′) & \overrightarrow{⟶} & q_{f}^{∗}(M′_{1}) \end{aligned}  (3)

Since two of the three vertical arrows in (3) are isomorphisms and the diagrams

q_{f}^{∗}(L) ⟶^{q_{f}^{∗}(𝔧)} q_{f}^{∗}(M) \overrightarrow{⟶} q_{f}^{∗}(M_{1})

and

q_{f}^{∗}(L′) ⟶^{q_{f}^{∗}(𝔧′)} q_{f}^{∗}(M′) \overrightarrow{⟶} q_{f}^{∗}(M′_{1})

are exact, the third one, q_{f}^{∗}(s′), is an isomorphism, or, equivalently, s′ ∈ Σ. This shows that

any morphism q_{f}^{∗}(N) ⟶^{g′} q_{f}^{∗}(M) which equalizes the pair q_{f}^{∗}(M \overrightarrow{⟶}^{p_{1}}_{p_{2}}M_{1}) factors uniquely

through the kernel of this pair. Notice that our argument shows the existence of this kernel.

(c) Let q_{f}^{∗}(L) ⟶^{ξ′} q_{f}^{∗}(M′) be a morphism in Σ^{−1}C_{X} and L ⟶^{ξ} M ⟵^{s} M′ morphisms such that s ∈ Σ and ξ′ = q_{f}^{∗}(s)^{−1}q_{f}^{∗}(ξ) (cf. (b) above). Consider the cokernel diagram

L ⟶^{ξ} M \overrightarrow{⟶}^{p_{1}}_{p_{2}} M_{1} = M ∐_{L} M

of the morphism L ⟶^{ξ} M. Since the localization functor q_{f}^{∗} preserves finite colimits, the diagram

q_{f}^{∗}(L ⟶^{ξ} M \overrightarrow{⟶}^{p_{1}}_{p_{2}} M_{1} = M ∐_{L} M)

is isomorphic to the cokernel diagram of the morphism q_{f}^{∗}(L) ⟶^{ξ′} q_{f}^{∗}(M′). Let L′ ⟶^{𝔧} M

denote the kernel of the pair of arrows M \overrightarrow{⟶}^{p_{1}}_{p_{2}} M_{1}, and let L ⟶^{t} L′ ⟶^{𝔧} M be the

canonical decomposition of L ⟶^{ξ} M.

It follows from the construction that q_{f}^{∗}(L) ⟶^{ξ′} q_{f}^{∗}(M′) is a strict monomorphism

iff t ∈ Σ, hence the morphism ξ′ is isomorphic to q_{f}^{∗}(L′) ⟶^{q_{f}^{∗}(𝔧)} q_{f}^{∗}(M). Therefore p_{f}^{∗}(ξ′) is

isomorphic to p_{f}^{∗}(q_{f}^{∗}(L′) ⟶^{q_{f}^{∗}(𝔧)} q_{f}^{∗}(M)) = f^{∗}(L′) ⟶^{f^{∗}(𝔧)} f^{∗}(M). By hypothesis, the functor f^{∗} preserves strict monomorphisms, hence p_{f}^{∗}(ξ′) is a strict monomorphism.

The following assertion is suggested by (the part (b) of) the argument of 4.2.

4.2.1. Proposition. Suppose the category C_{X} has kernels of pairs of arrows and for any arrow L ⟶ M, there exists a pushforward M ∐_{L} M. Then the following conditions on a left saturated multiplicative system Σ are equivalent:

(a) Σ is left exact.

(b) If in the commutative diagram

\begin{matrix}L & ⟶^{𝔧} & M & \overrightarrow{⟶}^{p_{1}}_{p_{2}} & M_{1} = M ∐_{L} M \\ s′↓ & & s ↓ & & ↓ t \\ L′ & ⟶^{𝔧′} & M′ & \overrightarrow{⟶}^{ϕ_{1}}_{ϕ_{2}} & M′′ \end{matrix}  (4)

the rows are exact and the vertical arrows s and t belong to Σ, then the left vertical arrow belongs to Σ too.

Proof. (a)⇒(b). The diagram (4) gives rise to the commutative diagram

\begin{matrix}L & ⟶^{𝔧} & M & \overrightarrow{⟶}^{p_{1}}_{p_{2}} & M_{1} & ⟶^{t} & M′′ \\ s′↓ & & s ↓ & & ↓ u & & ↓ id_{M′′} \\ L′ & ⟶^{𝔧′} & M′ & \overrightarrow{⟶}^{π_{1}}_{π_{2}} & M′_{1} & ⟶^{σ} & M′′ \end{matrix}  (5)

where M′_{1} = M′ ∐_{L′} M′, M_{1} ⟶^{σ} M′′ is uniquely determined by σπ_{i} = ϕ_{i}, i = 1, 2; and M_{1} ⟶^{u} M′_{1} is uniquely defined by the left square in (5) (due to the functoriality of

coproducts). Applying the localization functor C_{X} ⟶^{q^{∗}} Σ^{−1}C_{X} to the diagram (5), we obtain the diagram

\begin{matrix}q^{∗}(L) & \underrightarrow{_{𝔧}} & q^{∗}(M) & \overrightarrow{⟶}^{p′_{1}}_{p′_{2}} & q^{∗}(M_{1}) \\ q^{∗}(s′) ↓ & & q^{∗}(s) ↓ & & u′ ↓ ↑ σ′ \\ q^{∗}(L′) & \underrightarrow{_{𝔧′}} & q^{∗}(M′) & \overrightarrow{⟶}^{π′_{1}}_{π′_{2}} & q^{∗}(M′_{1}) \end{matrix}  (6)

where σ′ = q^{∗}(t)^{−1}q^{∗}(σ). Since the system Σ is left exact, the functor q^{∗} maps the diagrams L ⟶ M\overrightarrow{⟶}M_{1} and L′ ⟶ M′\overrightarrow{⟶}M′_{1} to exact diagrams. Therefore the commutative square

q^{∗}(M) \overrightarrow{⟶} q^{∗}(M_{1})
q^{∗}(s)^{−1}↑  ↑ σ′
q^{∗}(M) \overrightarrow{⟶} q^{∗}(M_{1})

and (6) yield a commutative square

q^{∗}(L) \underrightarrow{_{𝔧}} q^{∗}(M)
σ̃ ↑  ↑ q^{∗}(s)^{−1}
q^{∗}(L′) \underrightarrow{_{𝔧′}} q^{∗}(M′)

It follows from the universal property of kernels that σ̃q^{∗}(s′) = id_{q^{∗}(L)}, hence σ̃ is a strict epimorphism. On the other hand, the equality 𝔧̃σ̃ = q^{∗}(s)^{−1}𝔧̃′ (together with the fact that 𝔧̃′ is a monomorphism) implies that σ̃ is a monomorphism; hence it is an isomorphism.

(b)⇒(a). The argument is a repetition of the part (b) of the argument of 4.2. Details are left to the reader.

4.2.2. Example: left multiplicative systems in Sets. Let C_{E} = Sets. Then S^{𝔰}M_{ℓ}(E) = S^{𝔰}M_{ℓs}(E), i.e. every saturated left multiplicative system on Sets is left exact.

In fact, S^{𝔰}M_{ℓ}(E) consists of systems Σ_{α} and Σ_{α∗}, where α is an infinite cardinal (which belongs to a given universe). Recall that Σ_{α∗} = Σ_{α} ⋃{∅ ⟶ N | Card(N) < α} and Σ_{α} is the family of all maps M ⟶^{f} N (– morphisms of C_{E}) such that

(a) M ≠ ∅ if N ≠ ∅,

(b) Card(N − f(M)) < α,

(c) There exists a subset M′ of M such that Card(M − M′) < α and the restriction of the map f to M′ is injective (see 3.4).

Notice that the conditions (b) and (c) (defining Σ_{α∗}) are equivalent to the following condition which explains the meaning of Σ_{α∗} and is more convenient for our purposes:

(b’) There exists a subset M′ of M such that the restriction of the map f to M′ is injective and Card(M − M′) < α > Card(N − f(M′)).

Let Σ be Σ_{α} or Σ_{α∗}. Consider a commutative square

L ⟶^{𝔧} M
s ↓  ↓ t
L′ ⟶^{𝔧′} M′

in C_{E} = Sets whose horizontal arrows are monomorphisms and the right vertical arrow, M ⟶^{t} M′, belongs to Σ. Then L ⟶^{s} L′ belongs to Σ too.

Suppose first that M = ∅. Then L = ∅. If Σ = Σ_{α}, then L′ = M′ = ∅. In particular, L ⟶^{s} L′ belongs to Σ.

If M = ∅ and Σ = Σ_{α∗}, then Card(M′) < α. Since L′ ⟶^{𝔧′} M′ is an injective map, and Card(L′) ≤ Card(M′) < α. Therefore L ⟶^{s} L′ belongs to Σ_{α∗}.

Suppose now that M ≠ ∅ and L and L′ are subsets of resp. M and M′. The map M ⟶^{t} M′ belongs to Σ iff there exists a subset M′′ of M such that the restriction of t to M′′ is injective and Card(M − M′′) < α > Card(M′ − t(M′′)) (see (b’) above). Then the restriction of the map L ⟶^{s} L′ to L ∩ M′′ is injective and both Card(L − L ∩ M′′) and Card(L′ − s(L ∩ M′′)) are smaller than α. The latter means that s ∈ Σ.

Now it follows from 4.2.1 that Σ is a left exact multiplicative system.

4.3. Injective objects. Fix a ’space’ X. Let C_{X_{s}} denote the subcategory of C_{X} formed by all objects of C_{X} and split monomorphisms.

An object E of the category C_{X} is called injective (or strictly injective) if the functor C_{X}(−, E) : C^{op} ⟶ Sets maps strict epimorphisms (of C^{op}, i.e. strict monomoprhisms of C_{X}) to epimorphisms. We denote by C_{ℑ(X)} the subcategory of C_{X} formed by injective objects and split monomorphisms (or, what is the same, strict monomorphisms, see 1.2.3.3) between them. For any object E of the category C_{X}, let Σ_{E} denote the family of all arrows of C_{X} which the functor C_{X}(−, E) transforms into invertible morphisms.

4.3.1. Proposition. (a) Suppose that the category C_{X} has finite colimits. Then the map E ⟼ Σ_{E} extends to a functor C_{X_{s}}^{op} ⟶ S_{ℓ}^{𝔰}M(X).

(b) If, in addition, C_{X} has kernels of pairs of arrows. Then the map E ⟼ Σ_{E} defines a functor C_{ℑ(X)}^{op} ⟶ S_{ℓs}^{𝔰}M(X).

Proof. (a) Since the category C_{X} has finite colimits, it follows from [GZ, 1.3.4] that Σ_{E} belongs to S_{ℓ}^{𝔰}M(X) for every object E.

Let E_{1} ⟶^{u} E be a split monomorphism; and let L ⟶^{s} M be a morphism of C_{X} such that C_{X}(s, E) is an isomorphism. Then C_{X}(s, E_{1}) is an isomorphism. In fact, there exists a morphism E ⟶^{v} E_{1} such that v ∘ u = id_{E_{1}}. Thus, there are two commutative diagrams

C_{X}(M, E) ⟶^{C_{X}(s,E)} C_{X}(L, E)  C_{X}(M, E_{1}) ⟶^{C_{X}(s,E_{1})} C_{X}(L, E_{1})
C_{X}(M, v) ↓  ↓ C_{X}(L, v)  C_{X}(M, u) ↓  ↓ C_{X}(L, u)
C_{X}(M, E_{1}) ⟶^{C_{X}(s,E_{1})} C_{X}(L, E_{1})  C_{X}(M, E) ⟶^{C_{X}(s,E)} C_{X}(L, E)

such that the vertical arrows and the upper horizontal arrow of the first diagram are surjective (hence the remaining arrow, C_{X}(s, E_{1}) is surjective) and the vertical arrows and the lower horizontal arrow of the second diagram are injective, hence C_{X}(s, E_{1}) is injective. Therefore, C_{X}(s, E_{1}) is bijective, i.e. s ∈ Σ_{E_{1}}.

(b) If E is an injective object, then, by 4.2, the localization at Σ_{E} preserves strict monomorphisms, i.e. Σ_{E} belongs to S_{ℓs}^{𝔰}M(X).

4.4. Proposition. Let X ⟶^{f} Y be a continuous morphism with an inverse image functor f^{∗} and a direct image functor f_{∗}. Suppose that the functor f^{∗} maps strict monomorphisms to strict monomorphisms. Then f_{∗} maps injective objects to injective objects.

Proof. If E is an injective object in C_{Y}, then the functor C_{X}(f^{∗}(−), E) maps strict monomorphisms of the category C_{Y} to (strict) epimorphisms, because f^{∗} preserves strict monomorphisms and, since E is injective, C_{X}(−, E) maps strict monomorphisms to epimorphisms. But, C_{Y} (−, f_{∗}(E)) ≃ C_{X}(f^{∗}(−), E), hence the assertion.

4.5. Proposition. (a) Let Σ be a left multiplicative system in C_{X}. Then Σ ⊆ Σ_{E} for any Σ-torsion free injective object E.

(b) Suppose that for every morphism L ⟶ M in C_{X}, there exists a fibred coproduct M ∐_{L} M and the pair of coprojections M \overrightarrow{⟶} M ∐_{L} M has a kernel. Then for any Σ ∈ S_{ℓs}^{𝔰}M(X), the image of any injective Σ-torsion free object E in the quotient category Σ^{−1}C_{X} = C_{Σ^{−1}X} is an injective object.

(c) Let Σ ∈ S_{ℓs}^{𝔰}M(X) be a continuous multiplicative system such that Σ^{−1}C_{X} has a conservative family of injective objects. Then Σ = ⋂_{E∈𝔉} Σ_{E}, for some family 𝔉 of Σ-torsion free injective objects.

Proof. (a) Let Σ be a left multiplicative system and E a Σ-torsion free injective object. The claim is that for any arrow L ⟶^{s} M in Σ, the map

C_{X}(M, E) ⟶^{C_{X}(s,E)} C_{X}(L, E), f ⟼ f ∘ s,  (1)

is bijective. In fact, let L ⟶^{f} E be an arbitrary morphism. Since s ∈ Σ and Σ is a left multiplicative system, there exists a commutative diagram

M ⟶^{f′} M′
s ↑  ↑ t
L ⟶^{f} E

with t ∈ Σ. Since E is Σ-torsion free, the arrow E ⟶^{t} M′ is a strict monomorphism. Every strict monomorphism from an injective object splits, i.e. there exists a morphism M′ ⟶^{g} E such that g ∘ t = id_{E}. Therefore (g ∘ f′) ∘ s = g ∘ t ∘ f = f. This proves

the surjectivity of the map (1). Suppose now that M \overrightarrow{⟶}^{p_{1}}_{p_{2}}E is a pair of arrows such that

p_{1} ∘ s = p_{2} ∘ s. Since Σ is a left multiplicative system and s ∈ Σ, there exists an arrow E ⟶^{u} N in Σ such that u ∘ p_{1} = u ∘ p_{2}. The arrow u is a (strict) monomorphism, because E is Σ-torsion free, hence p_{1} = p_{2}. This shows that the map (1) is injective.

(b) Let q^{∗} be the localization functor C_{X} ⟶ Σ^{−1}C_{X}. Consider a diagram

q^{∗}(E) ⟵^{f} q^{∗}(L) ⟶^{𝔧′} q^{∗}(M)  (1)

such that 𝔧′ is a strict monomorphism. Since Σ is a left multiplicative system, the diagram (1) corresponds to the diagram

E ⟶^{s} K ⟵^{f′} L ⟶^{𝔧′′} M′ ⟵^{t} M.  (2)

Here s, t ∈ Σ and f = q^{∗}(s)^{−1}q^{∗}(f′) and q^{∗}(𝔧′) = q^{∗}(t)^{−1}q^{∗}(𝔧′′). Since E is Σ-torsion free, the arrow s is a strict monomorphism. It is a split monomorphism, because E is injective; i.e. there exists an arrow K ⟶^{h} E such that hs = id_{E}. In particular, g^{∗}(h) = g^{∗}(s)^{−1}. Thus, f = q^{∗}(𝔳), where 𝔳 = hf′ : L ⟶ E. Consider the decomposition

of the arrow L ⟶^{𝔧′′} M′ into L ⟶^{u′} L′ ⟶^{𝔧} M′, where L′ ⟶^{𝔧} M′ is the kernel of the pair

M′ \overrightarrow{⟶}^{p_{1}}_{p_{2}}M′ ∐_{L} M′. Since q^{∗}(𝔧′) = q^{∗}(t)^{−1}q^{∗}(𝔧′′) is a strict monomorphism, q^{∗}(𝔧′′) is a strict

monomorphism. The localization functor q^{∗} is right exact [GZ, I.3.1]; in particular, the diagram

q^{∗}(L ⟶^{𝔧′′} M′ \overrightarrow{⟶}^{p_{1}}_{p_{2}}M′ ∐_{L} M′)

is isomorphic to

q^{∗}(L) ⟶^{q^{∗}(𝔧′′)} q^{∗}(M′) \overrightarrow{⟶} q^{∗}(M′) ∐_{q^{∗}(L)} q^{∗}(M′).  (3)

Since q^{∗}(𝔧′′) is a strict monomorphism, the diagram (3) is exact. By hypothesis, q^{∗} maps strict monomorphisms to strict monomorphisms, hence the diagram

q^{∗}(L′) ⟶^{q^{∗}(𝔧)} q^{∗}(M′) \overrightarrow{⟶}^{q^{∗}(p_{1})}_{q^{∗}(p_{2})} q^{∗}(M′) ∐_{q^{∗}(L)} q^{∗}(M′).  (3′)

is exact too. By the universal property of kernels, this implies that q^{∗}(L) ⟶^{q^{∗}(u′)} q^{∗}(L′) is an isomorphism. Since Σ is saturated, u′ ∈ Σ. Therefore, there exists a commutative diagram

L ⟶^{𝔳} E
u′ ↓  ↓ u′′
L′ ⟶^{𝔳} E′

with u′′ ∈ Σ. Since E is Σ-torsion free and injective, there exists E′ ⟶^{w} E such that wu′′ = id_{E}. Thus, we obtain a diagram E ⟵^{𝔳′} L′ ⟶^{𝔧} M′ in which 𝔧 is a strict monomorphism. Therefore, there exists a morphism E ⟶^{g} M′ such that g𝔧 = 𝔳′. But, then γf = 𝔧′, where γ = q^{∗}(t)^{−1}q^{∗}(g) (see notations above).

(c) Fix Σ ∈ S_{ℓs}^{𝔰}M(X) such that Σ^{−1}C_{X} = C_{Σ^{−1}X} has a conservative family, 𝔉, of injective objects, i.e. {C_{Σ^{−1}X}(−, E) | E ∈ 𝔉} is a conservative family of functors. Fix a direct image functor C_{Σ^{−1}X} ⟶^{q_{_{Σ}∗}} C_{X} of the localization Σ^{−1}X ⟶^{q_{Σ}} X. Since the

localization functor C_{X} ⟶^{q_{Σ}^{∗}} C_{Σ^{−1}X} maps strict monomorphisms to strict monomorphisms, the functor q_{_{Σ}∗} maps injectives to Σ-torsion free injectives (see 4.4). By the (a), we have the inclusion Σ ⊆ ⋂_{E∈𝔉} Σ_{q_{_{Σ}∗}(E)}. The inverse inclusion follows from the conservativity of the

family {C_{Σ^{−1}X}(−, E) | E ∈ 𝔉}.

4.6. Example. Suppose that C_{X} is an abelian category with injective hulls (e.g. C_{X} is a Grothendieck category). Then the conditions of 4.5 hold. Moreover, every closed multiplicative system Σ is the intersection of the systems Σ_{E}, where E runs through a family of Σ-torsion free injectives.

In fact, closed multiplicative systems are in bijective correspondence with coreflective thick subcagories, Σ ⟼ 𝕋_{Σ}, where Ob𝕋_{Σ} = {M ∈ ObC_{X} | (0 ⟶ M) ∈ Σ}. If E is an injective object, then 𝕋_{Σ_{E}} = ^{⊥}E, i.e. Ob𝕋_{Σ_{E}} = {M ∈ ObC_{X} | C_{X}(M, E) = 0}. Notice that Σ_{𝕋}-torsion free objects are precisely 𝕋-torsion free objects, i.e. objects which have no nonzero subobjects from 𝕋.

Fix a coreflective thick subcategory 𝕋 and denote by q_{𝕋}^{∗} the localization functor C_{X} ⟶ C_{X}/𝕋, Let M be a nonzero 𝕋-torsion free object. If M ⟶ E is an essential monomorphism, then E is 𝕋-torsion free object too. Let 𝔉 be a family of 𝕋-torsion free objects such that {q_{𝕋}^{∗}(M) | M ∈ 𝔉} generates the quotient category C_{X}/𝕋 = C_{X/𝕋}, i.e. {q_{𝕋}^{∗}(M) | M ∈ 𝔉}^{⊥} = 0. For each M ∈ 𝔉 we chose an injective hull E(M) of M. It follows from 4.5(a) that Σ_{𝕋} ⊆ ⋂_{M∈𝔉} Σ_{E(M)}, or, equivalently, 𝕋 ⊆ ⋂_{M∈𝔉}^{⊥}E(M). We leave verifying

the inverse inclusion to the reader.

Suppose that C_{X} is a Grothendieck category. Then every closed multiplicative system, Σ, is flat and the corresponding quotient category C_{Σ^{−1}X}, is a Grothendieck category too. In particular, it has a set of generators, 𝔉. By the argument above, Σ = ⋂_{M∈𝔉} Σ_{E(M)} = Σ_{E_{𝔉}},

where E_{𝔉} = ∏_{M∈𝔉} E(M). Here we use the fact that injective hulls and small products exist in a Grothendieck category [BD, 6.3.1, 6.3.2]. Thus, we have recovered a well known assertion: every Serre subcategory of a Grothendieck category is of the form ^{⊥}E for some injective object E.

4.7. Example. The conditions of 4.5 hold if C_{X} is an elementary (Lawvere-Tierney) topos, in particular if C_{X} is a Grothendieck topos. In fact, by a Lawvere-Tierney theorem [J, 1.26], in a topos, all partial maps are representable. This means that for any object M, there exists a monomorphism M ⟶^{η_{M}} M̃ such that any diagram L ⟵^{𝔧} L′ ⟶^{f} M with a monomorphic left arrow is uniquely completed to a commutative square

\begin{matrix}L′ & ⟶^{f} & M \\ 𝔧 ↓ & & ↓ η_{M} \\ L & \underrightarrow{_{f}} & M̃ \end{matrix}  (1)

It follows from the uniqueness of f̃ in (1) that the map M ⟼ M̃ defines a functor C_{X} ⟶^{ℑ_{X}} C_{X} and η = {η_{M} | M ∈ ObC_{X}} is a functor morphism Id_{C_{X}} ⟶ ℑ_{X}. Moreover, for every object M, the object ℑ_{X}(M) = M̃ is injective [J, 1.27].

Note that ℑ_{X}(∏_{i∈I} M_{i}) ≃ ∏_{i∈I} ℑ_{X}(M_{i}) provided the product ∏_{i∈I} M_{i} exists, and if Y ⟶^{f}

X is a geometric morphism (that is f is continuous and f^{∗} is (left) exact), then there is a functor isomorphism f_{∗}ℑ_{Y} ≃ ℑ_{X}f_{∗}.

Let Σ be a flat multiplicative system in C_{X}. Set Y = Σ^{−1}X and denote by q the canonical morphism Y ⟶ X. The quotient category C_{Y} = Σ^{−1}C_{X} is a topos too. Let 𝔉 be a family of generators in C_{Y}. Then ⋂_{M∈𝔉} Σ_{ℑ_{Y} (M)} = Iso(C_{Y}). Therefore

Σ = ⋂_{M∈𝔉} Σ_{q_{∗}ℑ_{Y} (M)} = ⋂_{M∈𝔉} Σ_{ℑ_{X}q_{∗}(M)}.

Suppose now that C_{X} is a Grothendieck topos. Then C_{Y} = Σ^{−1}C_{X} is a Grothendieck topos. In particular, C_{Y} has small products and a set of generators, 𝔉. Therefore

Σ = ⋂_{M∈𝔉} Σ_{ℑ_{X}q_{∗}(M)} = Σ_{ℑ_{X}(M_{𝔉})}, where M_{𝔉} = ∏_{M∈𝔉} q_{∗}(M) ≃ q_{∗}( ∏_{M∈𝔉} M).

4.8. Note. The examples 4.6 and 4.7 suggest that abelian categories with injective hulls might be regarded as abelian versions of elementary toposes, and Grothendieck categories are abelian analogs of Grothendieck toposes.

4.9. Example. Let C_{E} = Sets (like in 4.2.2). Then all objects of C_{E} are injective (and projective). Fix an object E of C_{E}. If E is a one-element set, or the empty set, then Σ_{E} = HomC_{E}. If Card(E) ≥ 2, then Σ_{E} = Iso(C_{E}).

Evidently, the empty set and one-element sets are the only indecomposable injectives. Let Σ ⊆ HomC_{E}. By definition, an object M of C_{E} is Σ-torsion free if every mor-

phism M ⟶ M′ which belongs to Σ is a monomorphism. Suppose Σ is a saturated left multiplicative system, i.e. it coincides either with Σ_{α} or with Σ_{α∗} for some infinite cardinal number α (cf. 4.2.2). Then a set M is Σ-torsion free iff Card(M) ≤ 1; that is, again, either M = ∅, or M is a one-element set.

It follows from the definitions of Σ_{α} and Σ_{α∗} that objects of C_{E} having a morphism to a Σ-torsion free object are precisely sets N such that Card(N) < α. This shows that the only closed saturated left multiplicative systems on C_{E} are Iso(C_{E}) and HomC_{E}. Since, by [R4, 5.2.2], every continuous saturated left multiplicative system is closed, there are no non-trivial continuous saturated left multiplicative systems either.

4.10. The injective spectrum and the Gabriel spectrum.

4.10.1. Relatively maximal objects. Fix a functor 𝔊 ⟶^{F} ℌ. We call an object x of 𝔊 relatively maximal, or F -maximal, if F transforms any arrow x ⟶ y into an isomorphism. We denote by 𝔐𝔞𝔵(𝔊, F) the full subcategory of 𝔊 generated by relatively maximal objects.

4.10.2. Injective spectrum. Suppose that the category C_{X} has colimits of finite diagrams. Let F be the functor

C_{ℑ(X)}^{op} ⟶ S_{ℓs}^{𝔰}M(X), E ⟼ Σ_{E}

(see 4.3.1). We denote 𝔐𝔞𝔵(C_{ℑ(X)}^{op}, F) by ISpec(X). It follows that the objects of ISpec(X) are injective objects E such that Σ_{E} = Σ_{E_{1}} for every nontrivial split monomorphism E_{1} ⟶ E. We denote by ISpec(X) the full subcategory of the S_{ℓs}^{𝔰}M(X) generated by the image of ISpec(X) and call it the injective spectrum of X.

4.10.2.1. Note. The injective spectrum is introduced in [R, 6.5], in a slightly different way, in the case when C_{X} is an abelian category.

4.10.3. The Gabriel’s spectrum.

Recall that an object, E, of the category C_{X} is indecomposable if every nontrivial idempotent E ⟶^{𝔭} E is id_{E} (see 1.2.3.3).

We denote by ÎSpec(X) the groupoid Min(C_{ℑ(X)}) formed by indecomposable injectives and their isomorphisms. It follows that ÎSpec(X) ⊆ ISpec(X).

The Gabriel’s spectrum is the full subpreorder, ÎSpec(X), of the preorder S_{ℓs}^{𝔰}M(X) spanned by multiplicative systems Σ_{E}, where E runs through indecomposable injectives of C_{X}. In particular, the Gabriel’s spectrum is contained in the injective spectrum: ÎSpec(X) ⊆ ISpec(X).

4.10.3.1. Note. The Gabriel’s spectrum is introduced in [Gab] for a (locally noetherian) abelian category. Its elements are defined as isomorphism classes of indecomposable injectives. The preorder inherited from S^{𝔰}M(X) is opposite to the specialization preorder.

4.11. Injective spectrum of an abelian category. Fix an abelian category C_{X}. Let E ∈ ObC_{X}, and let ^{⊥}E be the full subcategory of the category C_{X} generated by all objects M which are left orthogonal to E, i.e. C_{X}(M, E) = 0. If E_{1} is a subobject of the object E, then ^{⊥}E ⊆ ^{⊥}E_{1}.

If E is an injective object, then ^{⊥}E is a Serre subcategory of the category C_{X} and the map E ⟼ ^{⊥}E is a functor 𝔍(X)^{op} ⟶ 𝔖𝔢(X). In particular, the spectrum ISpec(X) can be identified with the subpreorder of the preorder 𝔖𝔢(X) of Serre subcategories of C_{X} generated by the image of the map ISpec(X) ⟶ 𝔖𝔢(X), E ⟼ ^{⊥}E.

4.11.1. Proposition. Let C_{X} be an abelian category with the property (sup).

(a) If every object of C_{X} has an injective hull, then 𝔖𝔭𝔢𝔠^{1}(𝔖𝔢(X)) ⊆ ISpec(X). In particular, Spec_{𝔖𝔢}^{1}(X) ⊆ ISpec(X).

(b) If C_{X} has a Gabriel-Krull dimension, then

Spec_{𝔖𝔢}^{1}(X) = 𝔖𝔭𝔢𝔠^{1}(𝔖𝔢(X)) = ISpec(X) = ÎSpec(X).

Proof. (a) Let P be an object of 𝔖𝔭𝔢𝔠^{1}(𝔖𝔢(X)), i.e. there exists the smallest Serre subcategory, P^{𝔰}, properly containing P. Let M ∈ ObP^{𝔰} − ObP. Since, thanks to the property (sup), every Serre subcategory of C_{X}, in particular P, is coreflective, we can and will assume that M is P-torsion free. Let E(M) be an injective hull of the object M. Then E(M) is P-torsion free, because M is P-torsion free, hence P ⊆ ^{⊥}E(M). Notice that ^{⊥}E(M) cannot contain P properly, because if P ≠ ^{⊥}E(M), then ^{⊥}E(M), being a Serre subcategory, would contain P^{𝔰}, in particular, it would contain the object M which is not the case. This verifies the inclusion 𝔖𝔭𝔢𝔠^{1}(𝔖𝔢(X)) ⊆ ISpec(X). By 2.4.1, Spec_{𝔖𝔢}^{1}(X) ⊆ 𝔖𝔭𝔢𝔠^{1}(𝔖𝔢(X)), hence Spec_{𝔖𝔢}^{1}(X) ⊆ ISpec(X).

(b) If C_{X} has a Gabriel-Krull dimension, then, by [R4, 7.9.1], Spec_{𝔰}^{1}(X) = Spec^{−}(X) and by [R, 6.6.1.1, 6.6.1.2], Spec^{−}(X) = ISpec(X) = ÎSpec(X). The assertion follows now from the inclusions Spec^{−}(X) ⊆ Spec_{𝔖𝔢}^{1}(X) ⊆ 𝔖𝔭𝔢𝔠^{1}(𝔖𝔢(X)) ⊆ ISpec(X).

C. Complementary facts: relative support and associated points.

C.1. Relative support. Fix a functor 𝔊 ⟶^{F} ℌ. For an x ∈ Ob𝔊, we call 𝔖𝔲𝔭𝔭_{ℌ}(F (x)) the support of x in ℌ, or the relative support of x.

C.2. Weakly associated points. For any x ∈ Ob𝔊, we denote by Ass_{(𝔊,F)}^{1}(x) the full subcategory of 𝔖𝔭𝔢𝔠^{1}(ℌ) generated by all objects z for which there exists x̃ ⟶ x such that z ∈ Ob𝔖𝔲𝔭𝔭_{ℌ}(F (x̃)) and there is an arrow F (x̃) ⟶ z^{⋆}. As before, (z, z → z^{⋆}) denotes an initial object of (z\ℌ)^{1}.

It follows from this definition that Ass_{(𝔊,F)}^{1}(x) is a subcategory of the relative support, 𝔖𝔲𝔭𝔭_{ℌ}(F (x)), of the object x.

We call objects of Ass_{(𝔊,F)}^{1}(x) weakly associated points of x in (𝔊, F).

If F is the identical functor ℌ ⟶ ℌ, we shall write Ass_{ℌ}^{1}(x) instead of Ass_{(ℌ,Id_{ℌ})}^{1}(x) and call objects of this category weakly associated points of x. It follows that objects of Ass_{ℌ}^{1}(x) are z ∈ Ob𝔖𝔭𝔢𝔠^{1}(ℌ) such that there exist arrows z^{⋆} ⟵ x̃ ⟶ x and z belongs to 𝔖𝔲𝔭𝔭_{ℌ}(x̃).

C.3. Associated points. Fix a functor 𝔊 ⟶^{F} ℌ. For an object x of 𝔊, we denote by 𝔄𝔰𝔰_{(𝔊,F)}^{1}(x) the full subcategory of 𝔖𝔭𝔢𝔠^{1}(ℌ) generated by all objects z such that there exists x̃ ∈ Ob𝔊 having the following properties:

(a) there exist arrows x̃ ⟶ x and F (x̃) ⟶ z^{⋆};

(b) if there exists an arrow y ⟶ x̃, then ℌ(F (y), z) = ∅. We call objects of 𝔄𝔰𝔰_{(𝔊,F)}^{1}(x) associated points of the object x in (𝔊, F).

It follows from the condition (b) that z belongs to 𝔖𝔲𝔭𝔭_{ℌ}(F (x̃)). Together with the condition (a), this means that 𝔄𝔰𝔰_{(𝔊,F)}^{1}(x) ⊆ Ass_{(𝔊,F)}^{1}(x).

If F is the identical functor ℌ ⟶ ℌ, we shall write 𝔄𝔰𝔰_{ℌ}^{1}(x) instead of 𝔄𝔰𝔰_{(ℌ,Id_{ℌ})}^{1}(x). It follows that objects of 𝔄𝔰𝔰_{ℌ}^{1}(x) are z ∈ Ob𝔖𝔭𝔢𝔠^{1}(ℌ) for which there exists a pair of arrows z^{⋆} ⟵ x̃ ⟶ x with x̃ such that there is no diagram of the form z ⟵ y ⟶ x̃.

C.4. Associated points and weakly associated points in 𝔖𝔭𝔢𝔠^{0}(ℌ). We define 𝔄𝔰𝔰_{(𝔊,F)}^{0}(x), resp. Ass_{(𝔊,F)}^{0}(x), as full subcategories of ℌ generated by all z ∈ Ob𝔖𝔭𝔢𝔠^{0}(ℌ) such that ẑ is an object of 𝔄𝔰𝔰_{(𝔊,F)}^{1}(x), resp. an object of Ass_{(𝔊,F)}^{1}(x).

Consider the following two properties:

(sup1) If x ∈ Obℌ is the supremum of a filtered system, {x_{i} | i ∈ J}, of its subobjects, then for any morphism x̃ ⟶ x, there exists a cofinal subset I ⊆ J such that for every i ∈ I, there exists a fibre product, x̃_{i} = x̃ ×_{x} x_{i}, and the canonical arrow colim(x̃_{i}|i ∈ I) ⟶ x̃ is an isomorphism.

(sup2) If x ∈ Obℌ is the supremum of a filtered system, {x_{i} | i ∈ J}, of its subobjects, then for any morphism x̃ ⟶ x, there exists a diagram x̃ ⟵ y ⟶ x_{i} for some i ∈ J.

C.5. Proposition. (a) If x = colim(x_{i} | i ∈ J), then

⋃_{i∈J} Ass_{(𝔊,F)}^{1}(x_{i}) ⊆ Ass_{(𝔊,F)}^{1}(x) and ⋃_{i∈J} 𝔄𝔰𝔰_{(𝔊,F)}^{1}(x_{i}) ⊆ 𝔄𝔰𝔰_{(𝔊,F)}^{1}(x).

(b) Let x ∈ Obℌ be the supremum of a filtered system, {x_{i} | i ∈ J}, of its subobjects.

(i) If ℌ is a preorder with the property (sup1) and F preserves colimits of filtered systems, then

Ass_{(𝔊,F)}^{1}(x) = ⋃_{i∈J} Ass_{(𝔊,F)}^{1}(x_{i}).

(ii) Suppose 𝔊 possesses the property (sup2). Then

𝔄𝔰𝔰_{(𝔊,F)}^{1}(x) = ⋃_{i∈J} 𝔄𝔰𝔰_{(𝔊,F)}^{1}(x_{i}).

Proof. (a) Let z ∈ ObAss_{(𝔊,F)}^{1}(x_{i}), that is z belongs to 𝔖𝔭𝔢𝔠^{1}(ℌ) and there exists a pair of arrows, z^{⋆} ⟵ F (x̃_{i}) and x̃_{i} ⟶ x_{i}, such that z ∈ Ob𝔖𝔲𝔭𝔭_{ℌ}(F (x̃_{i})). Since there is an arrow x_{i} ⟶ x, same x̃_{i} serves for x. Therefore Ass_{(𝔊,F)}^{1}(x_{i}) ⊆ Ass_{(𝔊,F)}^{1}(x) for all i ∈ J. Similar argument shows the inclusion ⋃_{i∈J} 𝔄𝔰𝔰_{(𝔊,F)}^{1}(x_{i}) ⊆ 𝔄𝔰𝔰_{(𝔊,F)}^{1}(x).

(b)(i) Suppose that (𝔊, F) possesses the property (sup1). Let z be an object of Ass_{(𝔊,F)}^{1}(x); i.e. z belongs to 𝔖𝔭𝔢𝔠^{1}(ℌ) and there exists a pair of arrows, z^{⋆} ⟵ F (x̃) and x̃ ⟶ x, such that z ∈ Ob𝔖𝔲𝔭𝔭_{ℌ}(F (x̃)). By the property (sup1), a fibre product x̃_{i} = x̃ ×_{x} x_{i} exists for i ∈ I, where I is a cofinal subset of J, and the canonical arrow colim(x̃_{i}|i ∈ I) ⟶ x̃ is an isomorphism. If for every i ∈ I, there is an arrow F (x̃_{i}) ⟶ z, then there is an arrow F (x̃) ⟶ z, that is z does non belong to 𝔖𝔲𝔭𝔭_{ℌ}(F (x̃)), which contradicts to the hypothesis. Thus, z ∈ Ob𝔖𝔲𝔭𝔭_{ℌ}(F (x̃_{i})) for some i ∈ I, which means that z is an object of Ass_{(𝔊,F)}^{1}(x_{i}).

(ii) Let now (𝔊, F) have the property (sup2). Let z be an object of 𝔄𝔰𝔰_{(𝔊,F)}^{1}(x); i.e. z belongs to 𝔖𝔭𝔢𝔠^{1}(ℌ) and there exists a pair of arrows, z^{⋆} ⟵ F (x̃) and x̃ ⟶ x, such that if ℌ(F (y), z) ≠ ∅, then 𝔊(y, x̃) = ∅. By the property (sup2), for some i ∈ J, there exists a pair of arrows x̃ ⟵ x̃_{i} ⟶ x_{i}. It follows that if ℌ(F (y), z) ≠ ∅, then 𝔊(y, x̃_{i}) = ∅, hence z belongs to 𝔄𝔰𝔰_{(𝔊,F)}^{1}(x_{i}).

C.6. Example: supports and associated points of a family of arrows. Fix a category C_{X}. Let ℌ be the preorder S^{𝔰}M(X) of saturated multiplicative systems of C_{X}. Let 𝔊 be the preorder (with respect to inclusion) of non-empty families of arrows of the category C_{X}, and let 𝔊 ⟶^{F} ℌ be the functor which assigns to each family S the intersection, [S]_{•}, of all saturated multiplicative systems containing S.

The support, 𝔖𝔲𝔭𝔭(F (S)), of a family of arrows S in ℌ = S^{𝔰}M(X) consists of all saturated multiplicative systems Σ which do not contain S.

Weakly associated points of S are saturated multiplicative systems Σ such that there exists S̃ ⊆ S which is not contained in Σ, but is contained in Σ^{⋆}. Here Σ^{⋆} is the intersection of all saturated multiplicative systems of C_{X} properly containing Σ.

Thus, the preorder Ass_{(𝔊,F)}(S) of weakly associated points of a family of arrows S coincides with the preorder Ass_{𝔏}(S) of weakly associated points of S defined in [R4, 9.4.2].

Notice that in this case, associated points and weakly associated points of S coincide. In fact, by definition C.3, associated points of S are saturated multiplicative systems

Σ having the following property: there exists a non-empty subfamily, T, of S such that T ∩ Σ = ∅. Let Σ be a weakly associated point of S, i.e. there exists S̃ ⊆ S such that Σ ⊉ S̃ ⊆ Σ^{⋆}. The family T = S̃ − Σ is non-empty, and T ∩ Σ = ∅. This means, precisely, that Σ is an associated point of S.

C.7. Supports and associated points of objects. Let 𝔊, ℌ and F be same as in C.6; i.e. 𝔊 is a preorder of families of arrows of a category C_{X}, ℌ is the preorder S^{𝔰}M(X) of saturated multiplicative systems of C_{X}, and F maps every family S to the smallest saturated multiplicative system containing S. Fix a functor C_{Y} ⟶^{ϕ^{∗}} C_{X}. The functor ϕ^{∗}

determines, for every M ∈ ObC_{Y}, a functor C_{Y} /M ⟶^{ϕ_{M}^{∗}} C_{X}/ϕ^{∗}(M). The map

ObC_{Y} ⟶ 𝔊, M ⟼ ϕ_{M}^{∗}(Ob(C_{Y} /M)),

defines a functor C_{Y} ⟶^{Φ} 𝔊. Let C_{Y} ⟶^{Υ_{ϕ}} ℌ denote the composition of the functor Φ with the functor F. The functor Υ_{ϕ} provides the notions of the support, weakly associated points and associated points in ℌ = S^{𝔰}M(X) of any object M of the category C_{Y}.

The support of an object M consists of all saturated multiplicative systems Σ such that ϕ^{∗}(ξ) ∉ Σ for some arrow L ⟶^{ξ} M.

A saturated multiplicative system Σ of C_{X} is a weakly associated point of an object M of the category C_{Y} iff there exists a morphism L ⟶ M such that ϕ^{∗}(C_{Y} (N, L)) ⊆ Σ^{⋆} for all N ∈ ObC_{Y} and ϕ^{∗}(ξ) ∉ Σ for some arrow M′ ⟶^{ξ} M.

Finally, Σ is an associated point of M iff there exists a morphism L ⟶ M such that Σ ∌ ϕ^{∗}(ξ) ∈ Σ^{⋆} for every arrow N ⟶^{ξ} L.

C.8. A canonical setting. Let C_{Y} denote the subcategory of C_{X} formed by all non-initial objects of C_{X} and strict monomorphisms; and let ϕ^{∗} be the inclusion functor C_{Y} ↪ C_{X}. Applying the construction of C.7, we obtain non-trivial notions of the support and weakly associated and associated points of any non-initial object of the category C_{X}.

C.8.1. The case of an abelian category. If C_{X} is an abelian category, these notions are equivalent to those introduced in [R4, 9.4, 10.8]. The equivalence is given by the isomorphism between the preorder S^{𝔰}M(X) of saturated multiplicative systems and the preorder 𝔗𝔥(X) of thick subcategories of the category C_{X}. By definition, a saturated system Σ_{𝕋} corresponding to a thick subcategory 𝕋 belongs to the support of an object M of C_{X} iff there exists a monomorphism N ⟶^{g} M which does not belong to Σ_{𝕋}. This means, precisely, that Cok(g) does not belong to the subcategory 𝕋. Thus, Σ_{𝕋} belongs to the support of M iff M does not belong to 𝕋.

A multiplicative system Σ_{𝕋} is a weakly associated point of an object M if there exists a subobject N of M such that all monoarrows L ⟶ N belong to Σ_{𝕋}^{⋆} = Σ_{𝕋^{⋆}}, but some of them do not belong Σ_{𝕋}. This means that N is an object of the subcategory 𝕋^{⋆} which does not belong to 𝕋. Here 𝕋^{⋆} is the intersection of all thick subcategories of C_{X} properly containing 𝕋.

A multiplicative system Σ_{𝕋} is an associated point of an object M if there is nonzero subobject N of M such that every nonzero monoarrow L ⟶ N belongs to Σ_{𝕋^{⋆}} and does not belong Σ_{𝕋}. This means that the subobject N belongs to 𝕋^{⋆} and is 𝕋-torsion free.

C.8.2. The direct description. Fix an abelian category C_{X}. Let 𝔊 be the subcategory C_{𝔐^{⋆}(X)} of C_{X} formed by all nonzero monomorphisms and all nonzero objects of C_{X}. Let ℌ be the preorder 𝔗𝔥(X) of thick subcategories of the category C_{X}. The functor 𝔊 ⟶^{F} ℌ assigns to every object M of the category C_{𝔐^{⋆}(X)} the smallest thick subcategory, [M]_{•}, containing M.

The support, 𝔖𝔲𝔭𝔭(F (M)), of the object M in ℌ = 𝔗𝔥(X) consists of all thick subcategories 𝕋 such that F (M) = [M]_{•} ⊈ 𝕋, or, equvalently, M ∉ Ob𝕋.

A thick subcategory 𝕋 is a weakly associated point of a nonzero object M iff there exists a subobject M̃ of M which belongs to Ob𝕋^{⋆} − Ob𝕋.

A thick subcategory 𝕋 is an associated point of a nonzero object M iff there exists a nonzero subobject M̃ of M which belongs to 𝕋^{⋆} and is 𝕋-torsion free.

Thus the preorder Ass_{(𝔊,F)}^{1}(M) coincides with the preorder Ass_{𝔗𝔥}^{1}(M) of weakly associated points of M in the sense of [R4, 10.1]. The preorder 𝔄𝔰𝔰_{(𝔊,F)}^{1}(M) coincides with the preorder 𝔄𝔰𝔰_{𝔏}^{1}(M) introduced in [R4, 10.8].

Appendix

The purpose of this appendix is to remind the definitions of the spectra introduced in [R1], [R2], and [R, Ch.6]. The main reference is [R].

A1. The left spectrum of a ring. Let I_{ℓ}R denote the set of left ideals of an associative unital ring R. For two left ideals, 𝔪, 𝔫, we write 𝔪 ≤ 𝔫 if there exists a finite set, x, of elements of R such that (𝔪 : x) = {r ∈ R | rx ⊂ 𝔪} is contained in 𝔫. This relation is a preorder. If 𝔪 is a two-sided ideal, then 𝔪 ≤ 𝔫 iff 𝔪 ⊆ 𝔫. In particular, if the ring R is commutative, then the preorder ≤ coincides with the inclusion.

The left spectrum of the ring R is the preorder (Spec_{ℓ}R, ≤), where Spec_{ℓ}R consists of all left ideals, 𝔭, in R such that (𝔭 : r) ≤ 𝔭 for any r ∈ R − 𝔭.

Since for a commutative ring, the relation ≤ is the inclusion, the left spectrum of a commutative ring coincides with its prime spectrum endowed with the specialization preorder, ⊆.

A2. The preorder ≻ and the spectrum Spec(X). Let C_{X} be an abelian category. Given two objects, M and N, of the category C_{X}, we write M ≻ N if N is a subquotient of a finite coproduct of copies of M. The relation ≻ is a preorder on ObC_{X}.

For any object M ∈ ObC_{X}, let [M] (resp. ⟨M⟩) denote the full subcategory of C_{X} generated by all N ∈ ObC_{X} such that M ≻ N (resp. N ⊁ M). It is easy to see that M ≻ L ⇔ [L] ⊆ [M] ⇔ ⟨L⟩ ⊆ ⟨M⟩.

A2.1. Note. A full subcategory, 𝕋, of C_{X} is called topologizing if it contains all subquotients of its objects and is closed under finite coproducts. One can check [M] is the smallest topologizing subcategory of C_{X} containing the object M, and ⟨M⟩ is the union of all topologizing subcategories of C_{X} which do not contain M.

A.2.2. Spec(X). We denote by Spec(X) a subpreorder of (ObC_{X}, ≻) formed by all nonzero objects M of C_{X} such that L ≻ M for any nonzero subobject L of M.

The spectrum Spec(X) consists of subcategories [M], where M ∈ Spec(X). The preorder ≻ induces on Spec(X) the partial order ⊇.

Every simple object of C_{X} belongs to Spec(X), and if M, N are simple objects, then M ≻ N iff they are isomorphic. Moreover, if M is a simple object and N is a nonzero object, then M ≻ N iff N is a coproduct of finite number of copies of M.

This shows that simple objects are maximal objects of C_{X} for the preorder ≻, and isomorphism classes of simple objects are maximal elements of (Spec(X), ⊇).

A2.3. Connections with the left spectrum of a ring. Let C_{X} be the category R − mod of left modules over a ring R. Then for any pair of left ideals, 𝔪, 𝔫 the relation R/𝔪 ≻ R/𝔫 is equivalent to the relation 𝔪 ≤ 𝔫 (cf. A1).

By [R, Ch.3, 4.3], the map 𝔪 ⟼ R/𝔪 assigns to each element of Spec_{ℓ}R an object of Spec(X) and this map is an equivalence of preorders (Spec_{ℓ}R, ≤) ⟶ (Spec(X), ≻), that is the map 𝔪 ⟼ [R/𝔪] induces an isomorphism of the partial order associated with the left spectrum (Spec_{ℓ}R, ≤) onto the spectrum (Spec(X), ⊇) of X.

A3. Local categories. An abelian category C_{X} is called local if it has a nonzero object, P, such that M ≻ P for any nonzero object M. An objects P possessing this property is called (in [R]) quasifinal. All quasifinal objects belong to Spec(X) and are equivalent one to another. They define a unique maximal element of the spectrum Spec(X).

It follows that if C_{X} is local and has simple objects, then all of them are isomorphic one to another. If C_{X} is the category of modules over a commutative ring R, then C_{X} is local iff the ring R is local.

A4. The complete spectrum. The complete spectrum of a ’space’ X represented by an abelian category C_{X} is the partial order (Spec^{1}(X), ⊆) of all thick subcategories P of C_{X} such that the quotient category C_{X}/P is local.

A4.1. Note. The notions of a local category and a thick subcategory are selfdual. Therefore the complete spectrum of a category C_{X} is naturally isomorphic to the complete spectrum of its opposite category, C_{X}^{op}. Setting C_{X^{o}} = C_{X}^{op}, we can write this as Spec^{1}(X) ≃ Spec^{1}(X^{o}).

A5. S-spectrum. For any subcategory 𝕋 of an abelian category C_{X}, let 𝕋^{−} denote the full subcategory of C_{X} generated by all M ∈ ObC_{X} such that any nonzero subquotient of M has a nonzero subobject which is isomorphic to an object of 𝕋. This construction has the following properties ([R, Ch.3, 2.3.2.1):

(a) The subcategory 𝕋^{−} is thick.

(b) (𝕋^{−})^{−} = 𝕋^{−}. A subcategory 𝕋 is called a Serre subcategory if 𝕋^{−} = 𝕋. In particular, every Serre subcategory is thick.

The S-spectrum (in [R] it is called the flat spectrum), Spec^{−}(X) is formed by all Serre subcategories, P, of C_{X} such that the quotient category C_{X}/P is local.

A5.1. Proposition ([R, Ch.3, 3.3.2). Let C_{X} be an abelian category. For any object M of Spec(X), the full subcategory ⟨M⟩ defined by Ob⟨M⟩ = {N ∈ ObC_{X} | N ⊁ M} belongs to Spec^{−}(X).

Thus, Spec(X) is naturally embedded into Spec^{−}(X).

References.

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