SECONDARY DERIVED FUNCTORS AND THE ADAMS SPECTRAL SEQUENCE

HANS JOACHIM BAUES AND MAMUKA JIBLADZE

Abstract. Classical homological algebra takes place in additive categories. In homotopy theory such additive categories arise as homotopy categories of “additive groupoid enriched categories”, in which a secondary analog of homological algebra can be performed. We introduce secondary chain complexes and secondary resolutions leading to the concept of secondary derived functors. As a main result we show that the E_{3}-term of the Adams spectral sequence can be expressed as a secondary derived functor. This result can be used to compute the E_{3} term explicitly by an algorithm.

The paper introduces the secondary derived functor Ext^{n} obtained by secondary resolutions. This generalizes the concept of the classical derived functor Ext^{n}. The classical groups Ext^{n}, for example, are used to describe the E_{2}-term of the Adams spectral sequence. As a main application we show that the secondary Ext groups determine the E_{3}-term of the Adams spectral sequence. Using the theory in [2] this yields an algorithm for the computation of the E_{3}-term, see [4]. The algorithm is achieved by taking into account the track structure: one considers not just homotopy classes of maps between spectra, but instead maps and homotopy classes of homotopies between maps, termed tracks. These form a track category, that is, a category enriched in groupoids. It then turns out that in appropriate track categories secondary Ext groups can be defined which are unchanged if one replaces the ambient track category with a weakly equivalent one. In fact in [2] a manageable purely algebraically described track category weakly equivalent to the track category of Eilenberg-Mac Lane spectra has been completely determined. It is this algebraic model that will be used on the basis of the main result 7.3 below to compute explicitly the E_{3}-term of the Adams spectral sequence as a secondary Ext-group, see [4].

1. Derived functors

We first recall the notion of a resolution in an additive category from which we deduce (primary) derived functors. Later we introduce the secondary version of these notions in the context of an “additive track category”, see section 3.

Our initial data consist of an additive category A and a full additive subcategory a of A. The basic situation to have in mind is the category R-Mod of modules over a ring R and its subcategory R-mod of free (or projective) R-modules. As another motivating example, coming from topology, one considers for A the opposite of the stable homotopy category and for a its full subcategory on objects represented by finite products of Eilenberg-Mac Lane spectra over a fixed prime field _{p}; then a is equivalent to the category of finitely generated free modules over the mod p Steenrod algebra.

(1.1) Definition. A chain complex (A, d) in A is a sequence of objects and morphisms

... → A_{n+1} ⟶^{d_{n}} A_{n} ⟶^{d_{n−1}} A_{n−1} → ...

from A, with d_{n−1}d_{n} = 0 (n ∈ ).

A chain map f : (A, d) → (A′, d′) is a sequence of morphisms f_{n} : A_{n} → A′_{n} with f_{n}d_{n} = d′_{n}f_{n+1}, n ∈ . For two maps f, f′: (A, d) → (A′, d′), a chain homotopy h from f to f′is a sequence of morphisms h_{n} : A_{n−1} → A′_{n} satisfying f′_{n} = f_{n} + d′_{n}h_{n+1} + h_{n}d_{n−1}, n ∈ .

A chain complex (A, d) is called a-exact if for any object X from the subcategory a the (ordinary) chain complex Hom_{A}(X, A_{•}) of abelian groups

... → Hom_{A}(X, A_{n+1}) ⟶^{Hom_{A}(X,d_{n})} Hom_{A}(X, A_{n}) ⟶^{Hom_{A}(X,d_{n−1})} Hom_{A}(X, A_{n−1}) → ...

2000 Mathematics Subject Classification. Primary 18G10, 55T15, 55S20; Secondary 55U99, 18D05, 18G25, 18G40.

is acyclic, i. e. an exact sequence. Explicitly, this means that for any n ∈ , any object X from a and any morphism a_{n} : X → A_{n} with d_{n−1}a_{n} = 0 there exists a morphism a_{n+1} : X → A_{n+1} with a_{n} = d_{n}a_{n+1}.

A chain map f : A → A′ is an a-equivalence if for every X in a the chain map Hom_{A}(X, f) is a quasiisomorphism. Thus a chain complex (A, d) is a-exact if and only if the map (0, 0) → (A, d) is an a-equivalence.

(1.2) Definition. For an object A of A, an A-augmented chain complex A_{•}^{ϵ} is a chain complex of the form

... → A_{1} → A_{0} → A → 0 → 0 → ...,

i. e. with A_{−1} = A and A_{−n} = 0 for n > 1. We will consider such an augmented chain complex as a map between chain complexes, ϵ : A_{•} → A, where A_{•} is the complex ... → A_{1} → A_{0} → 0 → 0 → ... whereas A is considered as a complex concentrated in degree 0, with ϵ = d_{−1} : A_{0} → A called the augmentation.

An a-resolution of A is an a-exact A-augmented chain complex such that all A_{n} for n 0 belong to a. Thus an a-resolution A_{•}^{ϵ} of an object A is the same as a chain complex A_{•} in a together with an a-equivalence ϵ : A_{•} → A.

There are obvious dual notions of an A-coaugmented complex and a-coresolution of A. Namely, this means a complex (resp. a-exact complex) with A_{1} = A and A_{n} = 0 for n > 1.

(1.3) Lemma. Suppose • coproducts of any families of objects of a exist in A and belong to a again;

• there is a small subcategory g of a such that every object of a is a retract of a coproduct of a family

of objects from g. Then every object of A has an a-resolution.

Proof. We begin by taking

A_{0} = ∐_{\begin{matrix}G∈g \\ a:G→A \end{matrix}} G,

with the obvious map ϵ : A_{0} → A having a for the a-th component. Next we take

A_{1} = ∐_{\begin{matrix}G∈g \\ t_{0}:G→A_{0} \\ ϵt_{0}=0 \end{matrix}} G,

with a similar map d_{0} : A_{1} → A_{0} whose t_{0}-th component is t_{0} (so obviously ϵd_{0} = 0). One continues in this way, with

A_{n+1} = ∐_{\begin{matrix}G∈g \\ t_{n}:G→A_{n} \\ d_{n−1}t_{n}=0 \end{matrix}}G,

n 1, with d_{n} : A_{n+1} → A_{n} having t_{n}-th component equal to t_{n}. Once again, d_{n−1}d_{n} = 0 is obvious.

To prove exactness, first note that if Hom_{A}(X, A_{•}) is exact, then for any retract A of X Hom_{A}(A, A_{•}) is exact as well. Similarly if Hom_{A}(G_{i}, A_{•}) are exact, so is Hom_{A}(∐ _{i} G_{i}, A_{•}). Thus it suffices to show that Hom_{A}(G, A_{•}) is exact for any object G from g. Thus suppose given t_{n} : G → A_{n} with d_{n−1}t_{n} = 0. Then t_{n} = d_{n}t_{n+1}, where t_{n+1} : G → A_{n+1} is the canonical inclusion of the t_{n}-th component into the coproduct.

(1.4) Lemma. Suppose given two A-augmented chain complexes ϵ : A_{•} → A and ϵ′: A′_{•} → A. If A_{n} are in a for n 0 and A′_{•} is a-exact, then there exists a chain map f : A_{•} → A′_{•} over A (i. e. with f_{−1} equal to the identity of A). Moreover this map is unique up to a chain homotopy over A, i. e. for any two f, f′: A_{•} → A′_{•} over A there is a chain homotopy h_{•} from f to f′over A (which means h_{0} = 0).

Proof. Since A_{0} is in a, by a-exactness of A′_{•} the map Hom_{A}(A_{0}, ϵ′) is surjective; in particular, there is a morphism f_{0} : A_{0} → A′_{0} with ϵ′f_{0} = ϵ. Next, as A_{1} is also in a, and ϵ′f_{0}d_{0} = ϵd_{0} = 0, again by a-exactness of A′_{•} there is a map f_{1} : A_{1} → A′_{1} with f_{0}d_{0} = d′_{0}f_{1}. Continuing this way, one obtains a sequence of maps f_{n} : A_{n} → A′_{n} with d′_{n}f_{n} = f_{n−1}d_{n} for all n 0.

Suppose now given two such sequences f, f′. Take h_{0} = 0 : A → A′_{0}. Since ϵ′(f_{0} − f′_{0}) = 0, there is a h_{1} : A_{0} → A′_{1} with f_{0} − f′_{0} = d′_{0}h_{1} = d′_{0}h_{1} + h_{0}d_{0}. Next since d′_{0}(f_{1} − f′_{1}− h_{1}d_{0}) = (f_{0} − f′_{0})d_{0} − d′_{0}h_{1}d_{0} = 0, there is a h_{2} : A_{1} → A′_{2} with f_{1} − f′_{1}− h_{1}d_{0} = d′_{1}h_{2}. Continuing one obtains the desired chain homotopy h.

As an immediate corollary we obtain that any two a-resolutions A_{•}, A′_{•} of an object are chain homotopy equivalent, i. e. there are maps f : A′_{•} → A_{•}, f′ : A_{•} → A′_{•} with f f′ and f′f chain homotopic to identity maps. We thus see that all the standard ingredients for doing homological algebra are available. So we define

(1.5) Definition. a-relative left derived functors L_{n}^{a}F, n 0, of a functor F : A → 𝒜 from A to an abelian category 𝒜 are defined by

(L_{n}^{a}F)A = H_{n}(F(A_{•})),

where A_{•} is given by any a-resolution of A. Similarly, a-relative right derived functors of a contravariant functor F : A^{op} → 𝒜 are given by

(R_{a}^{n}F)A = H^{n}(F(A_{•})).

By the above lemmas, L_{n}^{a}F and R_{a}^{n}F are indeed functors and do not depend on the choice of resolutions. Note also that these constructions are functorial in F, i. e. a natural transformation F → F′induces natural transformations between the corresponding derived functors.

In particular, we have a-relative Ext-groups given by

Ext_{a}^{n}(A, X) = (R_{a}^{n}(Hom_{A}( , X)))A = H^{n}(Hom_{A}(A_{•}, X)),

for objects A, X of A and an a-exact a-resolution A_{•} of A. Note that these groups can be equipped with the Yoneda product

Ext_{a}^{m}(Y, Z) ⊗ Ext_{a}^{n}(X, Y) → Ext_{a}^{m+n}(X, Z).

On representing cocycles this product can be defined as follows: given a-exact a-resolutions X_{•} of X and Y_{•} of Y, we can represent elements of the Ext groups in question by maps f : Y_{m} → Z with f d_{m} = 0 and g : X_{n} → Y with gd_{n} = 0. Then similarly to the proof of 1.4, we can find maps h_{0} : X_{n+1} → Y_{0}, ..., h_{m−1} : X_{n+m} → Y_{m−1}, h_{m} : X_{n+m+1} → Y_{m} giving a map of complexes, and define [f][g] = [f h_{m}]. Standard homological algebra argument then shows that this product is well-defined, bilinear and associative.

(1.6) Examples. 1. A typical situation for the above is given by a ringoid g, with A being the category of g-modules, i. e. of linear functors from g to abelian groups. The abelian version of the Yoneda embedding identifies g with the full subcategory of A with objects the representable functors. The natural choice for a is then either the category of free g-modules, which is the closure of this full subcategory g ⊂ A under arbitrary coproducts, or that of projective g-modules — the closure under both coproducts and retracts. In particular, when g has only one object, we obtain the classical setup for homological algebra given by a ring R, with A being the category of R-modules and a that of free or projective R-modules.

2. When A has finite limits, we obtain the additive case of derived functors from [18].

(1.7) Remark. There is an obvious dual version of the above which one obtains by replacing A with the opposite category A^{op}. Explicitly, chain complexes get replaced by cochain complexes (with differentials having degree +1 rather than -1); exactness of the complex Hom_{A}(X, A_{•}) becomes replaced by that of Hom_{A}(A^{•}, X), etc.

(1.8) Example. Let A be the stable homotopy category of spectra and let a ⊂ A be the full subcategory consisting of finite products of Eilenberg-Mac Lane spectra over a fixed prime field _{p}. Let 𝒜 be the mod p Steenrod algebra. The mod p cohomology functor restricted to a yields an equivalence of categories for which the following diagram commutes

\xymatrix{A^{op} \ar[r]^{H^{∗}} & 𝒜 −Mod = A_{𝒜} \\ a^{op} \ar@{^{(}->}[u] \ar[r]^{∼} & 𝒜 −mod = a_{𝒜} . \ar@{^{(}->}[u]}

Here A^{op} denotes the opposite category of A, 𝒜 −Mod is the category of positively graded A-modules and 𝒜 −mod is its full subcategory of finitely generated free modules. Given a spectrum X, its a-coresolution (A_{•}^{ϵ}, d)

... ← A_{1} ← A_{0} ← X ← 0 ← 0 ← ...

is an X-coaugmented chain complex in A, with A_{n} in a for n 0, which is a-coexact, that is Hom_{A}(A_{•}^{ϵ}, A′) is acyclic for all A′ ∈ a. Hence (A_{•}^{ϵ}, d) is an a^{op}-resolution of X in A^{op} which is carried by the cohomology functor H^{∗} to an a_{𝒜} -resolution of H^{∗}(X) in A_{𝒜} above. For this reason we get for a spectrum Y the binatural equation

Ext_{a^{op}}^{m}(X, Y) = Ext_{a_{𝒜}}^{m} (H^{∗}(X), H^{∗}(Y)).

Here the left hand side Ext_{a^{op}}^{m}(X, Y) is defined in the additive category A^{op} which is the opposite of the stable homotopy category. Moreover the right hand side is the classical Ext group

Ext_{a_{𝒜}}^{m} (H^{∗}(X), H^{∗}(Y)) = Ext_{𝒜}^{m} (H^{∗}(X), H^{∗}(Y)).

2. Secondary resolutions

We have seen in section 1 that resolutions yield the notion of derived functors. We now introduce secondary resolutions from which we deduce secondary derived functors. For this we need the notion of tracks.

Recall that a track category is a category enriched in groupoids; in particular, for all of its objects X, Y their hom-groupoid X, Y is given, whose objects are maps f : X → Y and whose morphisms, denoted α : f ⇒ f′, are called tracks.

Equivalently, a track category is a 2-category all of whose 2-cells are invertible. For a track α : f ⇒ f′ above and maps g : Y → Y′, e : X′ → X, the resulting composite tracks will be denoted by gα : gf ⇒ gf′ and αe : f e ⇒ f′e. Moreover there is a vertical composition of tracks, i. e. composition of morphisms in the groupoids X, Y; for α : f ⇒ f′ and β : f′ ⇒ f′′, it will be denoted β α : f ⇒ f′′. An inverse of a track α with respect to this composition will be denoted α^{}.

By a track functor we will mean a groupoid enriched functor between track categories.

A track category B will be also depicted as B_{1} B_{0}. Here B_{0} being the underlying ordinary category of B obtained by forgetting about the tracks, whereas B_{1} is another ordinary category with the same objects but with morphisms from X to Y being tracks α : f ⇒ f′ with f, f′ : X → Y in B_{0}, composite of α and β in the diagram

[figure: Z; Y; f′; f; α; X; g′; g; β]

being

(2.1)  αβ = αg′ f β = f′β αg : f g ⇒ f′g′.

There are thus two functors B_{1} → B_{0} which are identity on objects and which send a morphism α : f ⇒ f′ to f, resp. f′.

A track category B has the homotopy category B_{≃} — an ordinary category obtained by identifying homotopic maps, i. e. maps f, f′ for which there exists a track f ⇒ f′. It is thus the coequalizer of B_{1} B_{0} in the category of categories.

We now assume given a track category B such that its homotopy category is an additive category like A from section 1,

B_{≃} = A,

and that moreover B has a strict zero object, that is, an object * such that for every object X of B, X, ∗ and ∗, X are trivial groupoids with a single morphism. It then follows that in each X, Y there is a distinguished map 0_{X,Y}obtained by composing the unique maps X → ∗ and ∗ → Y. The identity track of this map will be denoted just by 0. Note that 0_{X,Y}may also admit non-identity self-tracks; one however has

(2.2)  0_{Y,Z}β = 0 = α0_{X,Y}

for any α : f ⇒ f′, f, f′: Y → Z, β : g ⇒ g′, g, g′: X → Y.

In section 3 we introduce the notion of an “additive track category” which is the most appropriate framework for secondary derived functors and which has the properties of the track category B.

(2.3) Example. The most easily described example is the track category��_{A} whose objects are chain complexes in an additive category A, maps are chain maps, and tracks are chain homotopies.

Our basic example is the track category����_{A}; it is the full track subcategory of�� whose objects are chain complexes concentrated in degrees 0 and 1 only. Thus objects A of����_{A} are given by morphisms ∂_{A}: A_{1} → A_{0} in A, a map f from A to B is a pair of morphisms (f_{1} : A_{1} → B_{1}, f_{0} : A_{0} → B_{0}) in A making the obvious square commute, and a track f ⇒ f′ for f, f′ : A → B is a morphism ϕ : A_{0} → B_{1} in A satisfying ϕ∂_{A} = f_{1} − f′_{1}and ∂_{B}ϕ = f_{0} − f′_{0}.

(2.4) Remark. The secondary homology ℋ , as defined in [2], yields a track functor

ℋ :��_{A} →����_{A^{}}.

Here A is an abelian category, A^{}denotes the category of -graded objects in A, and for a chain complex (A, d) in A the n-th component of ℋ (A, d) is given by

ℋ_{n}(A, d) = (d_{n} : Coker(d_{n+1}) → Ker(d_{n−1})) .

(2.5) Example. A further basic example we have in mind is the track category B which is opposite to the category of spectra, stable maps, and tracks which are stable homotopy classes of stable homotopies.

Next we describe the secondary analogues of the notions of chain complex and resolution in 1.1, 1.2.

(2.6) Definition. A secondary chain complex (A, d, δ) in a track category B is a diagram of the form

[figure: ⋯; 0; A_{n+2}; d_{n+1}; 0; δ_{n}; A_{n+1}; d_{n}; 0; δ_{n−1}; A_{n}; d_{n−1}; 0; A_{n−1}; ⋯]

i. e. a sequence of objects A_{n}, maps d_{n} : A_{n+1} → A_{n} and tracks δ_{n} : d_{n}d_{n+1} ⇒ 0, n ∈ , such that for each n the tracks

\xymatrix{d_{n−1}d_{n}d_{n+1} \ar@{=>}[r]^{d_{n−1}δ_{n}} & d_{n−1}0 = 0}

and

\xymatrix{d_{n−1}d_{n}d_{n+1} \ar@{=>}[r]^{δ_{n−1}d_{n+1}} & 0d_{n+1} = 0}

coincide. Equivalently, the track δ_{n−1}d_{n+1} d_{n−1}δn^{}in hom_{A_{n+2},A_{n−1}}(0, 0) must be the identity.

It is clear that a track functor F : B → B′ between track categories as above (which preserves the zero object) carries a secondary chain complex in B to a secondary chain complex in B′.

(2.7) Examples.

1. In the example����_{A}, a secondary chain complex looks like

[figure: ...; A_{1,n+2}; d_{1,n+1}; ∂_{n+2}; A_{1,n+1}; d_{1,n}; ∂_{n+1}; A_{1,n}; d_{1,n−1}; ∂_{n}; A_{1,n−1}; ∂_{n−1}; ...; ...; δ_{n+1}; A_{0,n+2}; d_{0,n+1}; δ_{n}; A_{0,n+1}; d_{0,n}; δ_{n−1}; A_{0,n}; d_{0,n−1}; A_{0,n−1}; ...]

with the equations ∂_{n}d_{1,n} = d_{0,n}∂_{n+1}, d_{1,n−1}d_{1,n} = δ_{n−1}∂_{n+1}, d_{0,n−1}d_{0,n} = ∂_{n−1}δ_{n−1} and d_{1,n−1}δ_{n} = δ_{n−1}d_{0,n+1} satisfied for all n.

More generally for��_{A} what one obtains is a bigraded group A_{m,n} with differentials ∂_{m,n} : A_{m+1,n} → A_{m,n}, ∂_{m,n}∂_{m+1,n} = 0, and maps d_{m,n}: A_{m,n+1} → A_{m,n}, δ_{m,n}: A_{m−1,n+2} → A_{m,n}satisfying the similar equalities for all m and n.

One thus obtains a structure strongly related to what is called multicomplex or twisted complex in the literature; cf. [9, 10, 12]

2. In [17], the notion of complex of categories with abelian group structure is investigated. One can show that a slightly strictified version of their notion coincides with that of the secondary chain complex in an appropriate track category. On the other hand we could relax the requirement of existence of the strict zero object to that of a weak zero object; then the construction of [17] would be subsumed in full generality.

(2.8) Definition. A secondary chain map (f, ϕ) between secondary chain complexes (A, d, δ) and (A′, d′, δ′) is a sequence of maps and tracks as indicated

[figure: ...; A_{n+1}; d_{n}; 0; δ_{n−1}; f_{n+1}; A_{n}; d_{n−1}; f_{n}; ⇓ϕ_{n}; A_{n−1}; f_{n−1}; ⇓ϕ_{n−1}; ...; ...; A′_{n+1}; d′_{n}; 0; δ′_{n−1}; A′_{n}; d′_{n−1}; A′_{n−1}; ...]

such that pasting of tracks in this diagram yields the trivial track 0 : 0 ⇒ 0, that is, the resulting track diagrams

(2.9)  [figure: d′_{n−1}f_{n}d_{n}; d′_{n−1}ϕ_{n}; d′_{n−1}d′_{n}f_{n+1}; δ′_{n−1}f_{n+1}; f_{n−1}d_{n−1}d_{n}; ϕ_{n−1}d_{n}; f_{n−1}δ_{n−1}; f_{n−1}0; 0; 0f_{n+1}]

commute for all n ∈ .

For secondary chain maps (f, ϕ) : (A, d, δ) → (A′, d′, δ′) and (f′, ϕ′) : (A′, d′, δ′) → (A′′, d′′, δ′′), their composite is given by (f′_{n}f_{n}, ϕ′_{n}f_{n+1} f′_{n}ϕ_{n}), n ∈ . It is straightforward to check that this indeed defines a secondary chain map, and that the resulting composition operation is associative. Thus these operations determine the category of secondary chain complexes.

We now similarly to section 1 fix a full track subcategory b of B, with a = b_{≃}.

(2.10) Definition. For a secondary complex (A, d, δ) in B and an integer n, its b-chain of degree n is a map X → A_{n} for some object X of b. A b-cycle is a pair (c, γ) consisting of a b-chain c : X → A_{n} and a track γ : d_{n−1}c ⇒ 0 such that the track d_{n−2}γ : d_{n−2}d_{n−1}c ⇒ d_{n−2}0 = 0 is equal to δ_{n−2}c : d_{n−2}d_{n−1}c ⇒ 0c = 0. A b-cycle (b, β) of degree n is a b-boundary if there exists a b-chain a of degree n + 1 and a track α : b ⇒ d_{n}a such that the following diagram of tracks commutes:

[figure: d_{n−1}d_{n}a; δ_{n−1}a; 0a; d_{n−1}b; β; d_{n−1}α; 0.]

A secondary complex (A, d, δ) is called b-exact if all of its b-cycles are b-boundaries. In other words, every diagram consisting of solid arrows below

[figure: X; 0; c; a; ⟹^{γ}; ⋯; A_{n+1}; ==⇒^{α}; d_{n}; 0; δ_{n−1}; A_{n}; d_{n−1}; 0; δ_{n−2}; A_{n−1}; d_{n−2}; A_{n−2}; ⋯]

in which the pasted track from d_{n−2}0_{X,A_{n−1}} to 0_{A_{n},A_{n−2}}c is the identity track can be completed by the dashed arrows in such a way that the resulting pasted track from 0_{A_{n+1},A_{n−1}}a to 0_{X,A_{n−1}} is the identity track.

(2.11) Example. Consider the track category����_{A} from 2.3, with A the category of modules over a ring R, and choose for b the full track subcategory on the objects 0 → R^{n}, n 0. Then for a secondary chain complex as in 2.7, a secondary cycle of degree n is a pair (c, γ) ∈ A_{0,n} × A_{1,n−1} satisfying d_{0,n−1}c = ∂_{n−1}γ and δ_{n−2}c = d_{1,n−2}γ. Such a cycle is a boundary if there exist elements a ∈ A_{0,n+1} and α ∈ A_{1,n} with c = d_{0,n}a + ∂_{n}α and γ = δ_{n−1}a + d_{1,n−1}α.

Note that we can arrange for a total complex

... ← A_{0,n−1} ⊕ A_{1,n−2} \underleftarrow{( \begin{matrix}d_{0,n−1} & −∂_{n−1} \\ δ_{n−2} & −d_{1,n−2} \end{matrix} )} A_{0,n} ⊕ A_{1,n−1} \underleftarrow{( \begin{matrix}d_{0,n} & −∂_{n} \\ δ_{n−1} & −d_{1,n−1} \end{matrix} )} A_{0,n+1} ⊕ A_{1,n} ← ...

in such a way that secondary cycles and boundaries will become usual cycles and boundaries in this total complex. In particular then, secondary exactness of the secondary chain complex of type 2.7 is equivalent to the exactness in the ordinary sense of the above total complex.

We now turn to the secondary analog of the notion of resolution from 1.2.

(2.12) Definition. For an object B in B, a B-augmented secondary chain complex is a secondary chain complex (B, d, δ) with B_{−1} = B, B_{−n} = 0 for n > 1, and δ_{−n} equal to identity track for n > 1. For a full track subcategory b of B, a B-augmented secondary chain complex is called a b-resolution of B if it is b-exact as a secondary chain complex and moreover all B_{n} for n 0 belong to b.

As in the primary case, denoting ϵ = d_{−1}, ϵ̂ = δ_{−1}, a B-augmented secondary chain complex can be considered as a secondary chain map (ϵ, ϵ̂) : B_{•} → B from the secondary chain complex B_{•} given by ... → B_{1} → B_{0} → 0 → 0 → ... with δ_{−n} identities for all n > 0, to the secondary chain complex B concentrated in degree 0, with trivial differentials:

[figure: …; B_{2}; d_{1}; 0; δ_{0}; B_{1}; d_{0}; =; B_{0}; ϵ; ⇓ϵ̂; 0; =; …; …; 0; 0; B; 0; . . ..]

Accordingly such an augmented secondary chain complex will be denoted B_{•}^{ϵ̂,ϵ} , and the pair (ϵ, ϵ̂) will be called its augmentation.

Dually, we have the notion of a B-coaugmented secondary chain complex - the one satisfying B_{1} = B, B_{n} = 0 for n > 1, and δ_{n} equal to the identity track for n > 1. Accordingly, there is a notion of a b-coresolution of B.

To have the analog of 1.3 we need appropriate notion of coproduct; we might in principle use groupoid enriched, or strong coproducts, but for further applications more suitable is the less restrictive notion of weak coproduct, which we now recall.

(2.13) Definition. A family of maps (i_{k} : A_{k} → A)_{k∈K} in a track category is a weak (respectively, strong) coproduct diagram if for every object X the induced functor

A, X → ∏_{k∈K} A_{k}, X

is an equivalence (resp., isomorphism) of groupoids.

Thus being a weak coproduct diagram means two things:

1) for any object X and any maps x_{k} : A_{k} → X, k ∈ K, there is a map x : A → X and a family of tracks χ_{k} : x_{k} ⇒ xi_{k}, k ∈ K;

2) for any x, x′ : A → X and any family of tracks (χ_{k} : xi_{k} ⇒ x′i_{k})_{k∈K} there is a unique track

χ : x → x′satisfying χ_{k} = χi_{k} for all k ∈ K, whereas for a strong coproduct one must have

1’) for any object X and any maps x_{k} : A_{k} → X, k ∈ K, there is a unique map x : A → X satisfying x_{k} = xi_{k} for all k ∈ K and 2).

We can also weaken the notion of retract in 1.3: call an object X a weak retract of an object Y if there exist maps j : X → Y, p : Y → X and a track 1_{X} ⇒ pj.

(2.14) Lemma. Suppose • weak coproducts of any families of objects of b exist in B and belong to b again;

• there is a small track subcategory g of b such that every object of b is a weak retract of a weak

coproduct of a family of objects from g. Then every object of B has a b-resolution.

Proof. The first step is exactly as in the primary case: for an object B we take

B_{0} = ∐_{\begin{matrix}G∈g \\ b:G→B \end{matrix}} G,

where ∐ denotes weak coproduct. Thus in particular there is a map d_{−1} : B_{0} → B and a family of tracks ι_{b} : b ⇒ d_{−1}i_{b} for each b : G → B.

Suppose now given a −1-dimensional b-cycle (b, β) in the resolution. This means just a map b : X → B for an object X of b, since β : d_{−2}b ⇒ 0 is then necessarily the trivial track. By hypothesis we then can find some weak coproduct G = ∐ _{k∈K}G_{k} of objects from g, maps j : X → G and p : G → X, and a track θ : 1_{X} ⇒ pj. Then by the weak coproduct property, for the maps i_{bpi_{k}} : G_{k} → B_{0}, where i_{k} : G_{k} → G are the weak coproduct inclusions, there exists a map f_{0} : G → B_{0} and a family of tracks ι_{k} : i_{bpi_{k}} ⇒ f_{0}i_{k}, k ∈ K. This then gives composite tracks

[figure: d_{−1}f_{0}i_{k}; d_{−1}ι_{k}; d_{−1}i_{bpi_{k}}; ι_{bpi_{k}}; bpi_{k}.]

Again by the weak coproduct universality there is then a track ϕ : bp ⇒ d_{−1}f_{0} with d_{−1}ι_{k} ι_{bpi_{k}} = ϕi_{k} for all k ∈ K. Denoting f_{0}j by a, one then obtains a track α : b ⇒ d_{−1}a, namely the composite

\xymatrix{b \ar@{=>}[r]^{bθ} & bpj \ar@{=>}[r]^{ϕj} & d_{−1}f_{0}j,}

which means that (b, β) is a boundary, since both β and δ_{−2}a d_{−2}α are zero by trivial reasons.

We next take

B_{1} = ∐_{\begin{matrix}G∈g \\ t_{0}:G→B_{0} \\ τ:d_{−1}t_{0}⇒0 \end{matrix}} G.

Then by the weak coproduct property, for the family (t_{0} : G → B_{0})_{τ:d_{−1}t_{0}⇒0} there exists d_{0} : B_{1} → B_{0} and tracks ι_{τ} : t_{0} ⇒ d_{0}i_{τ}, where the i_{τ} : G → B_{1} are the coproduct structure maps. Moreover for the family

(\xymatrix{d_{−1}d_{0}i_{τ} \ar@{=>}[r]^{d_{−1}ι_{τ}^{}} & d_{−1}t_{0} \ar@{=>}[r]^{τ} & 0 = 0_{G,B} = 0_{B_{1},B}i_{τ}} )_{τ:d_{−1}t_{0}⇒0}

there exists δ_{−1} : d_{−1}d_{0} ⇒ 0 with

(†_{0})  δ_{−1}i_{τ} = τ d_{−1}ιτ^{}

for all τ : d_{−1}t_{0} ⇒ 0. Since δ_{−2} by definition must be the identity track of the zero map, whereas d_{−2} is the unique map to the zero object, the condition d_{n−1}δ_{n} = δ_{n−1}d_{n+1} from 2.6 is trivially satisfied at n = −1.

To prove exactness at B_{0}, suppose given b_{0} : X → B_{0} and β : d_{−1}b_{0} ⇒ 0, for some object X of b. By hypothesis, there is a weak retraction j : X → G, p : G → X, θ : 1_{X} ⇒ pj for some weak coproduct G = ∐ _{k∈K}G_{k} of objects G_{k} from g. Then for the family (i_{βpi_{k}} : G_{k} → B_{1})_{k∈K}, where i_{k} : G_{k} → G are the coproduct structure maps, there exists a map f_{1} : G → B_{1} and tracks ι_{k} : i_{βpi_{k}} ⇒ f_{1}i_{k}, k ∈ K. One thus obtains the composite tracks

[figure: d_{0}f_{1}i_{k}; d_{0}ι_{k}; d_{0}i_{βpi_{k}}; ι_{βpi_{k}}; b_{0}pi_{k}.]

Then again by the second property of weak coproducts there is a track ϕ_{0} : b_{0}p ⇒ d_{0}f_{1} with ϕ_{0}i_{k} = d_{0}ι_{k} ι_{βpi_{k}}, k ∈ K. One then gets a_{1} = f_{1}j and α = ϕ_{0}j b_{0}θ : b_{0} ⇒ d_{0}a_{1}. To prove that (a_{1}, α) exhibits (b_{0}, β) as a boundary, it remains to show β = δ_{−1}a_{1} d_{−1}α, i. e. β = δ_{−1}f_{1}j d_{−1}ϕ_{0}j d_{−1}b_{0}θ. Now we have

\begin{aligned}δ_{−1}f_{1}i_{k} d_{−1}ϕ_{0}i_{k} & = δ_{−1}f_{1}i_{k} d_{−1}d_{0}ι_{k} d_{−1}ι_{βpi_{k}} \\ & = δ_{−1}i_{βpi_{k}} d_{−1}ι_{βpi_{k}}. \end{aligned}

On the other hand by (†_{0}) one has δ_{−1}i_{βpi_{k}} = βpi_{k} d_{−1}ιβpi_{k}^{}, so one obtains

δ_{−1}f_{1}i_{k} d_{−1}ϕ_{0}i_{k} = βpi_{k}

for all k; by the weak coproduct property this then implies δ_{−1}f_{1} d_{−1}ϕ_{0} = βp, hence

δ_{−1}f_{1}j d_{−1}ϕ_{0}j d_{−1}b_{0}θ = βpj d_{−1}b_{0}θ
= 0θ β
= β.

Now take some n 1 and suppose all the B_{i}, d_{i−1} and δ_{i−2} have been already constructed for i n in such a way that the conditions of 2.6 and b-exactness are satisfied up to dimension n − 1. Moreover we can assume by induction that exactness is constructively established for b-cycles originating at g, that is, for each (n − 1)-cycle (t_{n−1} : G → B_{n−1}, τ_{n−2} : d_{n−2}t_{n−1} ⇒ 0), G ∈ g, with d_{n−3}τ_{n−2} = δ_{n−3}t_{n−1}, we are given explicit maps i_{τ_{n−2}} : G → B_{n−1} and tracks ι_{τ_{n−2}} : t_{n−1} ⇒ d_{n−1}i_{τ_{n−2}} satisfying τ_{n−2} = δ_{n−2}i_{τ_{n−2}} d_{n−2}ι_{τ_{n−2}}. At least this induction hypothesis is certainly satisfied for n = 1, by (†_{0}) above.

We then define

B_{n+1} = ∐_{\begin{matrix}G∈g \\ t_{n}:G→B_{n} \\ τ_{n−1}:d_{n−1}t_{n}⇒0 \\ d_{n−2}τ_{n−1}=δ_{n−2}t_{n} \end{matrix}}G.

Then for the family (t_{n} : G → B_{n})_{{τ_{n−1}:d_{n−1}t_{n}⇒0 | d_{n−2}τ_{n−1}=δ_{n−2}t_{n}}} there exists d_{n} : B_{n+1} → B_{n} and tracks ι_{τ_{n−1}} : t_{n} ⇒ d_{n}i_{τ_{n−1}}, where the i_{τ_{n−1}} : G → B_{n+1} are the coproduct structure maps. Moreover for the family

(\xymatrix{d_{n−1}d_{n}i_{τ_{n−1}} \ar@{=>}[r]^{d_{n−1}ι_{τ_{n−1}}^{}} & d_{n−1}t_{n} \ar@{=>}[r]^{τ_{n−1}} & 0 = 0i_{τ_{n−1}}} )_{\{τ_{n−1}:d_{n−1}t_{n}⇒0 | d_{n−2}τ_{n−1}=δ_{n−2}t_{n}\}}

there exists δ_{n−1} : d_{n−1}d_{n} ⇒ 0 with

(†_{n})  δ_{n−1}i_{τ_{n−1}} = τ_{n−1} d_{n−1}ιτ_{n−1}^{}

for all τ_{n−1} : d_{n−1}t_{n} ⇒ 0 with d_{n−2}τ_{n−1} = δ_{n−2}t_{n}. To prove the condition d_{n−2}δ_{n−1} = δ_{n−2}d_{n} from 2.6, it suffices by the weak coproduct property to prove

d_{n−2}τ_{n−1}i_{τ_{n−1}} d_{n−2}d_{n−1}ι_{τ_{n−1}} = δ_{n−2}d_{n}i_{τ_{n−1}} d_{n−2}d_{n−1}ι_{τ_{n−1}}

for each τ_{n−1} : d_{n−1}t_{n} ⇒ 0, t_{n} : G → B_{n}, with d_{n−2}τ_{n−1} = δ_{n−2}t_{n}. Now by (†_{n}) we have

d_{n−2}δ_{n−1}i_{τ_{n−1}} d_{n−2}d_{n−1}ι_{τ_{n−1}} = d_{n−2}τ_{n−1},

whereas by naturality we have

δ_{n−2}d_{n}i_{τ_{n−1}} d_{n−2}d_{n−1}ι_{τ_{n−1}} = 0ι_{τ_{n−1}} δ_{n−2}t_{n} = δ_{n−2}t_{n}.

Next note that the maps i_{τ_{n−1}} and tracks ι_{τ_{n−1}} fulfil the induction hypothesis, i. e. explicitly exhibit cycles with domains from g as boundaries. Finally to prove exactness at B_{n}, suppose given any X, any weak coproduct G = ∐ _{k∈K}G_{k} of objects from g, any weak retraction j : X → G, p : G → X, θ : 1_{X} ⇒ pj, and any b_{n} : X → B_{n}, β_{n−1} : d_{n−1}b_{n} ⇒ 0 with d_{n−2}β_{n−1} = δ_{n−2}b_{n}. Then for each coproduct inclusion i_{k} : G_{k} → G one has cycles given by b_{n}pi_{k} : G_{k} → B_{n}, β_{n−1}pi_{k} : d_{n−1}b_{n}pi_{k} ⇒ 0, hence for the family (i_{β_{n−1}pi_{k}} : G_{k} → B_{n+1})_{k∈K} there exists a map f_{n+1} : G → B_{n+1} and tracks ι_{k} : i_{β_{n−1}pi_{k}} ⇒ f_{n+1}i_{k}. We then can consider the composite tracks

[figure: d_{n}f_{n+1}i_{k}; d_{n}ι_{k}; d_{n}i_{β_{n−1}pi_{k}}; ι_{β_{n−1}pi_{k}}; b_{n}pi_{k}]

and by the weak coproduct property of G find for them a track ϕ_{n} : b_{n}p ⇒ d_{n}f_{n+1} with d_{n}ι_{k} ι_{β_{n−1}pi_{k}} = ϕ_{n}i_{k} for all k ∈ K. This gives us an (n + 1)-chain a_{n+1} = f_{n+1}j and a track α = ϕ_{n}j b_{n}θ : b_{n} ⇒ d_{n}a_{n+1}. To show that these exhibit (b_{n}, β_{n−1}) as a boundary, one has to prove β_{n−1} = δ_{n−1}a_{n+1} d_{n−1}α. The proof goes exactly as for the case n = 0 above.

Now to the analog of 1.4.

(2.15) Lemma. Suppose given two B-augmented secondary chain complexes B_{•} and B′_{•}. If all B_{n} belong to b and B′_{•} is b-exact, then there exists a secondary chain map (f, ϕ) : B_{•} → B′_{•} over B (i. e. with f_{−1} equal to the identity of B).

Proof. The pair d_{−1} : B_{0} → B, identity_{0} : d_{−2}d_{−1} ⇒ 0 can be considered as a (−1)-cycle in B′_{•}, so by

b-exactness of B′_{•} there exist f_{0} : B_{0} → B′_{0} and ϕ_{−1} : d_{−1} ⇒ d′_{−1}f_{0}. Next f_{0}d_{0}, δ_{−1} ϕ−1^{}d_{0} : d′_{−1}f_{0}d_{0} ⇒

d_{−1}d_{0} ⇒ 0 is a 0-cycle in B′_{•}, so again by exactness of B′_{•} there are f_{1} : B_{1} → B′_{1} and ϕ_{0} : f_{0}d_{0} ⇒ d′_{0}f_{1} with

(∗)  δ_{−1} ϕ−1^{}d_{0} = δ′_{−1}f_{1} d′_{−1}ϕ_{0},

which ensures the condition of 2.8 for n = 0. Then f_{1}d_{1}, f_{0}δ_{0} ϕ0^{}d_{1} : d′_{0}f_{1}d_{1} ⇒ f_{0}d_{0}d_{1} ⇒ f_{0}0 = 0 is a 1-cycle in B′_{•}. Indeed (∗) above implies δ′_{−1}f_{1}d_{1} = δ_{−1}d_{1} ϕ−1^{}d_{0}d_{1} d′_{−1}ϕ0^{}d_{1}; on the other hand

δ_{−1}d_{1} ϕ−1^{}d_{0}d_{1} = d_{−1}δ_{0} ϕ−1^{}d_{0}d_{1} = ϕ−1^{}0 d′_{−1}f_{0}δ_{0} = d′_{−1}f_{0}δ_{0}, so δ′_{−1}f_{1}d_{1} = d′_{−1}f_{0}δ_{0} d′_{−1}ϕ0^{}d_{1}, which pre-

cisely means that the cycle condition is fulfilled. One thus obtains f_{2} : B_{2} → B′_{2} and ϕ_{1} : f_{1}d_{1} ⇒ d′_{1}f_{2} such that f_{0}δ_{0} ϕ0^{}d_{1} = δ′_{0}f_{2} d′_{0}ϕ_{1}, so the condition of 2.8 at n = 1 is also satisfied.

It is clear that continuing in this way one indeed obtains a secondary chain map.

3. Additive track categories

The secondary analogue of an additive category is an additive track category considered in this section. For related conditions, see [8].

(3.1) Definition. A track category B is called additive if it has a strict zero object ∗, the homotopy category A = B_{≃} is additive and moreover B is a linear track extension

D → B_{1} B_{0} → A

of A by a biadditive bifunctor

D : A^{op} × A →�� .

Explicitly, this means the following: a biadditive bifunctor D as above is given together with a system of isomorphisms

(3.2)  σ_{f} : D(X, Y) → Aut_{X,Y}(f)

for each 1-arrow f : X → Y in B, such that for any f : X → Y, g : Y → Z, a ∈ D(X, Y), b ∈ D(Y, Z), α : f ⇒ f′one has

\begin{aligned}σ_{gf} (ga) & = gσ_{f} (a); \\ σ_{gf} (bf) & = σ_{g}(b)f; \\ α σ_{f} (a) & = σ_{f′}(a) α. \end{aligned}

(3.3) Remark. Using 3.2 we can identify the bifunctor D via the natural equation

D(X, Y) = Aut(0_{X,Y}),

where 0 = 0_{X,Y}: X → ∗ → Y is the unique morphism factoring through the zero object.

A strict equivalence between additive track categories B, B′ is a track functor B → B′ which induces identity on A and is compatible with the actions 3.2 above. Thus for fixed A and D as above, one obtains a category whose objects are additive track categories which are linear track extensions of A by D and morphisms are strict equivalences. This category will be denoted by Trext(A;D). For an additive category A and a biadditive bifunctor D on it, there is a bijection

π_{0}(Trext(A; D)) ≈ H^{3}(A; D),

where π_{0}( ) denotes the set of connected components of a small category . Two additive track categories are called equivalent if they are in the same connected component of Trext(A;D). Thus in particular (as shown in [3, 13]) each additive track category B as above determines a class ⟨B⟩∈ H^{3}(A; D).

As shown in [15], when A is the category of free finitely generated modules over a ring R and D is given by D(X, Y) = Hom_{R}(X, B ⊗_{R}Y) for some R-R-bimodule B, there are isomorphisms

H^{3}(A; D) HML^{3}(R; B) THH^{3}(HR; HB),

where HML^{∗} denotes Mac Lane cohomology, THH is the topological Hochschild cohomology, and HR and HB are the Eilenberg-Mac Lane spectra corresponding to R and B.

(3.4) Definition. An additive track category B is -additive if an additive endofunctor : A → A is given which left represents the bifunctor D, i. e. B is a linear track extension of A by the bifunctor

D(X, Y) = Hom_{A}( X, Y).

Dually, B is -additive if an additive endofunctor : A → A is given such that B is a linear track extension of A by the bifunctor

D(X, Y) = Hom_{A}(X, Y).

For objects X, Y in a -, resp. -additive track category B we will denote the group Hom_{A}( ^{m}X, Y), resp. Hom_{A}(X, ^{m}Y) by [X, Y]^{m}.

In examples from topology the functor is the suspension and the functor is the loop space, compare also [6].

- or -additivity of a track category enables one to relate secondary exactness of a secondary chain complex with exactness of the corresponding chain complex in the homotopy category.

(3.5) Lemma. Let B be a track category with the additive homotopy category A = B_{≃}, let b be a full track subcategory of B and denote a = b_{≃}. Suppose that one of the following conditions is satisfied:

a) B is -additive and a is closed under suspensions (i. e. for each X ∈ a one has X ∈ a); or

b) B is -additive and the functor is a-exact (i. e. for an a-exact complex A_{•} in A, A_{•} is also

a-exact). Then for any secondary chain complex (A, d, δ) in B, a-exactness of its image (A, [d]) in A implies bexactness of (A, d, δ).

If moreover (A, d, δ) is bounded below, then conversely its b-exactness implies a-exactness of (A, [d]).

Proof. Unraveling definitions, we have that for any a_{n} : X → A_{n} with X in b and for any track α_{n−1} : d_{n−1}a_{n} ⇒ 0 there exists a_{n+1} : X → A_{n+1} and a track α_{n} : a_{n} ⇒ d_{n}a_{n+1}. From this, we have then to deduce that for (a_{n}, α_{n−1}) as above with the additional property d_{n−2}α_{n−1} = δ_{n−2}a_{n} one can actually find (ã_{n+1}, α̃_{n}) as above with the additional property α_{n−1} = δ_{n−1} ã_{n+1} d_{n−1} α̃_{n}.

Now suppose given a_{n} : X → A_{n}, α_{n−1} : d_{n−1}a_{n} ⇒ 0 with d_{n−2}α_{n−1} = δ_{n−2}a_{n}, and any a_{n+1} : X → A_{n+1}, α_{n} : a_{n} ⇒ d_{n}a_{n+1}. Consider then the element ω_{n−1} ∈ Aut(d_{n−1}a_{n}) given by the composite

\xymatrix{d_{n−1}a_{n} \ar@{=>}[r]^{d_{n−1}α_{n}} & d_{n−1}d_{n}a_{n+1} \ar@{=>}[r]^{δ_{n−1}a_{n+1}} & 0a_{n+1} = 0 \ar@{=>}[r]^{α_{n−1}^{}} & d_{n−1}a_{n}.}

For this element one has d_{n−2}ω_{n−1} = 0. Indeed, this equality is equivalent to the equality

d_{n−2}δ_{n−1}a_{n+1} d_{n−2}d_{n−1}α_{n} = d_{n−2}α_{n−1}

of tracks Aut(0_{X,A_{n−2}}). But d_{n−2}α_{n−1} = δ_{n−2}a_{n}. Moreover by naturality there is a commutative diagram

[figure: d_{n−2}d_{n−1}a_{n}; d_{n−2}d_{n−1}α_{n+1}; δ_{n−2}a_{n}; d_{n−2}d_{n−1}d_{n}a_{n+1}; δ_{n−2}d_{n}a_{n+1}; 0a_{n}; 0α_{n+1}; 0d_{n}a_{n+1}; 0]

showing that δ_{n−2}a_{n} = δ_{n−2}d_{n}a_{n+1} d_{n−2}d_{n−1}α_{n+1}. It thus follows that d_{n−2}ω_{n−1} = 0 iff one has

d_{n−2}δ_{n−1}a_{n+1} d_{n−2}d_{n−1}α_{n+1} = δ_{n−2}d_{n}a_{n+1} d_{n−2}d_{n−1}α_{n+1},

which is clear since (A, d, δ) is a secondary complex.

Now if a) is satisfied, then there is a commutative diagram

\xymatrix{[ X, A_{n}] \ar[r]^{[d_{n−1}]} \ar[d]_{} & [ X, A_{n−1}] \ar[r]^{[d_{n−2}]} \ar[d]_{} & [ X, A_{n−2}] \ar[d]_{} \\ Aut(a_{n}) \ar[r]^{d_{n−1}} & Aut(d_{n−1}a_{n}) \ar[r]^{d_{n−2}} & Aut(d_{n−2}d_{n−1}a_{n})}

Similarly if b) holds, then one has the diagram

\xymatrix{[X, A_{n}] \ar[r]^{[d_{n−1}]} \ar[d]_{} & [X, A_{n−1}] \ar[r]^{[d_{n−2}]} \ar[d]_{} & [X, A_{n−2}] \ar[d]_{} \\ Aut(a_{n}) \ar[r]^{d_{n−1}} & Aut(d_{n−1}a_{n}) \ar[r]^{d_{n−2}} & Aut(d_{n−2}d_{n−1}a_{n}).}

In both cases, it follows that there exists ω_{n} ∈ Aut(a_{n}) such that ω_{n−1} = d_{n−1}ω_{n}.

Let us then choose

ã_{n+1} = a_{n+1},
α̃_{n} = α_{n} ωn^{}.

Then αn−1^{}δ_{n−1}a_{n+1} d_{n−1} α̃_{n} = ω_{n−1} d_{n−1}ωn^{}is the identity track of d_{n−1}a_{n}, i. e.

α_{n−1} = δ_{n−1}a_{n+1} d_{n−1} α̃_{n},

as desired.

For the converse, by boundedness we can assume by induction that (A, [d]) is exact in all degrees < n. Let us then consider any a-cycle [c] ∈ [X, A_{n}] in (A, [d]), choose a representative map c : X → A_{n} and a track γ : 0 ⇒ d_{n−1}c and consider the composite track ω = δ_{n−2}c d_{n−2}γ in Aut(0_{X,A_{n−2}}).

The track d_{n−3}ω is the identity track 0 of 0_{X,A_{n−3}}. Indeed d_{n−3}δ_{n−2} = δ_{n−3}d_{n−1} by definition of a secondary chain complex, so d_{n−3}ω = δ_{n−3}d_{n−1}c d_{n−3}d_{n−2}γ. Then by (2.1) for δ_{n−3} : d_{n−3}d_{n−2} ⇒ 0_{A_{n−1},A_{n−3}} and γ : 0_{X,A_{n−1}} ⇒ d_{n−1}c one has δ_{n−3}d_{n−1}c d_{n−3}d_{n−2}γ = 0_{A_{n−1},A_{n−3}}γ δ_{n−3}0_{X,A_{n−1}} and by (2.2) both of the constituents in the last composition are identity tracks.

Now by induction hypothesis (A, [d]) is a-exact in degree n − 2, hence if a), resp. b) holds, then the diagram

\xymatrix{[ X, A_{n−1}] \ar[r]^{[d_{n−2}]} \ar[d]_{} & [ X, A_{n−2}] \ar[r]^{[d_{n−3}]} \ar[d]_{} & [ X, A_{n−3}] \ar[d]_{} \\ Aut(0_{X,A_{n−1}}) \ar[r]^{d_{n−2}} & Aut(0_{X,A_{n−2}}) \ar[r]^{d_{n−3}} & Aut(0_{X,A_{n−3}}),}

resp.

\xymatrix{[X, A_{n−1}] \ar[r]^{[d_{n−2}]} \ar[d]_{} & [X, A_{n−2}] \ar[r]^{[d_{n−3}]} \ar[d]_{} & [X, A_{n−3}] \ar[d]_{} \\ Aut(0_{X,A_{n−1}}) \ar[r]^{d_{n−2}} & Aut(0_{X,A_{n−2}}) \ar[r]^{d_{n−3}} & Aut(0_{X,A_{n−3}}).}

shows that there exists α ∈ Aut(0_{X,A_{n−1}}) such that ω = d_{n−2}α. Then for γ̃ = γ α^{}one has δ_{n−2}c d_{n−2} γ̃ = δ_{n−2}c d_{n−2}γ d_{n−2}α = ω ω^{}= 0, so that (c, γ̃) is a secondary cycle. Then by secondary b-exactness of (A, d, δ) there is a b : X → A_{n+1} and β : c ⇒ d_{n}b, so [c] is the boundary of [b] in [X, (A, [d])]. Thus (A, [d]) is exact in degree n and we are done.

4. The secondary Ext

In this section we deduce from a secondary resolution a differential defined on “primary” derived functors as studied in section 1. This differential is the analogue of the d_{2}-differential in a spectral sequence. We use the secondary differential to define certain “secondary” derived functors.

Let B be an additive track category with the additive homotopy category A = B_{≃}. Let us furthermore fix a full additive subcategory a in A; it determines the full track subcategory b of B on the same objects. It is clear that if b satisfies the conditions of 2.14, then a will satisfy those of 1.3. We can then consider the a-derived functors in A. In particular, the Ext groups Ext_{a}^{n}(X, Y) are defined for any objects X, Y in B. Moreover if B is -, resp. -additive, then derived functors of the functor D(X, Y) = Aut(0_{X,Y}) are given by

D_{a}^{n}(X, Y) Ext_{a}^{n}( X, Y),

resp.

D_{a}^{n}(X, Y) Ext_{a}^{n}(X, Y).

We will use these isomorphisms to introduce the graded Ext groups Ext_{a}^{n}(X, Y)^{m} = Ext_{a}^{n}( ^{m}X, Y), resp. Ext_{a}^{n}(X, Y)^{m} = Ext_{a}^{n}(X, ^{m}Y). Evidently if B is both - and -additive, these groups coincide.

We will from now on assume in what follows that for the pair (B, a) one of the conditions in 3.5 is satisfied, i. e. either B is -additive and a is closed under or B is o -additive and preserves a-exactness of chain complexes in B_{≃}; moreover in the latter case we also assume that a is closed under .

We are going to define the secondary differential

d_{(2)} = d_{(2)}^{n,m}: Ext_{a}^{n}(X, Y)^{m} → Ext_{a}^{n+2}(X, Y)^{m+1}.

Replacing, if needed, X by ^{m}X (resp. Y by ^{m}Y) we might clearly assume m = 0 here. Moreover by 2.14 we may suppose that a b-exact b-resolution (X_{•}, d_{•}, δ_{•}) of X is given. Then by 3.5 it determines an a-exact a-resolution (X_{•}, [d_{•}]) of X in A. Hence an element of Ext_{a}^{n}(X, Y) gets represented by an n-dimensional cocycle in that resolution, i. e. by a homotopy class [c] : X_{n} → Y with [c][d_{n}] = 0. Thus we may choose a map c ∈ [c] and a track γ : 0 ⇒ cd_{n} in B, as in the diagram below:

(4.1)  [figure: X_{n+3}; d_{n+2}; 0; δ_{n+1}; X_{n+2}; d_{n+1}; 0; δ_{n}; X_{n+1}; d_{n}; 0; γ; X_{n}; c; Y]

Then the composite track cδ_{n} γd_{n+1} ∈ Aut(0_{X_{n+2},Y}) determines an element Γ = Γ_{c,γ} in the group Aut(0_{X_{n+2},Y}). One then has Γd_{n+2} = 0. Indeed

Γd_{n+2} = (cδ_{n} γd_{n+1})d_{n+2}
= cδ_{n}d_{n+2} γd_{n+1}d_{n+2}
= cd_{n}δ_{n+1} γd_{n+1}d_{n+2}
= γ0 0δ_{n+1}
= 0.

Thus Γ determines an (n + 2)-cocycle in Aut(0_{(X_{•},[d_{•}]),Y}) [(X_{•}, [d_{•}]), Y]^{1}. We then have

(4.2) Theorem. The above construction does not depend on the choice of c, γ and the resolution, up to coboundaries in [(X_{•}, [d_{•}]), Y]^{1}; hence the assignment [c] ↦ [Γ_{c,γ}] gives a well-defined homomorphism

d_{(2)}^{n,m}: Ext_{a}^{n}(X, Y)^{m} → Ext_{a}^{n+2}(X, Y)^{m+1}.

(4.3) Remark. Of course the above homomorphism depends on the additive track category B in which we define the secondary resolution. In fact, d_{(2)} depends only on the track subcategory b {X, Y}⊂ B obtained by adding to b the objects X and Y and all morphisms and tracks from Z, X , Z, Y for all objects Z from b. We shall see in section 5 below that additive track categories B, B′ with subcategories b, b′ such that the track categories b {X, Y}and b′{X, Y}are track equivalent yield the same differential d_{(2)}.

If the composites d_{(2)}^{n,m}d_{(2)}^{n−2,m−1} are all zero (as this is the case for examples derived from spectral sequences), we define the secondary Ext groups

(4.4)  Ext_{b}^{n}(X, Y)^{m} := ker(d_{(2)}^{n,m})/ im(d_{(2)}^{n−2,m−1}).

This then will be, in examples, the E_{3}-term of a spectral sequence. We point out that the secondary Extgroups are well defined and do not depend on the choice of the secondary resolution. We shall use the secondary Ext-groups for the computation of the E_{3}-term in the Adams spectral sequence, see [4].

Proof. We will first show that the cocycles corresponding to (c, γ) and (c, γ′) for γ, γ′ : 0 ⇒ cd_{n} are cohomologous. Indeed the first one is cδ_{n} γd_{n+1} and the second is

\begin{aligned}cδ_{n} γ′d_{n+1} & = cδ_{n} γd_{n+1} γ^{}d_{n+1} γ′d_{n+1} \\ & = cδ_{n} γd_{n+1} (γ^{}γ′)d_{n+1}, \end{aligned}

so these cocycles indeed differ by the coboundary of γ^{}γ′. Thus we obtain a map d_{(2)} from the group of n-cocycles of Hom_{A}((X_{•}, [d_{•}]), Y) to H^{n+2}(Aut(0_{(X_{•},[d_{•}]),Y})) Ext_{a}^{n+2}(X, Y)^{1}.

Next let us show that the map we just constructed is actually a homomorphism.

To see this, let us choose maps p_{1}, ∇, p_{2} : Y ⊕ Y → Y in the homotopy classes ([1_{Y}], 0), ([1_{Y}], [1_{Y}]), (0, [1_{Y}]) ∈ [Y ⊕ Y, Y] respectively. Thus for any two maps c_{1}, c_{2} : X → Y there is a map c_{1,2} : X → Y ⊕ Y such that there exist tracks π_{i} : p_{i}c_{1,2} ⇒ c_{i}, i = 1, 2, and moreover [c_{1}] + [c_{2}] = [∇c_{1,2}]. Now suppose c_{i} represent cocycles, then [c_{1,2}][d_{n}] = ([c_{1}][d_{n}], [c_{2}][d_{n}]) = (0, 0) ∈ [X_{n+1}, Y] × [X_{n+1}, Y] ≈ [X_{n+1}, Y ⊕ Y], so there is a track γ : 0 ⇒ c_{1,2}d_{n}. Consequently the cohomology class d_{(2)}([c_{1}] + [c_{2}]) = d_{(2)}([∇c_{1,2}]) can be represented by the cocycle

∇c_{1,2}δ_{n} ∇γd_{n+1} = ∇(c_{1,2}δ_{n} γd_{n+1}).

On the other hand d_{(2)}([f_{i}]), i = 1, 2, can be represented by

c_{i}δ_{n} π_{i}d_{n}d_{n+1} p_{i}γd_{n+1} = π_{i}0 p_{i}c_{1,2}δ_{n} p_{i}γd_{n+1}
= p_{i}c_{1,2}δ_{n} p_{i}γd_{n+1}
= p_{i}(c_{1,2}δ_{n} γd_{n+1})

(see the diagram below).

[figure: X_{n+2}; d_{n+1}; 0; δ_{n}; X_{n+1}; d_{n}; 0; γ; 0; X_{n}; c_{1,2}; c_{i}; π_{i}; Y ⊕ Y; p_{i}; ∇; Y; Y]

But by assumption Aut(0) is biadditive, which in particular means that the map

(p_{1} , p_{2} ) : Aut(0_{X,Y⊕Y}) → Aut(0_{X,Y}) × Aut(0_{X,Y})

is an isomorphism, and moreover addition in Aut(0_{X,Y}) is given by the composite of the left action ∇ with the inverse of that isomorphism. This obviously means d_{(2)}([c_{1}] + [c_{2}]) = d_{(2)}([c_{1}]) + d_{(2)}([c_{2}]).

It follows that in order to show that d_{(2)} factors through a homomorphism from the group Ext_{a}^{n}(X, Y) = H^{n}([(X_{•}, [d_{•}]), Y]) it suffices to show that d_{(2)} vanishes on coboundaries, i. e. on cocycles of the form [c] = [ad_{n−1}], for some map a : X_{n−1} → Y. But for such a cocycle we may choose the track γ : 0 ⇒ ad_{n−1}d_{n} to be aδn−1^{}, and then the value of d_{(2)} on it will be represented by the cocycle ad_{n−1}δ_{n} aδ_{n−1}d_{n} = 0 — see the diagram.

[figure: X_{n+2}; d_{n+1}; 0; δ_{n}; X_{n+1}; d_{n}; 0; δ_{n−1}^{}; 0; X_{n}; d_{n−1}; X_{n−1}; a; =; Y]

Finally we must show that d_{(2)} does not depend on the choice of the secondary resolution. Indeed consider any two b-exact b-resolutions (X_{•}, d_{•}, δ_{•}) and (X′_{•}, d′_{•}, δ′_{•}) of X. By 2.15 there is a secondary chain map (f, ϕ) between them over X. Obviously then [f] determines a chain map between (X_{•}, [d_{•}]) and (X′_{•}, [d′_{•}]) inducing isomorphisms f^{∗} on cohomology of the cochain complexes obtained by applying [ , Y] and Aut(0_{,Y}). We must then show that the diagrams

\xymatrix{H^{n}([(X_{•}, [d_{•}]), Y]) \ar[r]^{d_{(2)}} \ar[d]_{f^{∗}} & H^{n}(Aut(0_{(X_{•},[d_{•}]),Y})) \ar[d]^{f^{∗}} \\ H^{n}([(X′_{•}, [d′_{•}]), Y]) \ar[r]^{d_{(2)}} & H^{n}(Aut(0_{(X′_{•},[d′_{•}]),Y}))}

commute. This can be seen from the diagram

[figure: X′_{n+2}; d′_{n+1}; 0; δ′_{n}; f_{n+2}; X′_{n+1}; d′_{n}; f_{n+1}; ⇓ϕ_{n+1}; X′_{n}; f_{n}; ⇓ϕ_{n}; X_{n+2}; d_{n+1}; 0; δ_{n}; X_{n+1}; d_{n}; 0; \underrightarrow{γ}; X_{n}; c; Y;]

in more detail, one observes the track diagram

[figure: cf_{n}0; cf_{n}δ′_{n}; cf_{n}d′_{n}d′_{n+1}; cϕ_{n}^{}d′_{n+1}; cd_{n}f_{n+1}d′_{n+1}; γf_{n+1}d′_{n+1}; cd_{n}ϕ_{n+1}; 0f_{n+1}d′_{n+1}; 0ϕ_{n+1}; 0; 0; c0f_{n+2}; cδ_{n}f_{n+2}; cd_{n}d_{n+1}f_{n+2}; γd_{n+1}f_{n+2}; 0d_{n+1}f_{n+2}]

whose left part commutes by 2.9 and the right part by naturality. Now the lower composition of this diagram is

(cδ_{n} γd_{n+1})f_{n+2} = f^{∗}d_{(2)}([c]),

whereas the upper one is

(cf_{n})δ′_{n} (cϕn^{}γf_{n+1})d′_{n+1},

which represents d_{(2)}(f^{∗}([c])), since we might choose for γ′ : 0 ⇒ f^{∗}(c)d′_{n} the track cϕn^{}γf_{n+1} : 0 ⇒ cf_{n}d′_{n} = f^{∗}(c)d′_{n}.

5. Invariance of the secondary differential in the equivalence class of the track extension

In this section we will prove that the secondary differential

d_{(2)}^{n,m}: Ext_{a}^{n}(X, Y)^{m} → Ext_{a}^{n+2}(X, Y)^{m+1}

constructed from an additive track category B depends only on the class

⟨B⟩∈ H^{3}(A; D).

More precisely one has

(5.1) Theorem. Suppose given additive track categories B and B′ with B_{≃} = A = B′_{≃}. Then for any additive subcategory a ⊂ A, the secondary differentials d_{(2)}^{n,m}constructed from B and B′ coincide provided

there is a strict equivalence of track subcategories b {X, Y}⟶^{∼} b′{X, Y} .

Proof. Recall the construction of

d_{(2)}^{n,m}: Ext_{a}^{n}(X, Y)^{m} → Ext_{a}^{n+2}(X, Y)^{m+1}.

Let b ⊂ B, b′ ⊂ B′ be the full track subcategories in B, resp. B′, on objects from a. Then F(b) ⊂ b′. One starts from a b-exact b-resolution (X_{•}, d_{•}, δ_{•}) of X in B; according to 4.2, the resulting d_{(2)} does not depend on the choice of such a resolution. Suppose now given an element in Ext_{a}^{n}(X, Y) represented by a Y-valued n-cocycle [c] ∈ [X_{n}, Y] in the a-exact a-resolution (X_{•}, [d_{•}]) of X in A. By our construction, value on this element of the d_{(2)} corresponding to B is obtained by choosing a representative [c] ∋ c : X_{n} → Y and a track γ : 0 ⇒ cd_{n} in B, as in 4.1. One then has

d_{(2)}([c]) = [cδ_{n} γd_{n+1}].

But it is clear that F(X_{•}, d_{•}, δ_{•}) is a b′-exact b′-resolution of X in B′. We then might choose F(c) and F(γ) for the corresponding data in B′, which would give us the element of Aut_{B′}(0_{X_{n+2},Y}) equal to F(c)F(δ_{n}) F(γ)F(d_{n−1}) = F(cδ_{n} γd_{n_{1}}). Since by assumption F induces identity on Aut(0), theorem follows.

6. Resolutions of the Adams type

Let B be a track category with a strict zero object and homotopy category B_{≃} = A.

(6.1) Definition. For an object X of A, an X-coaugmented sequence ℛ is a diagram in A of the form

ℛ : \xymatrix{... & Y_{n+1} \ar[l] & A_{n} \ar[l]_{p̄_{n}} & Y_{n} \ar[l]_{ī_{n}} & ... \ar[l] & Y_{2} \ar[l] & A_{1} \ar[l]_{p̄_{1}} & Y_{1} \ar[l]_{ī_{1}} & A_{0} \ar[l]_{p̄_{0}} & Y_{0} = X \ar[l]_{ī_{0}}}

satisfying

p̄_{n}ī_{n} = 0

in A for all n = 0, 1, 2, .... The associated X-coaugmented cochain complex of such a sequence is then defined to be

C_{A}(ℛ) :  \xymatrix{... & A_{n} \ar[l]_{^{ī_{n+1} p̄_{n}}} & ... \ar[l]_{^{ī_{n}p̄_{n−1}}} & A_{1} \ar[l]_{^{ī_{2}p̄_{1}}} & A_{0} \ar[l]_{^{ī_{1}p̄_{0}}} & X. \ar[l]_{ī_{0}}}

For an additive subcategory a ⊂ A = B_{≃}, an X-coaugmented sequence ℛ as above will be called an a-sequence if A_{n} belongs to a for all n. Moreover it will be called a-exact if for any object A from a, the induced sequence

Hom_{A}(Y_{n+1}, A) → Hom_{A}(A_{n}, A) → Hom_{A}(Y_{n}, A)

is a short exact sequence of abelian groups for all n 0. Thus in this case, the chain complex C_{A}(ℛ) is aexact in the sense of 1.1. In fact, for any object A in a the differential d̄_{n}: Hom_{A}(A_{n+1}, A) → Hom_{A}(A_{n}, A) in Hom_{A}(C_{A}(ℛ), A) is then Hom_{A}(ī_{n+1} p̄_{n}, A), and one has

ker(d̄_{n−1}) = im(d̄_{n}) = Hom_{A}(Y_{n+1}, A)

for all n and all A ∈ a.

(6.2) Proposition. For each X-coaugmented sequence ℛ in A, any choice of representatives i_{n} ∈ ī_{n}, p_{n} ∈ p̄_{n} in B_{0} and of tracks α_{n} : p_{n}i_{n} ⇒ 0_{Y_{n},Y_{n+1}}determines an X-coaugmented secondary chain complex in B of the form

C_{B}(ℛ) :  [figure: ⋯; A_{3}; i_{4}p_{3}; A_{2}; i_{3}p_{2}; 0; i_{4}α_{3}p_{2}; A_{1}; i_{2}p_{1}; 0; i_{3}α_{2}p_{1}; A_{0}; i_{1}p_{0}; 0; i_{2}α_{1}p_{0}; X.; i_{0}; 0; i_{1}α_{0}]

Proof. Consider the diagram

[figure: ⋯; Y_{5}; i_{5}; A_{4}; p_{4}; Y_{4}; i_{4}; 0; α_{4}; A_{3}; p_{3}; 0; Y_{3}; i_{3}; 0; α_{3}; A_{2}; p_{2}; 0; Y_{2}; i_{2}; 0; α_{2}; A_{1}; p_{1}; 0; Y_{1}; i_{1}; 0; α_{1}; A_{0}; p_{0}; 0; X.; i_{0}; 0; α_{0}; 0]

That this diagram yields on C_{B}(ℛ) above the structure of a secondary chain complex, is equivalent to the identities

i_{n+1}p_{n}i_{n}α_{n−1}p_{n−2} = i_{n+1}α_{n}p_{n−1}i_{n−1}p_{n−2}.

These are satisfied since one actually has

p_{n}i_{n}α_{n−1} = α_{n}p_{n−1}i_{n−1},

as the next lemma shows.

(6.3) Lemma. For any maps f : X → Y, f′: Y → Z and tracks α : f ⇒ 0_{X,Y}, α′: f′ ⇒ 0_{Y,Z}one has

f′α = α′f.

Proof. This is a particular case of 2.1.

(6.4) Remark. Strictly speaking, C_{B}(ℛ) depends on the choice of the i_{n}, p_{n} and α_{n}; however it will be harmless in what follows to suppress these from the notation.

(6.5) Example. Let B be a track category and suppose that A = B_{≃} is equipped with the structure of a triangulated category. Thus there is an endofunctor : A → A which is a self-equivalence, with an inverse equivalence ^{−1}, and one has a distinguished class of diagrams of the form

A ← B ← C ← A,

called exact triangles, which satisfy certain axioms. A fiber tower 𝒯 over an object X is a diagram in A

(6.6)  \xymatrix{X & X_{0} \ar[d] & X_{1} \ar[l] \ar[d] & X_{2} \ar[l] \ar[d] & ... \ar[l] \\  & A_{0} & A_{1} & A_{2} & ⋯}

such that each A_{n} ← X_{n} ← X_{n+1} ← A_{n} is an exact triangle in A. In particular, the composites ^{−1}X_{n+1} ← A_{n} ← X_{n} are zero maps in A. For an additive subcategory a in A, call a fiber tower 𝒯 a-exact if A_{i} ∈ a for all i and moreover each of its exact triangles induces a short exact sequence

0 → Hom( ^{−1}X_{n+1}, A) → Hom(A_{n}, A) → Hom(X_{n}, A) → 0

for all A ∈ a.

A fiber tower yields a system of coaugmented sequences in A of the form

(6.7)  \xymatrix{ &  &  &  & A_{0} & X \ar[l] \\  &  & A_{1} & X_{1} \ar[l] & A_{0} \ar[l] & X \ar[l] \\ A_{2} & X_{2} \ar[l] & A_{1} \ar[l] & X_{1} \ar[l] & ^{2}A_{0} \ar[l] & ^{2}X \ar[l] \\  &  &  & ⋮}

which via delooping in A yields the X-coaugmented sequence

ℛ(𝒯 ) : ... ← ^{−2}A_{2} ← ^{−2}X_{2} ← ^{−1}A_{1} ← ^{−1}X_{1} ← A_{0} ← X.

Thus by 6.2 each fiber tower over X gives rise to an X-coaugmented secondary chain complex C_{B}(ℛ(𝒯 )).

(6.8) Remark. Before the authors obtained the construction from 6.2, a direct topological proof that Adams resolutions give rise to a secondary complex has been kindly provided to them by Birgit Richter [16].

One then has

(6.9) Theorem. Assume either B is -additive and A ∈ a implies A ∈ a, or B is -additive and preserves a-exactness of complexes in A (cf. 3.5). Then for any a-exact fiber tower over an object X, any X-coaugmented secondary chain complex associated to it (as in 6.5 and 6.2) is a b-coresolution of X. Hence for any object Y there is a secondary differential

d_{(2)} : Ext_{a^{op}}^{n}(X, Y)^{m} → Ext_{a^{op}}^{n+2}(X, Y)^{m+1},

where Ext_{a^{op}}^{∗}(X, Y)^{m} denotes either Ext_{a^{op}}^{∗}( ^{m}X, Y) or Ext_{a^{op}}^{∗}(X, ^{m}Y) in A^{op}. The differential d_{(2)} is welldefined by the cohomology class ⟨B⟩∈ H^{3}(A; D) with D in 3.4.

Proof. This follows directly from 3.5.

7. The E_{3} term of the Adams spectral sequence

As in example 1.8 let A be the stable homotopy category of spectra and let a ⊂ A be the full subcategory of finite products of Eilenberg-Mac Lane spectra over a fixed prime field _{p}. Let X be a spectrum of finite type, that is, for which the cohomology groups H^{i}(X; _{p}) are finite dimensional _{p}-vector spaces for all i. Then the Adams fiber tower of X is given by

(7.1)  [figure: X; X_{0}; X_{1}; X_{2}; ...; H ∧ X_{0}; H ∧ X_{1}; H ∧ X_{2}; · · ·.]

Here H = H _{p} is the Eilenberg-Mac Lane spectrum, the map X_{i} → H ∧ X_{i} is given by smashing S^{0} → H with X_{i}, and X_{i+1} is the fiber of this map. Since X is of finite type all spectra H ∧ X_{i} can be considered to be objects of a. By construction the Adams fiber tower is a-exact.

Since the category of spectra is a Quillen model category we know that A is the homotopy category of all spectra which are fibrant and cofibrant. Using the cylinder of such spectra we obtain the additive track category B. That is, B consists of spectra which are fibrant and cofibrant, of maps between such spectra, and tracks between such maps. Then B is -additive (and also Ω-additive) and A ∈ a implies A ∈ a. Therefore we can apply theorem 6.9 to the Adams fiber tower where b is the full track subcategory of B on objects from a. Hence we get for a spectrum Y the following diagram whose top row is defined by any secondary b-coresolution of X and the bottom row is the differential d_{(2)} in the Adams spectral sequence.

(7.2)  \xymatrix{Ext_{a^{op}}^{n}(X, Y)^{m} \ar[r]^{d_{(2)}} \ar[d]^{} & Ext_{a^{op}}^{n+2}(X, Y)^{m+1} \ar[d]^{} \\ Ext_{𝒜}^{n} (H^{∗}X, H^{∗}Y)^{m} \ar[r]^{d_{(2)}} & Ext_{𝒜}^{n+2}(H^{∗}X, H^{∗}Y)^{m+1}.}

The vertical isomorphisms in this diagram are defined in example 1.8.

(7.3) Theorem. The diagram 7.2 commutes.

This shows that d_{(2)}d_{(2)} = 0 so that the secondary Ext in section 4 coincides with the E_{3} term of the Adams spectral sequence. In the book [2] a pair algebra ℬ is computed which can be used to describe algebraic models for secondary b-coresolutions. This, in fact, yields an algorithm computing the d_{(2)} differential in the Adams spectral sequence since we can use theorem 7.3.

Proof. In our terms the second differential of the Adams spectral sequence can be understood in the following way: one is given a fiber tower 𝒯 like 6.6 or 7.1 over an object X in the stable homotopy category, with the associated X-coaugmented sequence ℛ(𝒯 ) as in 6.5. To it corresponds by 6.1 the associated X-coaugmented cochain complex

C_{A}(ℛ(𝒯 )) :  \xymatrix{... & Ω^{−n−1} A_{n+1} \ar[l] & Ω^{−n}A_{n} \ar[l]_{Ω^{−n}d^{n}} & ... \ar[l] & Ω^{−1}A_{1} \ar[l]_{Ω^{−1}d^{1}} & A_{0} \ar[l]_{d^{0}} & X \ar[l]}

where d^{n} : A_{n} → Ω^{−1}A_{n+1} are the composites A_{n} → Ω^{−1}X_{n+1} → Ω^{−1}A_{n+1} of maps in the exact triangles X_{n+1} → X_{n} → A_{n} → Ω^{−1}X_{n+1} and A_{n+1} → Ω^{−1}X_{n+2} → Ω^{−1}X_{n+1} → Ω^{−1}A_{n+1}. Here all A_{n} are _{p}- module spectra, i. e. Eilenberg-MacLane spectra of _{p}-vector spaces, and moreover the sequences X_{n} → A_{n} → Ω^{−1}X_{n+1} are _{p}-exact, i. e. applying H^{∗}( ; _{p}) to them yields short exact sequences. In particular, H^{∗}(C_{A}(ℛ(𝒯 )); _{p}) is an 𝒜 -projective resolution of H^{∗}(X; _{p}).

Now choose new spectra B_{n} fitting in exact triangles

\xymatrix{B_{n} \ar[r]^{i^{n}} & A_{n} \ar[r]^{d^{n}} & Ω^{−1}A_{n+1} \ar[r] & Ω^{−1}B_{n}}

and observe that by the octahedron axiom there is a commutative diagram of (co)fibre sequences of the form

(7.4) \xymatrix{X_{n+2} \ar[rr] \ar[dr] &  & X_{n} \ar[rr] \ar[dr] &  & A_{n} \ar[rr] \ar[dr] &  & Ω^{−1}A_{n+1} \ar[rr] \ar[dr] &  & ⋯ \\  & X_{n+1} \ar[ur] \ar[dr] &  & B_{n} \ar[ur] \ar[dr] &  & Ω^{−1}X_{n+1} \ar[ur] \ar[dr] &  & Ω^{−1}B_{n} \ar[ur] \ar[dr] & ⋯ \\  &  & A_{n+1} \ar[rr] \ar[ur] &  & Ω^{−1}X_{n+2} \ar[rr] \ar[ur] &  & Ω^{−1}X_{n} \ar[rr] \ar[ur] &  & ⋯}

so that in particular the original fiber tower 𝒯 “doubles” to give two new fiber towers 𝒯 ^{(2)} starting at X_{0}, resp. X_{1}, of the form

\xymatrix{... & X_{n} \ar[d] \ar[l] & X_{n+2} \ar[l] \ar[d] & X_{n+4} \ar[l] \ar[d] & ... \ar[l] \\ ⋯ & B_{n} & B_{n+2} & B_{n+4} & · · ·.}

The associated sequences ℛ(𝒯 ^{(2)}) and the cochain complexes

C_{A}(ℛ(𝒯 ^{(2)})) :  \xymatrix{... & Ω^{−n−2} B_{n+4} \ar[l] & Ω^{−n−1} B_{n+2} \ar[l] & Ω^{−n}B_{n} \ar[l]_{Ω^{−n}d(2)^{n}} & ..., \ar[l]}

where d(2)^{n} : B_{n} → Ω^{−1}B_{n+2} is the composite B_{n} → Ω^{−1}X_{n+2} → Ω^{−1}B_{n+2}, are then obtained as in 6.5.

Let us now take any spectrum Y and apply the stable homotopy classes functor {Y, }to the whole business. Because of the exact triangles B_{n} → A_{n} → Ω^{−1}A_{n+1} → Ω^{−1}B_{n}, there are isomorphisms

im({Y, B_{n}}→ {Y, A_{n}} ) ker({Y, A_{n}}→ {Y, Ω^{−1}A_{n+1}}) .

On the other hand it is known (see e. g. [11]) that the canonical maps

(7.5)  {Y, A_{n}}⟶^{}Hom_{𝒜} (H^{∗}(A_{n}; _{p}), H^{∗}(Y; _{p}))

are isomorphisms; it thus follows that the groups

E_{2}^{s,t}(Y, X) = im({Y, Ω^{t−s}B_{s}}⟶^{{Y,Ω^{t−s}i_{s}}} {Y, Ω^{t−s}A_{s}}) im({Y, Ω^{t−s+1}A_{s−1}}⟶^{{Y,Ω^{t−s+1} d^{s−1}}} {Y, Ω^{t−s}A_{s}})

are isomorphic to Ext_{𝒜}^{s} (H^{∗}(X; _{p}), H^{∗}(Y; _{p}))^{t}.

Moreover (see again [11]) the Adams differential E_{2}^{s,t} → E_{2}^{s+2,t+1} is induced by the map

im({Y, Ω^{t−s}B_{s}}→ {Y, Ω^{t−s}A_{s}}) → im({Y, Ω^{t−s−1}B_{s+2}}→ {Y, Ω^{t−s−1}A_{s+2}})

which sends the class of a stable map

Y → Ω^{t−s}B_{s} → Ω^{t−s}A_{s}

to the class of the composite

Y → Ω^{t−s}B_{s} ⟶^{Ω^{t−s}d(2)^{s}} Ω^{t−s−1}B_{s+2} → Ω^{t−s−1}A_{s+2}

or, which by 7.4 is the same, the composite

Y → Ω^{t−s}B_{s} → Ω^{t−s−1}X_{s+2} → Ω^{t−s−1}A_{s+2}.

To see then that the differential so defined coincides with the secondary differential as constructed in 4.2, 4 and 6.9, let us choose zero tracks α_{n} for the composites X_{n} → A_{n} → Ω^{−1}X_{n+1} and switch from C_{A}(ℛ(𝒯 )) to the X-coaugmented secondary cochain complex C_{B}(ℛ(𝒯 )) as defined in 6.2. Then according to 4.2, given an element ⟨c⟩of Ext_{𝒜}^{s} (H^{∗}(X; _{p}), H^{∗}(Y; _{p})))^{t}, the corresponding element d_{(2)} ⟨c⟩∈ Ext_{𝒜}^{s+2}(H^{∗}(X; _{p}), H^{∗}(Y; _{p}))^{t+1} is constructed in the following way. First represent ⟨c⟩ by a cocycle in C_{A}(ℛ(𝒯 )), i. e. by a homomorphism of 𝒜 -modules [c] : H^{∗}(Ω^{t−s}A_{s}; _{p}) → H^{∗}(Y; _{p}) with [c] ∘ H^{∗}(Ω^{t−s}d^{s}; _{p}) = 0. By 7.5, this homomorphism is in turn induced by a map c : Y → Ω^{t−s}A_{s} such that d^{s} ∘ c is nullhomotopic. Choosing a homotopy γ : 0 ⇒ d^{s} ∘ c, according to 4.2 the class d_{(2)} ⟨c⟩is represented by the map Y → Ω^{t−s−1}A_{s+2} = ΩΩ^{t−s−2}A_{s+2} which corresponds to the composite homotopy

\xymatrix{0 = Ω^{t−s−1}d^{s+1} ∘ 0 \ar@{=>}[r]^{Ω^{t−s−1} d^{s+1} γ} & Ω^{t−s−1}d^{s+1} ∘ Ω^{t−s}d^{s} ∘ c \ar@{=>}[r]^{δc} & 0c = 0}

from the zero map Y → Ω^{t−s−2}A_{s+2} to itself, as in

[figure: Y; c; 0; γ; Ω^{t−s}A_{s}; Ω^{t−s}d^{s}; 0; δ; Ω^{t−s−1}A_{s+1}; Ω^{t−s−1} d^{s+1}; Ω^{t−s−2}A_{s+2}.]

Now according to the construction of C_{B}(ℛ(𝒯 )) given in 6.2, this diagram reduces to the following diagram

(7.6)  [figure: Y; c; 0; γ; Ω^{t−s}A_{s}; Ω^{t−s−1}X_{s+1}; 0; α; Ω^{t−s−1}A_{s+1}; Ω^{t−s−2}X_{s+2}; Ω^{t−s−2}A_{s+2}.]

Next note that because of the fibre sequence

Ω^{t−s}B_{s} → Ω^{t−s}A_{s} → Ω^{t−s−1}A_{s+1} → Ω^{t−s−1}B_{s},

choosing γ : 0 ⇒ Ω^{t−s}d^{s} ∘ c is equivalent to choosing a lift of c to a map Y → Ω^{t−s}B_{s}. Similar correspondences between homotopies and liftings of maps take place further along the sequence, as can be summarized in the following diagram

[figure: Y; Ω^{t−s}B_{s}; Ω^{t−s−1}X_{s+2}; Ω^{t−s−1}X_{s+2}; Ω^{t−s−1}A_{s+2}; Y; γ; Ω^{t−s}A_{s}; Ω^{t−s−1}X_{s+1}; α; ∗; ∗; ∗; Ω^{t−s−1}A_{s+1}; Ω^{t−s−1}A_{s+1}; Ω^{t−s−2}X_{s+2}; Ω^{t−s−2}A_{s+2},]

in which the columns form fiber sequences and the upper horizontal maps are liftings corresponding to the homotopies indicated in lower squares. That the resulting upper horizontal composite is indeed the lifting corresponding to the composite homotopy in 7.6 now follows from the following standard homotopytheoretic lemma which can be found e. g. in [1, (2.9) on p. 263]:

(7.7) Lemma. Given a diagram

[figure: F; F′; F′′; E; e; γ; E′; δ; E′′; B; B′; b; B′′]

whose columns are fiber sequences and upper horizontal maps are liftings corresponding to the indicated homotopies, then the composite F → F′′is the lifting corresponding to the composite homotopy δe bγ.

Acknowledgements

The authors are grateful to Birgit Richter for 6.8 and to Gerald Gaudens for discussions on several aspects of the paper.

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Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

   E-mail address: baues@mpim-bonn.mpg.de

Razmadze Mathematical Institute, M. Alexidze st. 1, Tbilisi 0193, Georgia

Current address: Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

   E-mail address: jib@rmi.acnet.ge