Homotopy Category K(A)

The homotopy category $K(\mathcal{A})$ is obtained from the category of chain complexes by identifying chain homotopic maps. It is the first step in the construction of the derived category: we mod out by the "irrelevant" data of chain homotopies, keeping only the cohomological information. The homotopy category is a triangulated category, but it is not abelian.

Definition

Definition4.10Homotopy Category

The homotopy category $K(\mathcal{A})$ of an abelian category $\mathcal{A}$ has:

Objects: Cochain complexes in $\mathcal{A}$ (same as $\mathbf{Ch}(\mathcal{A})$ ).
Morphisms: Chain maps modulo chain homotopy: $\mathrm{Hom}_{K(\mathcal{A})}(A^\bullet, B^\bullet) = \mathrm{Hom}_{\mathbf{Ch}}(A^\bullet, B^\bullet) / \sim$ .
Composition: Inherited from $\mathbf{Ch}(\mathcal{A})$ (well-defined since homotopy is compatible with composition).

We also write $K^+(\mathcal{A})$ , $K^-(\mathcal{A})$ , $K^b(\mathcal{A})$ for the full subcategories of bounded below, bounded above, and bounded complexes.

Triangulated Structure

Definition4.11Distinguished triangles in K(A)

For a chain map $f : A^\bullet \to B^\bullet$ , the standard triangle is

$A^\bullet \xrightarrow{f} B^\bullet \xrightarrow{\iota} \mathrm{Cone}(f) \xrightarrow{p} A^\bullet[1]$

A triangle in $K(\mathcal{A})$ is distinguished if it is isomorphic (in $K(\mathcal{A})$ ) to a standard triangle.

Theorem4.5K(A) is triangulated

With the shift functor $[1]$ and the distinguished triangles above, $K(\mathcal{A})$ is a triangulated category.

Examples

ExampleK(A) is not abelian

$K(\mathcal{A})$ is generally not abelian. For instance, in $K(\mathbb{Z}\text{-}\mathbf{Mod})$ , the map $\mathbb{Z}[0] \xrightarrow{2} \mathbb{Z}[0]$ has no kernel in $K(\mathcal{A})$ . The failure of abelianness is precisely why we need the derived category.

ExampleK(A) is additive

$K(\mathcal{A})$ is additive: morphisms can be added (addition of chain maps is compatible with homotopy), and biproducts exist (direct sums of complexes).

ExampleCohomology factors through K(A)

Since homotopic maps induce equal maps on cohomology, the functor $H^n : \mathbf{Ch}(\mathcal{A}) \to \mathcal{A}$ factors through $K(\mathcal{A})$ : we get $H^n : K(\mathcal{A}) \to \mathcal{A}$ .

ExampleHomotopy equivalence = isomorphism in K(A)

A chain map $f$ is a homotopy equivalence iff it becomes an isomorphism in $K(\mathcal{A})$ . A quasi-isomorphism becomes an isomorphism in $D(\mathcal{A})$ but not necessarily in $K(\mathcal{A})$ .

ExampleMapping cone triangle

For any $f : A^\bullet \to B^\bullet$ , the triangle $A \to B \to \mathrm{Cone}(f) \to A[1]$ is distinguished. The long exact sequence in cohomology $\cdots \to H^n(A) \to H^n(B) \to H^n(\mathrm{Cone}(f)) \to H^{n+1}(A) \to \cdots$ is the long exact sequence associated to this distinguished triangle.

ExampleK(Proj) and K(Inj)

$K^-(\mathrm{Proj}\, R)$ (bounded above complexes of projectives) is equivalent to $D^-(\mathcal{A})$ via the natural functor. Similarly, $K^+(\mathrm{Inj}\, \mathcal{A})$ is equivalent to $D^+(\mathcal{A})$ . These equivalences are the foundation for computing derived functors.

ExampleDerived Hom in K(A)

In $K(\mathcal{A})$ , for complexes $A^\bullet, B^\bullet$ of injectives (resp. projectives), $\mathrm{Hom}_{K(\mathcal{A})}(A^\bullet, B^\bullet)$ computes the "derived Hom" up to a shift. This motivates the enrichment of $K(\mathcal{A})$ to a DG category.

ExampleNull-homotopic maps form an ideal

The null-homotopic maps $\mathrm{Ht}(A^\bullet, B^\bullet) \subseteq \mathrm{Hom}_{\mathbf{Ch}}(A^\bullet, B^\bullet)$ form a two-sided ideal. The quotient $\mathrm{Hom}_K = \mathrm{Hom}_{\mathbf{Ch}} / \mathrm{Ht}$ gives the hom-sets of $K(\mathcal{A})$ .

ExampleSplit exact complexes are zero in K(A)

A split exact complex (one that is contractible) becomes isomorphic to zero in $K(\mathcal{A})$ . However, an exact (acyclic) but non-split complex may be nonzero in $K(\mathcal{A})$ — it becomes zero only in $D(\mathcal{A})$ .

ExampleRotation of triangles

If $A \to B \to C \to A[1]$ is a distinguished triangle, then $B \to C \to A[1] \to B[1]$ is also distinguished (with appropriate signs). This is one of the axioms of a triangulated category.

ExampleK(A) and model categories

The homotopy category $K(\mathcal{A})$ can be obtained from the model category structure on $\mathbf{Ch}(\mathcal{A})$ (projective or injective model structure) by inverting weak equivalences. The Quillen model structure gives $K(\mathcal{A})$ as the homotopy category of a model category, and $D(\mathcal{A})$ as the localization at weak equivalences.

ExampleExt via K(A)

$\mathrm{Ext}^n(M, N) \cong \mathrm{Hom}_{K(\mathcal{A})}(P_\bullet, N[n])$ where $P_\bullet \to M$ is a projective resolution. This gives a "derived" interpretation of Ext within the homotopy category.

RemarkK(A) as an intermediate step

The homotopy category sits between $\mathbf{Ch}(\mathcal{A})$ and $D(\mathcal{A})$ :

$\mathbf{Ch}(\mathcal{A}) \twoheadrightarrow K(\mathcal{A}) \to D(\mathcal{A})$

The first arrow mods out by homotopy; the second inverts quasi-isomorphisms. The composition gives the localization functor $Q : \mathbf{Ch}(\mathcal{A}) \to D(\mathcal{A})$ .