Complexes, homotopies and contractibility in an additive category

Additive setting. Fix an additive category $\mathcal{A}$ (Additive category): every hom-set is an abelian group, finite biproducts exist, and every object has an identity morphism. All morphisms below are morphisms of $\mathcal{A}$, sums and negatives are taken in the hom-groups, and $0$ denotes the zero morphism between the indicated objects.

Cochain complexes. A cochain complex $C^{\bullet }=(C^n,d^n{)}_{n\in \mathbb{Z}}$ consists of objects $C^n$ of $\mathcal{A}$ and morphisms $d^n:C^n\to C^{n+1}$ with $d^{n+1}{d}^n=0\text{for all }n\in \mathbb{Z}\mathrm{.}$ The morphisms $d^n$ are the differentials of $C^{\bullet }$. No boundedness, finiteness or nonvanishing condition is imposed on the family of objects.

Cochain maps. A cochain map $f:C^{\bullet }\to D^{\bullet }$ is a family of morphisms $f^n:C^n\to D^n$ with $f^{n+1}{d}_C^n=d_D^n{f}^n\text{for all }n\in \mathbb{Z}\mathrm{.}$ Identities and composites of cochain maps are cochain maps, so cochain complexes and cochain maps form a category; this category is written ${\mathrm{Ch}}^{\bullet }(\mathcal{A})$ when the ambient category needs to be recorded.

Homotopies. A homotopy $h:f\simeq g$ between cochain maps $f,g:C^{\bullet }\to D^{\bullet }$ is a family of morphisms $h^n:C^n\to D^{n-1}$, one in each degree, such that $f^n-g^n=d_D^{n-1}{h}^n+h^{n+1}{d}_C^n\text{for all }n\in \mathbb{Z}\mathrm{.}$ Thus $h$ has degree $-1$, and the right-hand side is the $n$-th component of the graded map $dh+hd:=(d_D^{n-1}{h}^n+h^{n+1}{d}_C^n{)}_{n\in \mathbb{Z}},$ written with $d$ on both sides. A null homotopy of a cochain map $f:C^{\bullet }\to D^{\bullet }$ is a homotopy $f\simeq 0$ to the zero map with the same source and target; $D^{\bullet }$ need not be the zero complex. A map admitting such a homotopy is null-homotopic.

Homotopy equivalence. A cochain map $f:C^{\bullet }\to D^{\bullet }$ is a homotopy equivalence when there is a cochain map $g:D^{\bullet }\to C^{\bullet }$ with $gf\simeq 1_{C^{\bullet }}$ and $fg\simeq 1_{D^{\bullet }}$; the complexes are then homotopy equivalent. The maps $f$ and $g$ are homotopy inverses of one another.

Contractibility. A cochain complex $C^{\bullet }$ is contractible when its identity is null-homotopic, $1_{C^{\bullet }}\simeq 0$: that is, when there is a family of morphisms $h^n:C^n\to C^{n-1}$ with $1_{C^n}=d^{n-1}{h}^n+h^{n+1}{d}^n\text{for all }n\in \mathbb{Z}\mathrm{.}$ A complex is contractible exactly when it is homotopy equivalent to the zero complex, since a homotopy equivalence onto the zero complex is a pair of null-homotopies of the identities.

Degreewise biproducts. If $C^{\bullet }$ and $D^{\bullet }$ are cochain complexes, then defining $(C\oplus D{)}^n:=C^n\oplus D^n,d_{C\oplus D}^n:=d_C^n\oplus d_D^n$ gives a cochain complex, because $(d_C^{n+1}\oplus d_D^{n+1})(d_C^n\oplus d_D^n)=0\oplus 0=0$. The degreewise injections and projections are cochain maps, and the biproduct identities hold in each degree and are therefore identities of cochain maps; hence $C\oplus D$ is a biproduct of $C$ and $D$. Iterating, finite direct sums of cochain complexes are formed degreewise, and finite direct sums of cochain maps and of homotopies are formed degreewise as well. Under the reindexing below this is the additive structure on complexes over an additive category recorded in The category of complexes in an additive category is additive.

Dictionary with the published chain convention. Reindex by $C_n:=C^{-n}$ and $d_n:=d^{-n}$. Then $d_n:C_n\to C_{n-1}$ and $d_{n-1}{d}_n=d^{-(n-1)}{d}^{-n}=d^{-n+1}{d}^{-n}=0,$ so $(C_{\bullet },d_{\bullet })$ is an ordinary chain complex. A cochain map $f$ becomes the chain map with components $f_n:=f^{-n}$, and a homotopy $h:f\simeq g$ becomes the chain homotopy $s_n:=h^{-n}:C_n\to D_{n+1}$, because substituting $n\mapsto -n$ into the displayed homotopy equation produces exactly $f_n-g_n=d_{n+1}^D{s}_n+s_{n-1}{d}_n^C\mathrm{.}$ Consequently, when $\mathcal{A}$ is abelian, these definitions restrict under this dictionary to the published A chain homotopy and A contractible complex, which are stated for chain complexes in an abelian category. Reindexing reverses the sign of the differential degree ($+1$ for cochains, $-1$ for chains) and of the homotopy degree ($-1$ for cochains, $+1$ for chains), and introduces no further sign.

What is not asserted. The definitions use only zero morphisms, addition, negatives, composition and identities. No kernel, cokernel, image, homology object or exactness is assumed or defined, no linear structure on the hom-groups beyond the additive one is used, and no homology object is attached to a complex in an arbitrary additive category; contractibility is the existence of the displayed family $h$, not the vanishing of homology.