Theory and Applications of Categories, Vol. 9, No. 2, 2001, pp. 17–42.

A NOTE ON EXACTNESS AND STABILITY IN HOMOTOPICAL ALGEBRA

MARCO GRANDIS

ABSTRACT.

Exact sequences are a well known notion in homological algebra. We investigate here the more vague properties of “homotopical exactness”, appearing for instance in the fibre or cofibre sequence of a map. Such notions of exactness can be given for very general “categories with homotopies” having homotopy kernels and cokernels, but become more interesting under suitable “stability” hypotheses, satisfied - in particular - by chain complexes. It is then possible to measure the default of homotopical exactness of a sequence by the homotopy type of a certain object, a sort of “homotopical homology”.

Introduction

The purpose of this work is to investigate the notion of “homotopically exact” sequence in categories equipped with homotopies, pursuing a project of developing homotopical algebra as an enriched version of homological algebra [6, 7]. Well known instances of such sequences are:

(a) the cofibre sequence of a map f: A → B, or Puppe sequence [18], for topological spaces or pointed spaces

A → B → Cf → ΣA → ΣB → ΣCf → …  (1)

(Cf is the h-cokernel of f, or standard homotopy cokernel, or mapping cone; Σ denotes the suspension) where every map is, up to homotopy equivalence, an h-cokernel of the preceding one;

(b) the fibre sequence of a map of pointed spaces, which has a dual construction and properties;

(c) the fibre-cofibre sequence of a map f: A → B of chain complexes (Kf is the h-kernel)

… → ΩA → ΩB → Kf → A → B → Cf → ΣA → ΣB → …  (2)

where both the aforementioned exactness conditions are satisfied, and each three-term part is homotopy equivalent to a componentwise-split short exact sequence of complexes. The drastic simplification of exactness properties in the last example is a product of the

Work supported by MIUR Research Projects Received by the editors 2000 August 29 and, in revised form, 2001 November 28. Published on December 5 in the volume of articles from CT2000. 2000 Mathematics Subject Classification: 55U35, 18G55, 18D05, 55P05, 55R05, 55U15. Key words and phrases: Homotopy theory, abstract homotopy theory, 2-categories, cofibrations, fibre

spaces, chain complexes.

\begin{matrix}c \\ ◯ \end{matrix} Marco Grandis, 2001. Permission to copy for private use granted.

homotopical stability of chain complexes: the suspension and loop endofunctor (which in general just form an adjunction Σ ⊣ Ω), are inverse and take the sequence (2) to itself, by a three-step shift forward or backward, so that the properties of its left part reflect on the right part and vice versa. Triangulated categories abstract these facts in the notion of “exact triangle” [19, 20, 21, 11].

For the sake of simplicity, let us go on considering this simple but relevant situation: the category of chain complexes Ch_{∗}D over an additive category, even though the following notions are studied below in a much more general frame. An h-differential sequence (f, g; α) consists of two consecutive maps of chain complexes f, g together with a nullhomotopy α of their composite (represented by a dotted arc)

[figure: B; f; α; A; g; C; α: 0 ≃ gf: B → C.]  (3)

This sequence (f, g; α) will be said to be h-exact if the h-kernel of g is homotopically equivalent to the h-kernel of the h-cokernel of f, or equivalently if the dual condition is satisfied; other conditions, of left and right h-exactness, are equivalent to the previous one in the stable case (thm. 2.3). One can measure the default of exactness by the homotopy type of a chain complex H(f, g; α), called the homotopical homology of the sequence (2.5, 3.4).

The construction is a homotopical version of an obvious construction of ordinary homology (in an abelian category), as presented in the left diagram below for a differential sequence (f, g); the construction only uses kernels and cokernels, and the sequence is exact if and only if H(f, g) = 0

[figure: B; B; f; 0; B; u_{α}; B; f; α; 0; Kerg; A; g; C; Kg; kg; cu_{α}; A; g; cf; C; H(f, g); Cokf; C; H(f, g; α); kv_{α}; Cf; v_{α}; C] (4)

In a similar way, we can construct in Ch_{∗}D the right diagram above, replacing (co)kernels with standard homotopy (co)kernels: we start now from the h-differential sequence (f, g; α), construct the h-kernel Kg (with a structural nullhomotopy 0 ≃ g·kg), the h-cokernel Cf, two canonical maps u_{α}: B → Kg and v_{α}: Cf → C (coherently with all previous maps and homotopies), and, rather surprisingly, find that there is one chain complex which is at the same time the h-kernel of Cf → C and the h-cokernel of B → Kg, namely:

\begin{matrix}(H(f, g; α))_{n} = B_{n−1} ⊕ A_{n} ⊕ C_{n+1}, \\ ∂(b, a, c) = (−∂b, −fb + ∂a, −αb + ga − ∂c). \end{matrix}  (5)

Now, the sequence (f, g; α) is h-exact if and only if H(f, g; α) is contractible. Note that, in contrast with ordinary homology, all this requires the additive structure of D, instead of its kernels and cokernels; moreover, monics are replaced with fibrations and epis with cofibrations.

The abstract frame we will use for “categories with homotopies” is a notion of homotopical category, developed in previous papers and recalled in Subsections 1.1, 1.2: it is a sort of lax 2-category with suitable comma and cocomma squares. The main reference, [7], is cited as Part I (and I.7 means Section 7 therein).

The links of this setting with triangulated categories or the Freyd embedding are dealt with in [8, 10], respectively; its parallelism with homological algebra in [6]. Finally, it can be noted that the Freyd embedding of a stable homotopy category into an abelian category [3, 4, 5] provides a notion of exactness for a sequence (f, g) of maps with gf ≃ 0, which is too weak for the present purposes (1.5, 2.1).

Outline. Section 1 introduces our notions of homotopical exactness of an h-differential sequence. Then, in Section 2, various properties of stability are considered (2.1, 2.2), with their links with h-exactness of sequences (2.3) and the introduction of the “homotopical homology” (2.5). Section 3 deals with the categories of chain complexes (a stable case) and positive complexes (left h-stable); these homotopical categories have a homotopical homology, which characterises h-exactness in both cases (3.4-3.6). The last two sections are devoted to bounded complexes between fixed degrees: Ch_{0}^{p}D is an h-semistable homotopical category. In Section 4 we consider the very particular case Ch_{0}^{1}D = D^{2} of maps of D, viewed as chain complexes and equipped with chain homotopies; again, homotopical homology measures the default of h-exactness. Finally, Section 5 studies the general bounded case Ch_{0}^{p}D on an abelian basis D; here, similar results hold up to weak equivalences (the chain maps which induce isomorphism in homology): the weak homotopy type of H(f, g; α) measures the weak exactness of the sequence.

1. Exactness in homotopical categories

After a study of homotopy (co)kernels and of homotopical exactness of sequences (1.5), we prove that the fibre and cofibre sequences of a map are, respectively, left and right h-exact (1.7).

1.1. Homotopical categories. In the whole paper, A is a pointed homotopical category, as defined in I.7 (i.e., Section 7 of [7]); we just review its main aspects, also to fix the (slightly different) notation used here.

To begin with, A is a sort of lax 2-category, quite different from bicategories. It has objects, maps, and homotopies (2-cells) α: f → g: A → B, with part of the usual structure of 2-dimensional categories, plus an equivalence relation α ≃_{2} α′ between parallel homotopies α, α′: f → g, called 2-homotopy (and corresponding to the relative homotopy of paths, with fixed endpoints). The vertical composition of homotopies (or concatenation) will be written additively, α+β: f → h (for β: g → h); thus, the vertical identity (or trivial homotopy) of a map is written as 0: f → f or 0_{f}, and a reverse homotopy as −α: g → f. The vertical structure behaves categorically (“groupoidally”) up to 2-homotopy. There is no assigned horizontal composition of cells, but only a whisker composition v·α·u for homotopies and maps, with reduced interchange property up to ≃_{2}. (A strict interchange would allow us to derive one horizontal composition from the previous operations; but, even for chain complexes, this is not the case.)

The homotopy relation f ≃ g (meaning that there is some homotopy α: f → g) is a congruence; the quotient HoA = A/≃ is the homotopy category, while Ho_{2}A = A/≃_{2} (same objects, same maps, and classes of homotopies up to 2-homotopy) is the track 2category of A, with invertible cells. Generally, one cannot reduce the study of homotopy to Ho_{2}A, somehow in the same way as higher dimensional category theory cannot be reduced to 2-categories.

Homotopies can usually be represented by a cylinder functor, or dually by a path functor; often by both via their adjunction I ⊣ P; then, all higher homotopies are automatically produced by their powers. This gives a more powerful abstract frame, studied in [9] in a form which will be marginally used here.

1.2. Homotopy kernels. As a second main aspect, the pointed homotopical category A is assumed to have regular homotopy kernels and homotopy cokernels, with respect to a zero object 0. The latter is defined by a 2-dimensional universal property: every object X has precisely one map t: X → 0 and one homotopy t → t (necessarily 0_{t}); and dually.

Also to fix the present notation, the standard homotopy kernel, or h-kernel, of the map f: A → B is a triple hkerf = (Kf, kf, κf), as in the left diagram below, determined up to isomorphism by the following universal property (of comma squares)

[figure: A; f; B; 0; B; Kf; kf; κf; 0; ΩB; ωB; 0]  (6)

for every similar triple (X, x, ξ), where x: X → A and ξ: 0 → fx: X → B, there is a unique u: X → Kf such that kf ·u = x and κf ·u = ξ.

In particular, for A = 0 (as in the right diagram above), we obtain the loop-object ΩB = K(0 → B), with a structural homotopy ωB: 0 → 0: ΩB → B.

The h-kernel is assumed to be regular, i.e. to satisfy also the following 2-dimensional property (called h4-regularity in I.2.5): given two maps u, v: X → Kf and a coherent homotopy β: kf ·u → kf ·v (κf ·u + fβ ≃_{2} κf ·v), there is some homotopy α: u → v which lifts β (i.e., kf ·α = β)

[figure: X; 0; κf; 0; κf·u; f ·kf ·u; fβ; X; u; v; Kf; kf; B; ≃_{2}; 0; κf·v; f ·kf ·v; X; β; A; f]  (7)

The dual universal properties define the h-cokernel hcokf = (Cf, cf, γf)

[figure: A; f; B; cf; A; 0; 0; γf; Cf; 0; σA; ΣA]  (8)

which reduces to the suspension ΣA when B = 0.

The h-kernel of A → 0 is 1: A → A. But note that f ≃ 0 does not imply that kf: Kf → A be a homotopy equivalence, as shown by ΩB = K(0 → B). If f: A → B is a homotopy equivalence, then Kf and Cf are contractible, i.e. homotopy equivalent to 0 (I.3.7; the converse holds under stability hypotheses, 2.3). Thus, for every object A, the cocone KA = K(idA) and the cone CA = C(idA) are contractible

[figure: A; A; A; A; cA; KA; kA; κA; 0; 0; γA; CA]  (9)

1.3. Elements and coelements. The kernel-cokernel adjunction of abelian categories corresponds here to an adjunction given by h-kernels and h-cokernels. We consider first its strict version, then - in the next subsection - its coherent version.

Let us work in the category A^{2} of morphisms of A: an object is an A-map x: X′ → X′′, a morphism f = (f′, f′′): x → y is a commutative square of A (as in the diagram below); the composition is obvious: a pasting of commutative squares.

By h-kernels and h-cokernels, the morphism f: x → y yields a commutative diagram in A

\xymatrix{Kx \ar[r]^{kx} \ar[d]_{K(f)} & X′ \ar[r]^{x} \ar[d]_{f′} & X′′ \ar[r]^{cx} \ar[d]^{f′′} & Cx \ar[d]^{C(f)} \\ Ky \ar[r]_{ky} & Y′ \ar[r]_{y} & Y′′ \ar[r]_{cy} & Cy}  (10)

where K(f) and C(f) are defined by

\begin{matrix}ky·K(f) = f′·kx,  κy·K(f) = f′′·κx, \\ C(f)·cx = cy·f′′,  C(f)·γx = γy·f′. \end{matrix}  (11)

This gives two adjoint endofunctors c ⊣ k: A^{2} → A^{2}, whose action on maps is obvious (and displayed in the diagram above): k(f) = (K(f), f′): kx → ky and c(f) = (f′′, C(f)): cx → cy. The unit and counit are determined as follows

(u_{x}, 1): x → kcx,  (1, v_{y}): cky → y,  (12)

\xymatrix{• \ar[r]^{x} \ar[d]_{u_{x}} & A \ar[r]^{cx} & Cx & Ky \ar[r]^{ky} & A \ar[r]^{y} \ar[dr]_{cky} & • \\ Kcx \ar[ur]_{kcx} &  &  &  &  & Cky \ar[u]_{v_{y}}}

kcx·u_{x} = x,  κcx·u_{x} = γx,  v_{y}·cky = y,  v_{y}·γky = κy.  (13)

(The action of the functors c, k on arrows - c(f), k(f) - is written with parentheses, to avoid ambiguity with the structural morphisms cx, ky.) Keeping fixed the object A, all this restricts to an adjunction between the slice categories of elements and coelements of A, denoted as

ElA = A/A = (A ↓ A),  ClA = A\A = (A ↓ A),  (14)
\xymatrix{c : ElA \ar[r] & ClA : k, \ar[l] & u_{x}: x → kcx, & v_{y}: cky → y,}  (15)

where (with the usual abuse of notation) c takes the morphism f = (f, 1_{A}): x → x′ to c(f) = C(f, 1_{A}): cx → cx′, and k takes g = (1_{A}, g): y → y′ to k(g) = K(1_{A}, g): ky → ky′

[figure: X; x; f; A; cx; Cx; C(f,1); KY; ky; K(1,g); A; y; Y; g; X′; x′; A; cx′; Cx′; KY′; ky′; A; y′; Y′]  (16)

For A = 0, we get El(0) = A = Cl(0) and (15) becomes the classical suspension-loop adjunction Σ ⊣ Ω

\xymatrix{Σ : A \ar[r] & A : Ω, \ar[l] & u_{X}: X → ΩΣX, & v_{Y}: ΣΩY → Y.}  (17)

1.4. Homotopy elements. The previous point just requires “1-dimensional homotopical properties” of A (I.7.1). But the fact that A is homotopical allows for a more interesting version of elements, “up to homotopy” (cf. [12, 13], for coherent categories of objects over a space).

Let us replace A^{2} = (2, A) with the coherent homotopy category of morphisms [2, A]: an object is still an A-map x: X′ → X′′, but a morphism [f] = [f′, f′′; ϕ] : x → y derives from a triple f = (f′, f′′; ϕ) forming a homotopy-commutative square ϕ: f′′x ≃ yf′

[figure: X′; x; f′; ϕ; X′′; f′′; Y′; y; Y′′]  (18)

up to identifying [f′, f′′; ϕ] = [g′, g′′; ψ] when there exists a coherent pair of homotopies α′: f′ → g′, α′′: f′′ → g′′, as in the diagram below

[figure: X′; g′; α′; X′; x; f′; ϕ; X′′; f′′; ≃_{2}; X′; x; g′; ψ; X′′; g′′; α′′; X′′; f′′; Y′; Y′; y; Y′′; Y′; y; Y′′; Y′′]  (19)

As a relevant fact, well known for the coherent category of objects over a fixed space ([12], 1.3), and related to a classical theorem of Dold ([2], 6.1), the map [f′, f′′; ϕ]: x → y is an isomorphism if and only if f′ and f′′ are homotopy equivalences. In fact, in this case, one can choose an h-adjoint equivalence for f′

α′: 1 → g′f′, β′: f′g′ → 1 (f′α′ + β′f′ ≃_{2} 0_{f′}, α′g′ + g′β′ ≃_{2} 0_{g′}),

(by Vogt’s Lemma [22]: given an arbitrary equivalence, replace the homotopy β′ with (−β′f′g′ − f′αg′) + β′ and verify the triangle identities); and similarly for f′′. Finally, one constructs an inverse [g′, g′′, ψ] with a suitable homotopy ψ: g′′y → xg′; namely, ψ = (−g′′yβ′ − g′′ϕg′) − α′′xg′.

Replacing the diagram (10) with the following one, where f = (f′, f′′; ϕ), K(f) is defined below, and C(f) is similarly defined

[figure: Kx; kx; K(f); X′; x; f′; ϕ; X′′; cx; f′′; Cx; C(f); Ky; ky; Y′; y; Y′′; cy; Cy]  (20)

ky·K(f) = f′·kx,  κy·K(f) = f′′·κx + ϕ·kx,  (21)

we have two adjoint endofunctors c, k: [2, A] → [2, A], with (strict) unit and counit defined essentially as above, [u_{x}, 1; 0]: x → kcx and [1, v_{y}; 0]: cky → y.

For a fixed A, we have a restricted adjunction between h-elements x: • → A and h-coelements y: A → •

\xymatrix{c : hElA \ar[r] & hClA : k, \ar[l] & [u_{x}]: x → kcx, & [v_{y}]: cky → y;}  (22)

and again, for A = 0, this reduces to a suspension-loop adjunction (induced by (17)), for the homotopy category HoA = hEl(0) = hCl(0)

\xymatrix{Σ : HoA \ar[r] & HoA : Ω, \ar[l] & [u_{X}]: X → ΩΣX, & [v_{Y}]: ΣΩY → Y.}  (23)

1.5. Homotopical exactness. An h-differential sequence (f, g; α) consists of a nullhomotopy α: 0 ≃ gf. The properties we want to investigate concern a diagram constructed via h-kernels and h-cokernels; the vertical maps are defined below

[figure: B; u_{α}; u_{f}; f; C; α; •; kcf; u; A; g; ckg; cf; •; v_{g}; α: 0 ≃ gf: B → C; Kg; kg; Cf; v; v_{α}; u = K(v_{α}), v = C(u_{α}),]  (24)

\begin{matrix}kg·u_{α} = f, & κg·u_{α} = α, & v_{α}·cf = g, & v_{α}·γf = α, \\ kcf ·u_{f} = f, & κcf ·u_{f} = γf, & v_{g}·ckg = g, & v_{g}·γkg = κg, \\ kg·u = kcf, & κg·u = v_{α}·κcf, & v·cf = ckg, & v·γf = γkg·u_{α}, \end{matrix}  (25)

and the diagram commutes: u·u_{f} = u_{α}, v_{g}·v = v_{α}. (Note that u_{f} = u_{γf}, v_{f} = v_{κf}). We say that the h-differential sequence (f, g; α) is:

(a) left h-exact if the map [u_{α}]: f → kg (in hElA) determined by the universal property of the h-kernel is an isomorphism; or, equivalently, if u_{α}: B → Kg is a homotopy equivalence in A;

(b) right h-exact if, dually, v_{α}: Cf → C is a homotopy equivalence;

(c) strongly h-exact if it is both left and right h-exact;

(d) h-exact if k[v_{α}]: kcf → kg and c[u_{α}]: cf → ckg are isomorphisms; or, equivalently, if K(v_{α}): Kcf → Kg and C(u_{α}): Cf → Ckg are homotopy equivalences of A.

Strong h-exactness implies h-exactness. The condition (c) is often too strong to be of interest, but is of use in the stable case (2.3); we shall see that the last condition becomes simpler, and is implied by left (or right) h-exactness, as soon as A is homotopically “semistable” (2.3a).

It is interesting to note what happens when the middle object is zero: in this case (which would be trivial in homological algebra), our data reduce to a triple (X, Y; α), formed of two objects and a nullhomotopy α

[figure: X; u_{α}; u_{X}; Y; α; ΩΣX; u; 0; ΣΩY; v_{Y}; α: 0 ≃ 0: X → Y; ΩY; ΣX; v; v_{α}; u = Ω(v_{α}), v = Σ(u_{α}).]  (26)

The typical left h-exact sequence of this type is (ΩY, Y; ωY) and the typical right hexact one is (X, ΣX; σX). This case also shows that a right h-exact sequence (X, ΣX; σX) need not be h-exact: in the homotopical category Top_{∗} of pointed spaces, take X = S^{0}; then ΣX = S^{1} is not homotopy equivalent to ΣΩS^{1} (apply H_{1}).

Marginally, we shall also consider a condition of pseudo exactness, meaning that gf ≃ 0 and cf·kg ≃ 0, which just concerns a pair (f, g) of consecutive arrows; any h-exact sequence is also pseudo exact, but the converse is far from being true: a sequence X → 0 → Y is trivially pseudo exact, but (for any choice of α) cannot be left h-exact unless X is homotopy equivalent to ΩY. In the stable case, this notion will be shown to be equivalent to ordinary exactness within the associated Freyd’s abelian category (2.1).

1.6. Definition and Theorem. (Coherence)

Two h-differential sequences (f, g; α) and (x, y; β) will be said to be coherently equivalent if there is a coherent diagram linking them

[figure: A′; f; u′; ϕ; α; A; g; u; ψ; A′′; u′′; α: 0 ≃ gf,  β: 0 ≃ yx,; ϕ: uf ≃ xu′,  ψ: u′′g ≃ yu,; B′; x; β; B; y; B′′; βu′ − yϕ ≃_{2} u′′α + ψf,]  (27)

where the vertical arrows u, u′, u′′ are homotopy equivalences.

In these hypotheses, the upper row is left h-exact if and only if the lower row is so (and similarly for all the h-exactness conditions considered in 1.5).

Proof. Let us assume, for instance, that the upper row is left h-exact and prove that also the lower one is so. The map [u, u′′; ψ]: g → y is iso in [2, A], whence k[u, u′′; ψ] is iso, and w = K(u, u′′; ψ) is a homotopy equivalence. We form the diagram

[figure: A′; s; u′; ϕ′; Kg; kg; w; κg; A; g; u; ψ; A′′; u′′; B′; t; Ky; ky; κy; B; y; B′′]  (28)

\begin{matrix}ky·w = u·kg, & κy·w = u′′·κg + ψ·kg, \\ kg·s = f,  κg·s = α, & ky·t = x,  κy·t = β, \end{matrix}  (29)

where s = u_{α} is a homotopy equivalence, by hypothesis, and we want to prove that also t = u_{β} is so; plainly, it suffices to prove the existence of a homotopy ϕ′: ws → tu′, which can be derived from the 2-dimensional property of the h-pullback hker(y) = (Ky, ky, κy). In fact, the maps ws, tu′: A′ → Ky have a homotopic projection on B, coherently with κy: 0 ≃ y·ky (as a consequence of the coherence hypothesis, in (27))

\begin{matrix}ky·ws = u·kg·s = uf,  ky·tu′ = xu′,  ϕ: ky·ws → ky·tu′, \\ κy·ws + yϕ = (u′′·κg + ψ·kg)·s + yϕ = (u′′α + ψf) + yϕ \\ ≃_{2} βu′ = κy·tu′. \end{matrix}  (30)

1.7. Theorem. (The fibre and cofibre sequences of a map)

If A is pointed homotopical, the cofibre sequence of a map f: A → B

[figure: A; f; γf; B; cf; 0; Cf; δ; σf; ΣA; Σf; γ′f; ΣB; Σ(cf); ΣCf; …]  (31)

is right h-exact at any point, with non-canonical nullhomotopies

σf: 0 ≃ Σf ·δ, γ′f: 0 ≃ Σ(cf)·Σf

and so on. In the stronger assumption of a pointed IP4-homotopical structure for A [9], there are canonical nullhomotopies σf, Σ(γf),... having that effect.

Dually, the fibre sequence of a map is left h-exact

[figure: …; ΩKf; Ω(kf); κ′f; ΩA; Ωf; ωf; ΩB; ∂; 0; Kf; kf; κf; A; f; B]  (32)

Proof. The first part of the statement is essentially proved in I.5 (and only needs a right homotopical category). To begin with, it is proved (I.5.4.11) that there exists some nullhomotopy σf satisfying

σf: 0 ≃ Σf ·δ: Cf → ΣB,  σf ·cf = σB: 0 ≃ 0: B → ΣB.  (33)

Then, applying the homotopy invariant functor Σ (I.4.5) we obtain, again in a noncanonical way, the subsequent nullhomotopies γ′f,... We have now the contracted cofibre diagram (I.5.6.3)

[figure: A; f; B; cf; Cf; δ; ΣA; Σf; ≃; ΣB; Σ(cf); ≃; ΣCf; …; A; f; B; x_{1}; B_{2}; x_{2}; B_{3}; x_{3}; u_{0}; B_{4}; x_{4}; u_{1}; B_{5}; u_{2}; …]  (34)

linking the cofibre sequence of f to the sequence of its iterated h-cokernels

\begin{matrix}x_{1} = cf, & x_{i+1} = cx_{i}, \\ hcok(x_{i}) = (B_{i+1}, x_{i+1}, γx_{i}: 0 ≃ x_{i+1}·x_{i}) & (i ≥ 1). \end{matrix}  (35)

All squares are h-commutative, all vertical arrows are homotopy equivalences. Moreover, the diagram is coherent (as in (27)), with respect to the nullhomotopies of the upper row (as in (31) and the structural nullhomotopies γx_{i} of the lower one: this follows from the construction of u_{i}’s in I.5.4-6.

Finally, if A is pointed IP4-homotopical, in the sense of [9], the cone functor C: A → A inherits an induced monad (as proved more in detail in [8], 3.7)

c: 1 → C,  g: C^{2} → C  (g·cC = id = g·Cc),  (36)

whose operation g (induced by a “connection” of the cylinder functor) represents a natural nullhomotopy of the cone CX, via gX ·γCX: 0 ≃ 1: CX → CX. We can now deduce a structural nullhomotopy σf: 0 ≃ Σf ·δ; in fact, Σf ·δ factors through CB, as proved by the following (canonical) diagram

[figure: B; B; cB; 0; γB; A; f; f; B; cf; 0; 0; CB; p; ΣB; 0; γf; Cf; δ; m; ΣA; Σf]  (37)

In the left cube, the front and back face are h-pushouts (the h-cokernels of f and 1_{B}); in the right cube, the front and back face are ordinary pushouts, so that the front and back rectangle are h-pushouts, by the pasting property I.2.2: σA = δ·γf and σB = p·γB; thus Σf ·δ = (Cf → CB → ΣB) = pm, and we can define

σf = p·gB·γCB·m: 0 ≃ pm,
σf ·cf = p·gB·γCB·m·cf = p·gB·γCB·cB = p·gB·CcB·γB
= p·γB = σB.

(38)

Finally, Σ: A → A has a canonical extension to homotopies (as proved for the cylinder functor, in [10], 2.9).

2. The links between stability and exactness

Under suitable stability hypotheses (2.1, 2.2), the exactness properties behave in a simple way (2.3) and the default of h-exactness can be measured by the homotopy type of suitable objects (2.5).

2.1. Stability. Homotopical stability essentially requires that the suspension-loop adjunction be an equivalence. As in Part I, we will use a stronger definition, better related with (co)fibre sequences. Let A be pointed homotopical and f: A → B a map. Its fibrecofibre sequence can be inserted as the central row of a homotopy-commutative adjunction fibre-cofibre diagram (I.7.5)

[figure: …; ΣΩ^{2}B; v_{ΩB}; ΣΩKf; v_{Kf}; ΣΩA; ΣΩf; v_{A}; ΣΩB; Σ∂; v_{B}; ≃; ΣKf; Σkf; V_{f}; ≃; ΣA; Σf; …; …; Ωf; ΩB; ∂; ≃; Kf; kf; U_{f}; ≃; A; f; u_{A}; B; cf; u_{B}; Cf; δ; u_{Cf}; ΣA; Σf; u_{ΣA}; …; …; Ωf; ΩB; ΩCf; ΩCf; Ωδ; ΩΣA; ΩΣf; ΩΣB; ΩΣCf; ΩΣ^{2}A; …]  (39)

whose upper and lower rows are obtained by letting the functors Σ and Ω operate on the original sequence, and shifting the result of three steps. The vertical arrows consist of the (co)units u, v of the adjunction Σ ⊣ Ω, together with the adjoint maps U_{f}, V_{f} provided by the following structural nullhomotopy ρ_{f}

U_{f}: Kf → ΩCf,  V_{f}: ΣKf → Cf,

ρ_{f} = cf ·κf − γf ·kf: 0 → 0: Kf → Cf. (40)

A is defined to be h-stable (I.7.8) if all these maps U_{f}, V_{f} are homotopy equivalences. Then also the (co)unit-maps u_{A} and v_{A} are so: in fact (I.7.6)

u_{A} ≃ U_{f} (f: A → 0),  v_{A} ≃ V_{g} (g: 0 → A),  (41)

with coincidence whenever the trivial homotopies are strict identities for concatenation, as it happens within chain complexes. (Ch_{∗}D is even strictly stable: all the vertical arrows of (39) are isos and all the squares commute.)

In the h-stable case, the homotopy category HoA has a canonical embedding in an abelian category F = F r(HoA), introduced by Freyd [3, 4, 5]; this provides a notion of exactness which coincides with pseudo exactness, as defined at the end of 1.5: the sequence (f, g) of A is pseudo exact if and only if ([f], [g]) is exact in F.

(We only sketch the proof of this fact, which will not be used here. The last property means that Im_{F}[f] = Ker_{F}[g]; but kernels in F derive from weak kernels in HoA, and the latter from h-kernels of A, so that Ker_{F}[g] = Im_{F}[kg]. Finally, subobjects in F amount to weak subobjects in HoA [10]; thus, Im_{F}[f] = Im_{F}[kg] means that [f] factors through [kg] in HoA, i.e. gf ≃ 0 in A, and conversely, i.e. cf ·kg ≃ 0.)

2.2. Other stability properties. The adjunction c ⊣ k between h-elements f: • → A and g A → • (1.4) yields other stability conditions, which we prove below to be weaker than h-stability (2.4). The pointed homotopical category A will be said to be:

(a) left h-stable if, for every object A, the adjunction c ⊣ h (c: hElA → hClA) is a coreflection, i.e. all components u_{f}: Dom(f) → Kcf are homotopy equivalences;

(b) right h-stable if the dual condition holds, i.e. all components v_{g}: Ckg → Cod(g) are homotopy equivalences (if both conditions hold, the adjunction c ⊣ h is an equivalence);

(c) h-semistable if, for every object A, the adjunction c ⊣ h is idempotent, i.e. the following equivalent conditions hold:

- all components C(u_{f}): Cf → Ckcf are homotopy equivalences,

- all components v_{cf}: Ckcf → Cf are homotopy equivalences,

- all components u_{kg}: Kg → Kckg are homotopy equivalences,

- all components K(v_{g}): Kckg → Kg are homotopy equivalences.

The equivalence comes from a general result (of 2-category theory): in an adjunction, the four natural transformations which appear in the triangle identities are invertible whenever any of them is so ([16], Part 0, Lemma 4.3); such adjunctions are called “idempotent” (or “exact”) because they are also characterised by the idempotence of the associated monad, cf. [1], Section 6. (Any adjunction between ordered sets is so, as well as any reflection or coreflection between categories.)

Plainly, left (or right) h-stable implies h-semistable. There are parallel strict notions (left stable, etc.), for strict elements and coelements, where these components are required to be isos.

Recall that the adjunction Σ ⊣ Ω is a particular instance of the adjunction c ⊣ k (22); therefore, if A is left h-stable (resp. left and right h-stable, h-semistable), then, at the level of the homotopy category HoA, the ΣΩ-adjunction is a coreflection (resp. an endoequivalence, an idempotent adjunction).

(We shall see that Ch_{∗}D, being stable, is also left and right h-stable. Its subcategory of positive complexes is just left h-stable (3.6); negative complexes yield a right h-stable case and the bounded ones an h-semistable category.)

2.3. Theorem. (Stability and exactness)

(a) If A is h-semistable, the two conditions defining h-exactness (1.5d) are equivalent: K(v_{α}) is a homotopy equivalence iff C(u_{α}) is so; therefore, left or right h-exact implies h-exact.

(b) If A is left h-stable a sequence is h-exact iff it is left h-exact. A is left h-stable iff all sequences (f, cf; γf) are strongly h-exact, iff every right h-exact sequence is strongly h-exact. If A is left h-stable, a map f is a homotopy equivalence iff the object Cf is contractible, and this implies that Kf is also so.

(c) A is left and right h-stable iff all the conditions of h-exactness of 1.5 are equivalent. In this case, any fibre-cofibre sequence is strongly h-exact; a map f is a homotopy equivalence iff Kf is contractible, iff Cf is so.

Proof. It suffices to prove (a) and (b). We use the notation of diagram (24); recall the relations u·u_{f} = u_{α}, v_{g}·v = v_{α}.

(a) By hypothesis, C(u_{f}) is a homotopy equivalence; if also u is so, the same is true of v = C(u_{α}) = C(u)·C(u_{f}). And dually.

(b) If A is left h-stable, u_{f}: B → Kcf is a homotopy equivalence; then, u_{α} = u·u_{f} shows that the sequence (f, g; α) is h-exact if and only if it is left h-exact. Thus, also by (a), if A is left h-stable, right h-exact implies strongly h-exact. In this case, any sequence (f, cf; γf) is strongly h-exact. But, if this holds, the canonical map u_{f}: B → Kcf is a homotopy equivalence and A is left h-stable.

Finally, let A be h-stable; if f is a homotopy equivalence, we know that Cf is contractible (1.2); conversely, in this case, the invariance of h-exactness (1.6) shows that kcf is a homotopy equivalence, whence also f = kcf.u_{f} is so.

2.4. Corollary. If the pointed homotopical category A is h-stable, then it is also left and right h-stable.

Proof. Let A be h-stable. Because of this, the fibre-cofibre sequence of f is strongly h-exact, as follows from the adjunction fibre-cofibre diagram (39), the invariance of hexactness (1.6), and the fact that Σ and Ω, being quasi-reciprocal, preserve left and right h-exactness. Thus, the sequences (f, cf; γf) and (kf, f; κf) are strongly h-exact, which implies the thesis, by 2.3c.

2.5. Homotopical homology. The default of h-exactness of h-differential sequences can be measured by the homotopy type of suitable objects. Starting from an h-differential sequence (f, g; α), there is a commutative, coherent diagram (for i, see below)

[figure: B; u_{α}; B; f; α; 0; γu_{α}; γf; α: 0 ≃ gf; Kg; kg; cu_{α}; κg; A; g; cf; C; κg: 0 ≃ g·kg; γf: 0 ≃ cf ·f; Cu_{α}; i; κv_{α}: 0 ≃ v_{α}·kv_{α}; Kv_{α}; kv_{α}; κv_{α}; Cf; v_{α}; C; γu_{α}: 0 ≃ cu_{α}·u_{α}.]  (42)

It provides the left homotopical homology and the right homotopical homology of the sequence

H^{−}(f, g; α) = Cu_{α},  H^{+}(f, g; α) = Kv_{α},  (43)

linked by a non-canonical comparison i satisfying the following relations

kv_{α}·i·cu_{α} = cf ·kg,  kv_{α}·i·γu_{α} = γf,  κv_{α}·i·cu_{α} = κg;  (44)

(its existence is proved below; a precise determination is possible in the stronger setting of IP4-homotopical categories [10]).

In the left h-stable case, the homotopy type of H^{−}(f, g; α) measures the default of h-exactness of (f, g; α): the sequence is h-exact iff it is left h-exact (2.3c), iff u_{α} is a homotopy equivalence, iff Cu_{α} is contractible (2.3c, again). Dually, in the right h-stable case, the object H^{+}(f, g; α) measures the same default of h-exactness (which coincides now with right h-exactness). Finally, if A is left and right h-stable, both objects H^{−}(f, g; α) and H^{+}(f, g; α) measure the default of h-exactness.

It is therefore of interest to study the property that the canonical map i be a homotopy equivalence. When this is the case, H(f, g; α) will denote the homotopy type of H^{−}(f, g; α) and H^{+}(f, g; α); or their isomorphism type, if i is an iso, as happens for chain complexes (3.4, 3.5).

Now, to prove the existence of i, consider first that there is one map x: Kg → Kv_{α} such that kv_{α}·x = cf·kg, κv_{α}·x = κg. Second, xu_{α} ≃ 0, by the 2-dimensional property of the h-kernel (Kv_{α}, kv_{α}, κv_{α}); in fact, kv_{α}·xu_{α} = cf ·kg·u_{α} = cf ·f, and the nullhomotopy γf: 0 ≃ cf ·f is (strictly) coherent with κv_{α}: 0 ≃ v_{α} ·kv_{α}, in the sense that κv_{α} ·xu_{α} = κg·u_{α} = α = v_{α}·γf (and κv_{α}·0 is the trivial endohomotopy).

There is thus some homotopy ρ: 0 ≃ xu_{α} such that kv_{α}·ρ = γf. (In the IP4-homotopical case, we may determine ρ via a double homotopy produced by the homotopy κv_{α}·xu_{α} = v_{α}·γf together with a connection of the cylinder or path functor.) Finally, this homotopy ρ produces a map i: Cu_{α} → Kv_{α} such that i·cu_{α} = x, i·γu_{α} = ρ, and (44) holds.

3. Stability and homotopical exactness for chain complexes

This section deals with the usual homotopical structure of the category Ch_{∗}D of unbounded chain complexes over an additive category D; positive chain complexes are briefly considered (3.5).

3.1. Notation. A morphism f: ⨁ A_{i} → ⨁ B_{j} between two finite biproducts in the additive category D is determined by its components f_{ji}: A_{i} → B_{j} and can be written as a matrix f = (f_{ji}). But we prefer to write it as an “expression in m variables”, as one would do in a category of modules

f(a_{1}, a_{2}, …, a_{m}) = (∑ f_{1i}a_{i}, ∑ f_{2i}a_{i}, …, ∑ f_{ni}a_{i}).  (45)

(Viewing a_{i} as the i-th projection of A = ⨁ A_{i}, the expression is correct: (a_{1}, a_{2}, …, a_{m}) is the identity of A, with specified names for projections.)

3.2. The homotopical structure. The category Ch_{∗}D of (unbounded) chain complexes over the additive category D is equipped with the usual homotopies of chain maps

\begin{matrix}α = (f, g, (α_{n})): f → g: A → B, \\ α_{n}: A_{n} → B_{n+1}, & −f_{n} + g_{n} = α_{n−1}∂_{n} + ∂_{n+1}α_{n}. \end{matrix}  (46)

These come from an IP4-homotopical structure, based on well known cylinder and path functors (cf. [9], 6.5-6.8). The opposite structure is isomorphic to Ch_{∗}D^{op}; this duality provides a choice of h-kernels and h-cokernels which reduces the structural isomorphisms to identities.

Let f: A → B be a map of chain complexes. The left homotopical structure of Ch_{∗}D is computed as follows (note that any “shift of degree” leads to a change of sign)

\begin{matrix}(Kf)_{n} = A_{n} ⊕ B_{n+1}, & ∂(a, b) = (∂a, fa − ∂b), \\ k: Kf → A, & k(a, b) = a, \\ κ: 0 ≃ fk: Kf → B, & κ(a, b) = b, \end{matrix}  (47)

ΩA = K(0 → A),  (ΩA)_{n} = A_{n+1},  ∂_{n}^{Ω} = −∂_{n+1}.  (48)

Analogously, the right homotopical structure is described by:

\begin{matrix}(Cf)_{n} = A_{n−1} ⊕ B_{n}, & ∂(a, b) = (−∂a, −fa + ∂b), \\ c: B → Cf, & c(b) = (0, b), \\ γ: 0 ≃ cf: A → Cf, & γ(a) = (−a, 0), \end{matrix}  (49)
ΣA = C(A → 0),  (ΣA)_{n} = A_{n−1},  ∂_{n}^{Σ} = −∂_{n−1}.  (50)

3.3. Stability. This homotopical structure is strictly stable, as is well known (for the fact that Σ and Ω are inverse) or follows easily from the previous expressions (for the additional condition in 2.1). Thus, Ch_{∗}D is also left and right h-stable (2.4); note, however, that it is not left stable in the strict sense: the previous computations show that u_{f}: A → Kcf is not an isomorphism, generally.

By 2.3, the fibre-cofibre sequence of the map f is strongly h-exact and sent to itself by Σ and Ω. Each h-exact sequence is thus coherently equivalent to a sequence B → Cf → ΣA, which means a componentwise-split short exact sequence of complexes (as in the triangulated structure).

3.4. Theorem. (Homotopical homology of chain complexes)

In the category Ch_{∗}D of chain complexes over an additive category, the left and the right homotopical homology of an h-differential sequence (2.5) coincide, yielding a chain complex whose homotopy type vanishes if and only if the sequence is h-exact (or, equivalently, strongly h-exact)

\begin{matrix}(H(f, g; α))_{n} = B_{n−1} ⊕ A_{n} ⊕ C_{n+1}, \\ ∂(b, a, c) = (−∂b, −fb + ∂a, −αb + ga − ∂c). \end{matrix}  (51)

Proof. It is an easy computation, based on 3.2.

3.5. Positive chain complexes. Let the additive category D have finite limits. The category Ch_{p}D of positive chain complexes (null in negative degree) is also homotopical (actually IP4-homotopical).

The h-cokernel of the chain map f: A → B in Ch_{p}D is computed as in Ch_{∗}D, since (Cf)_{n} = A_{n−1} ⊕ B_{n} is null in negative degree; the h-kernel is computed by coreflection and differs in degree 0, where we get a pullback

(Kf)_{n} = A_{n} ⊕ B_{n+1} (n > 0), (Kf)_{0} = pb(f, ∂).  (52)

Ch_{p}D is left h-stable but not right h-stable, as we prove below (3.6). The relation Cu_{α} = Kv_{α} providing the homotopical homology H(f, g; α) of an h-differential sequence (2.5) holds also here, with

\begin{matrix}H_{0}(f, g; α) = (Kg)_{0} = (Cu_{α})_{0} = (Kv_{α})_{0} \\ = pb(g_{0} : A_{0} → C_{0} ← C_{1} : ∂_{1}). \end{matrix}  (53)

From left h-stability, we deduce that the h-differential sequence (f, g; α) is h-exact iff it is left h-exact, iff H(f, g; α) is contractible (2.3b).

But being right h-exact is a stronger condition (even up to weak equivalence). For instance, the typical left h-exact sequence ΩA → 0 → A (with nullhomotopy ωA) is right h-exact iff the counit v_{A}: ΣΩA → A is a homotopy equivalence, which requires that H_{0}(A) = H_{0}(ΣΩA) = 0. On the other hand, the homotopical homology H(ΩA, A; ωA) of that sequence consists of the h-cokernel of id: ΩA → ΩA, i.e. the cone C(ΩA), which is always contractible, independently of right h-exactness.

3.6. Proposition. Ch_{p}D is left h-stable. It is not right h-stable, unless D is trivial.

Proof. Given f: X → A, we show that u_{f}: X → Kcf is a deformation retract, where K = Kcf is computed below, as a pullback in degree 0 (and Cf in 3.2.4)

\begin{matrix}K_{n} = A_{n} ⊕ X_{n} ⊕ A_{n+1} (n > 0), \\ K_{0} = pb((cf)_{0}, ∂_{1}) ⊂ A_{0} ⊕ X_{0} ⊕ A_{1}, \\ ∂(a, x, a′) = (∂a, ∂x, a − ∂a′ + fx), \\ k(a, x, a′) = a, κ(a, x, a′) = (x, a′), & u_{f}(x) = (fx, −x, 0), \end{matrix}  (54)

\xymatrix{X \ar[r]^{f} \ar[d]_{u_{f}} & A \ar[r]^{cf} & Cf & γ = γf: 0 ≃ cf ·f: X → Cf, \\ Kcf \ar@{-->}[u]_{u′} \ar@{-->}[ur]_{k} &  &  & κ = κcf: 0 ≃ cf ·k: K → Cf.}

Now, take u′: K → X, u′(a, x, a′) = −x. It is a chain map, with u′u_{f} = 1; and there is a homotopy σ: u_{f}u′ ≃ 1_{K}, namely σ(a, x, a′) = (a′, 0, 0).

On the other hand, Ch_{p}D is not right h-stable: the coreflexive adjunction Σ ⊣ Ω (ΩΣ = 1) is not an equivalence, even up to homotopy, since (ΣΩA)_{0} = 0 and H_{0}(ΣΩA) = 0.

3.7. Negative and bounded chain complexes. Dually, the category of negative chain complexes, on an additive category D with finite colimits, is pointed homotopical and right h-stable.

A sequence (f, g; α) (as in 2.5) is h-exact iff it is right h-exact, iff H(f, g; α) is contractible; left h-exactness is a stronger condition. The homotopical homology is computed as a pushout in degree 0

H_{0}(f, g; α) = (Cf)_{0} = (Cu_{α})_{0} = (Kv_{α})_{0} = po(B_{−1} ← B_{0} → A_{0}).  (55)

If D is finitely complete and cocomplete, the category Ch_{0}^{p}D of chain complexes concentrated in degrees n ∈ [0, p] (p ≥ 1) is homotopical and h-semistable, as it follows from computations similar to the previous ones. The object H(f, g; α) = Cu_{α} = Kv_{α} is still well-defined, and computed by a pullback in degree 0 and a pushout in degree p

\begin{matrix}H_{0}(f, g; α) = pb(A_{0} → C_{0} ← C_{1}), \\ H_{p}(f, g; α) = po(B_{p−1} ← B_{p} → A_{p}). \end{matrix}  (56)

The particular case p = 1 (where a complex is reduced to a map) will be treated in detail in the next section, where we show that the homotopy type of H(f, g; α) still measures the default of h-exactness of the sequence. For p > 1, this holds in a weaker sense, as we show in Section 5.

Finally, let us also consider the “degenerate” case A = Ch_{0}^{0}D = D, where homotopies are trivial (i.e., identities), and h-(co)kernels are ordinary (co)kernels. If D is pointed, with kernels and cokernels, the adjunction c ⊣ k between elements and coelements is always idempotent (and induces the usual duality between normal subobjects and normal quotients): in other words, A is h-semistable. If D is abelian (but, actually, much more generally), left and right homology coincide and determine exact sequences; the present analysis of homotopical exactness reduces to pure homological algebra.

4. Categories of morphisms

The case Ch_{0}^{1}D = D^{2} gives a simple h-semistable category having homotopical homology.

4.1. The homotopy structure. In this section, D is always an additive category with finite limits and colimits. A = Ch_{0}^{1}D = D^{2} is the category of morphisms of D, equipped with chain homotopies, as already considered above (3.7) and examined below; this makes it a pointed homotopical category.

(D can be an abelian category, or the category of topological abelian groups, or Banach spaces, etc. Note also that Ab^{2} is equivalent to the category of internal categories in abelian groups, by the Dold-Kan theorem [14]; one can show that,in this case, the present analysis of h-exactness agrees with notions recently introduced for symmetric cat-groups, in [15].)

An object will be written as A = ∂_{A}: A′ → A′′ and a morphism as f = (f′, f′′): A → B. A nullhomotopy a: 0 ≃ f is determined by a diagonal map α: A′′ → B′

\xymatrix{A′ \ar[r]^{f′} \ar[d]_{∂} & B′ \ar[d]^{∂} & f′ = α∂, \\ A′′ \ar[r]_{f′′} \ar@{-->}[ur]_{α} & B′′ & f′′ = ∂α,}  (57)

while a homotopy α: f ≃ g: A → B is given by a nullhomotopy α: 0 ≃ −f + g (i.e. −f′ + g′ = α∂, −f′′ + g′′ = ∂α). As an exception in the family Ch_{0}^{p}D, we have here a 2-category (essentially because 2-dimensional homotopies vanish, being produced by morphisms of degree 2): given two consecutive nullhomotopies α: 0 ≃ f and β: 0 ≃ g, the reduced interchange holds strictly: gα = βf (both homotopies are determined by the same diagonal g′α = β∂α = βf′′: A′′ → C′′).

Plainly, an object ∂_{A}: A′ → A′′ is contractible if and only if ∂_{A} is an isomorphism, if and only if H_{∗}(A) = 0 (where H_{∗} is the graded homology group, with two components). Thus, a homotopy equivalence need not be an iso; however, there is a relevant case where the two notions coincide.

4.2. Lemma. If, in the morphism f: A → B, either f′ or f′′ is an iso, then f is a homotopy equivalence if and only if it is an isomorphism.

Proof. Let f′ = 1_{Z} (Z = A′ = B′) and choose an adjoint equivalence between A and B

[figure: Z; ∂_{A}; Z; g′; Z; Z; ∂_{B}; α: 1_{A} ≃ gf,  β: fg ≃ 1_{B},; A′′; f′′; α; B′′; g′′; β; A′′; f′′; B′′; fα + βf = 0,  αg + gβ = 0.]  (58)

Then f′′ admits an inverse mapping: h = g′′ + ∂_{A}β. In fact, f′′h = f′′g′′ + f′′∂_{A}β = 1−∂_{B}β +∂_{B}β = 1 and hf′′ = g′′f′′ +∂_{A}βf′′ = 1+∂_{A}a−∂_{A}a = 1 (use the triangle identity fα + βf = 0). Similarly, if f′′ = 1_{Z} (Z = A′′ = B′′), then f′ has an inverse mapping, namely g′ − a∂_{B}: B′ → A′.

4.3. Homotopy kernels and cokernels. If f: A → B is the commutative square (57), form the pullback K of (f′′, ∂_{B}) and the pushout C of (∂_{A}, f′); this yields an inner bicartesian square

\xymatrix{ &  & A′′ \ar[dr]_{γ} \ar[drr]^{f′′} \\ A′ \ar[r]|{k′} \ar[urr]^{∂} \ar[drr]_{f′} & K \ar@{-->}[ur]_{k′′} \ar[dr]^{κ} &  & C \ar[r]|{c′′} & B′′ \\  &  & B′ \ar@{-->}[ur]^{c′} \ar[urr]_{∂}}  (59)

from which the h-kernel and the h-cokernel of f are constructed

[figure: A′; k′; A′; f′; B′; c′; C; c′′; Kf = k′, kf = (1, k′′), κf = κ,; K; k′′; κ; A′′; f′′; γ; B′′; B′′; Cf = c′′, cf = (c′, 1), γf = γ.]  (60)

Now, the h-cokernel of kf (the h-normal coimage of f) is ckf = (k′, 1): A → k′′, and its h-kernel is again kf: k′ → A (left solid diagram below). This shows that the adjunction between h-elements and h-coelements of A is exact (2.2c), and A is h-semistable

[figure: A′; k′; A′; f′; K; κ; k′′; B′; c′; B′; c′; C; c′′; K; k′′; 1; A′′; A′′; γ; C; c′′; 1; B′′; B′′]  (61)

Symmetrically, the right solid diagram shows how the h-kernel of cf (the h-normal image of f) is kcf = (1, c′′): c′ → B, whose h-cokernel is again cf: B → c′′. Inserting the dotted morphism in the middle, the three central squares are a factorisation of f: A → B, the h-normal factorisation (much in the same way as for the ordinary normal factorisation, in a pointed category with kernels and cokernels).

4.4. Left and right exactness. Given an h-differential sequence (f, g; α)

\xymatrix{X′ \ar[r]^{f′} \ar[d]_{∂} & A′ \ar[r]^{g′} \ar[d] & Y′ \ar[d]^{∂} & α: 0 ≃ gf: X → Y, \\ X′′ \ar[r]_{f′′} \ar@{-->}[urr]^{α} & A′′ \ar[r]_{g′′} & Y′′}  (62)

its properties of h-exactness are characterised as follows, up to coherent equivalence (1.6)

(a) in the typical left h-exact sequence, (X′′, α, f′′) is the pullback of (g′′, ∂_{Y}) and f′ = 1_{A′},

(b) in the typical right h-exact sequence, (Y′, α, g′) is the pushout of (f′, ∂_{X}) and g′′ = 1′′_{A},

(c) the typical strongly h-exact sequence is of the following type, determined by an arbitrary factorisation ∂_{A} = h′′h′

[figure: A′; h′; A′; h′; H; h′′; H; h′′; 1; A′′; A′′]  (63)

4.5. Theorem. (Homotopical homology of morphisms)

Given an h-differential sequence a: 0 ≃ gf, its homotopical homology exists and is computed as

H(f, g; α) = Cu_{α} = Kvα = (w: B → Z),  (64)

from the diagram below, where (B, c, γ) is the pushout of (∂_{X}, f′), while (Z, k, κ) is the pullback of (∂_{Y} , g′′) and w: B → Z is the induced morphism (think of B as h-boundaries, of Z as h-cycles)

\xymatrix{X′ \ar[rr]^{f′} \ar[dd]_{∂} &  & A′ \ar[rr]^{g′} \ar[dl]_{c} \ar[dd]^{∂} \ar[dr]^{z} &  & Y′ \ar[dd]^{∂} \\  & B \ar@{-->}[rr]^{w} \ar[dr]_{b} &  & Z \ar[dl]^{k} \ar[ur]_{κ} \\ X′′ \ar[rr]_{f′′} \ar@{-->}[ur]^{γ} &  & A′′ \ar[rr]_{g′′} &  & Y′′}  (65)

Moreover, our sequence is h-exact iff K(v_{α}) = (1_{A′}, w) is a homotopy equivalence, iff w is an isomorphism of D, iff H(f, g; α) is contractible, iff its homology is null: H_{∗}(H(f, g; α)) = 0.

Proof. We already know that H(f, g; α) exists and how to compute it (3.7), but let us write down everything, in this simple case. Computing h-kernels and h-cokernels, as in 4.3, the following diagram shows that Cu_{α} and Kv_{α} are indeed the object w: B → Z

[figure: X′; f′; ∂; f′; A′; g′; ∂; c; Y′; ∂; A′; z; c; B; b; κw; B; w; X′′; f′′; wγ; A′′; g′′; Y′′; Z; k; A′′; g′′; Z; k]  (66)

hker(g) = (z, (1, k): z → ∂_{A}, κ), hcok(f) = (b, (c, 1): ∂_{A} → b, γ),
u_{α} = (f′, wγ): X → z,  v_{α} = (κw, g′′): b → Y,
hcok(u_{α}) = (w, (c, 1): z → w, γ), hker(v_{α}) = (w, (1, k): w → b, κ).

(67)

Finally, the diagram (24) becomes

[figure: X; u_{α}; u_{f}; f; Y; α; u_{f} = (f′, γ), u = (1, w),; c; (1,b); u; A; g; (z,1); (c,1); k; v_{g}; v_{g} = (κ, g′′), v = (w, 1).; z; (1,k); b; v; v_{α}]  (68)

This proves that our sequence is h-exact iff K(v_{α}) = (1_{A′}, w) is a homotopy equivalence. The other equivalent conditions follow immediately from 4.2 and 4.1.

5. Bounded chain complexes and weak exactness

For the category A = Ch_{0}^{p}D bounded chain complexes on an abelian category D, a notion of weak exactness, controlled by homology, is more effective.

5.1. Homology. Let D be abelian. Recall that the category A = Ch_{0}^{p}D of chain complexes concentrated in degrees n ∈ [0, p] (p ≥ 1) is homotopical and h-semistable (3.7).

The ordinary homology functors H_{n}: A → D (0 ≤ n ≤ p) satisfy the following self-dual axioms, combining properties of homology and homotopy theories

(i) homotopy invariance: if f ≃ g in A, then H_{n}(f) = H_{n}(g);

(ii) exactness: for every map f: A → B in A, the sequences

H_{0}(Kf) → H_{0}(A) → H_{0}(B), H_{p}(A) → H_{p}(B) → H_{p}(Cf),  (69)

are exact in D; (iii) stability: there are two natural isomorphisms

ω: H_{n}(ΩA) → H_{n+1}(A), σ: H_{n}(A) → H_{n+1}(ΣA) (0 ≤ n < p),  (70)

which are coherent with the morphisms U_{f}, V_{f} (for every map f: A → B), i.e. form a commutative diagram (in Ab)

\xymatrix{H_{n}(Kf) \ar[r]^{H_{n}(U_{f})} \ar[d]_{σK} \ar@{-->}[dr]^{i_{f}} & H_{n}(ΩCf) \ar[d]^{ωC} \\ H_{n+1}(ΣKf) \ar[r]_{H_{n+1}(V_{f})} & H_{n+1}(Cf)}  (71)

yielding a natural transformation i_{f}: H_{n}(Kf) → H_{n+1}(Cf) (0 ≤ n < p).

Abstracting from this situation, we shall consider the following setting: a pointed homotopical category A with Ω^{p+1} = 0 = Σ^{p+1}, equipped with a theory H_{n}: A → E with values in an abelian category and satisfying the axioms above for 0 ≤ n ≤ p. (This can be easily adapted to the positive or unbounded cases.)

5.2. Theorem. In this setting:

(a) ω and σ are also coherent with the morphisms induced in homology by the unit u_{A} and counit v_{A} of the adjunction Σ ⊣ Ω: they form commutative diagrams in E

\xymatrix{H_{n}(A) \ar[rr]^{H_{n}(u_{A})} \ar@{-->}[dr]_{σA} &  & H_{n}(ΩΣA) \ar@{-->}[dl]^{ωΣA} & H_{n+1}(ΣΩA) \ar[rr]^{H_{n+1}(v_{A})} &  & H_{n+1}(A) \\  & H_{n+1}(ΣA) &  &  & H_{n}(ΩA) \ar@{-->}[ul]^{ωΣA} \ar@{-->}[ur]_{ωA}}  (72)

(b) The morphisms i_{f}, H_{n}(U_{f}), H_{n+1}(V_{f}), H_{n}(u_{A}), H_{n+1}(v_{A}) are iso (0 ≤ n < p).

(c) every map f: A → B produces two exact sequences in E

[figure: H_{p}(Kf); H_{p}(A); H_{1}(B); H_{0}(Kf); i_{f}; H_{0}(A); H_{0}(B); H_{p}(A); H_{1}(B); H_{1}(Cf); H_{0}(A); H_{0}(B); H_{0}(Cf)] (73)

which are actually one, modulo the isomorphisms i_{f} and the commutative squares above: the fibre-cofibre homology sequence of f.

Proof. (a) The diagrams (72) derive from (71), writing u_{A} and v_{A} as in (41).

(c) For the upper row, apply H_{0} to the fibre sequence of f, taking into account that H_{n}(A) ≅ H_{0}(Ω^{n}A) for 0 ≤ n ≤ p and that Ω^{p+1} = 0; exactness follows from the properties of the fibre sequence (1.7), together with the axioms of exactness and homotopy invariance for H_{0}. Similarly for the lower row. The commutativity of (73) follows from the commutativity of the adjunction fibre-cofibre diagram (39).

(b) The Five Lemma in E proves now that all i_{f} are iso; the rest follows from the commutative diagrams (71), (72).

5.3. Weak equivalences. Now, say that a map f: A → B is a weak equivalence (with respect to the theory H_{∗}) if H_{n}(f) is iso for all n; similarly, the object A is weakly null (or weakly contractible) if H_{n}(A) = 0 for all n. Plainly, by the usual properties of exact sequences in abelian categories:

(a) Kf is weakly null if and only if H_{n}(f) is iso for n > 0 and mono for n = 0,

(b) Cf is weakly null if and only if H_{n}(f) is iso for n < p and epi for n = p. Thus, for p = 2, the complex A = (A_{2} → A_{1} → A_{0}) is a weakly null object iff it forms

a short exact sequence; but it is easy to see that it is nullhomotopic iff this sequence splits.

5.4. Lemma. In the general setting mentioned at the end of (5.1), let an h-differential sequence (f, g; α) be given. Then the map i: Cu_{α} = Kv_{α} (2.5) is a weak equivalence.

Proof. The diagram (42) gives a commutative diagram in the abelian category E

[figure: X_{n}; X_{n}; 0; X_{n−1}; X_{n}; X_{n}; 0; X_{n−1}; Xn^{′}; A_{n}; Y_{n}; Xn−1^{′}; Xn^{′}; A_{n}; Y_{n}; Xn−1^{′}; K_{n}; Yn^{′}; Y_{n}; K_{n−1}; C_{n}; i_{∗}; Yn^{′}; Y_{n}; C_{n−1}; i_{∗}]  (74)

with C_{n} = H_{n}(Cu_{α}) and K_{n} = H_{n}(Kv_{α}).

Its rows and columns are exact, except - possibly - the lowest front row and the left back column, which are just known to be of order two. Moreover, in each row or column, each term C_{n} or K_{n} has at least two terms on each side (possibly null). By a sort of variation of the 3×3-Lemma, it follows that all H_{n}(i) are iso. (Of course, one only needs to verify this for modules, by diagram chasing.)

5.5. Definition and Theorem. (Weak exactness)

In the general setting mentioned at the end of 5.1, let an h-differential sequence (f, g; α) be given, and consider its diagram (24). The following conditions are equivalent:

(a) the map u: Kcf → Kg is a weak equivalence (with respect to H_{∗}),

(b) H_{n}(v_{α}): H_{n}(Cf) → Hn(C) is iso for n > 0, and mono for n = 0,

(c) the object H^{+}(f, g; α) = Kv_{α} is weakly null.

(d) the map v: Cf → Ckg is a weak equivalence (with respect to H_{∗}),

(e) H_{n}(u_{α}): H_{n}(B) → H_{n}(Kg) is iso for n < p, and epi for n = p, (f) the object H^{−}(f, g; α) = Cu_{α} is weakly null,

If they are satisfied, the sequence (f, g; α) will be said to be w-exact. This notion is strictly weaker than h-exactness, provided that p ≥ 2 and E has some non-split short exact sequence.

Proof. The preceding lemma shows that (c) and (f) are equivalent, while the equivalence of (b) and (c) has been considered in 5.3. Therefore, we only need to prove that (a) and (b) are equivalent. This follows from the Five Lemma in E: take the diagram

[figure: Kcf; kcf; u; A; cf; Cf; v_{α}; Kg; kg; A; g; C]  (75)

and apply the (natural) fibre-cofibre homology sequence (73), to cf and g

[figure: 0; H_{p}(Kcf); u_{∗}; H_{p}(A); H_{0}(Kcf); u_{∗}; H_{0}(A); H_{0}(Cf); (v_{α})_{∗}; 0; 0; H_{p}(Kg); H_{p}(A); H_{0}(Kg); H_{0}(A); H_{0}(C); H_{0}(Cg); 0]  (76)

Finally, it suffices to consider a sequence 0 → A → 0, which is h-exact (resp. w-exact) if and only if A is nullhomotopic (resp. w-null), which - under our assumptions - are indeed non-equivalent conditions by a remark at the end of 5.3.

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