Category of chain complexes

See Category.

A chain complex is a sequence of abelian groups and their homomorphisms: $$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} \newcommand{\la}[1]{\!\!\!\!\!\!\!\xleftarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\ua}[1]{\left\uparrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{ccccccc} ... & \ra{} & S_{k+1} & \ra{\partial_{k+1}} & S_{k} & \ra{\partial_{k}} & S_{k-1} & \ra{\partial_{k-1}} & S_{k-2} & \ra{} & ...\\ \end{array} $$ that satisfies: $$\partial _k \partial _{k+1} = 0.$$ This homomorphism is called the boundary operator or differential.

We use the notation $$(S,\partial)$$ for the complex given by $$S=\bigoplus _i S_i,\partial=\bigoplus _i \partial _i : S \rightarrow S,$$ or, interchangeably, by $$S=\{ S_i \},\partial=\{ \partial _i : S_i \rightarrow S_{i-1} \}.$$

A (degree $0$) chain map between two chain complexes $$f: (S,\partial ^S) \rightarrow (T,\partial ^T)$$ is a sequence of homomorphisms $$f_i: S_i \rightarrow T_i$$ that commute with the boundary operators, i.e., this diagram is commutative: $$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} \newcommand{\la}[1]{\!\!\!\!\!\!\!\xleftarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\ua}[1]{\left\uparrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{ccccccc} ... & \ra{} & S_{m+1} & \ra{\partial_{m+1}^S} & S_{m} & \ra{\partial_{m}^S} & S_{m-1} & \ra{\partial_{m-1}^S} & S_{m-2} & \ra{} & ...\\ & & \da{f_{m+1}} & & \da{f_{m}} & & \da{f_{m-1}} & & \da{f_{m-2}} & & \\ ... & \ra{} & T_{m+1} & \ra{\partial_{m+1}^T} & T_{m} & \ra{\partial_{m}^T} & T_{m-1} & \ra{\partial_{m-1}^T} & T_{m-2} & \ra{} & ...\\ \end{array} $$

We use ${\mathscr Comp}$ to denote the category of chain complexes and chain maps:

• $\text{Obj}(Ch(\mathcal{Mod}) = \{(S,\partial ^S) \},$
• $\text{Hom}(Ch(\mathcal{Mod}) =\{ f:(S,\partial ^S) \rightarrow (T,\partial ^T) \}.$

The composition of morphisms is "coordinate-wise": $$(fg)_i=f_ig_i.$$

The commutativity condition $$f_{m-1} \partial_m^K = \partial_m^L f_m,$$ or simply: $$ f \partial = \partial f,$$ requires that these homomorphisms preserve boundaries. In this sense it is the algebraic analogue of continuity. The algebraic analogue of homotopy is chain homotopy.

Suppose we have two chain maps between two chain complexes $$f,g: (S,\partial ^S) \rightarrow (T,\partial ^T).$$ Then a chain homotopy between $f$ and $g$ is a sequence $$h=\{h_m \colon S_m \to T_{m + 1}\}$$ of homomorphisms (not a chain map but similar to a "degree 1 chain map") such that $$f_m - g_m = \partial ^T_{m + 1} h_m + h_{m - 1} \partial ^S_m,$$ or simply $$f - g = \partial h + h \partial.$$

The equation is illustrated by this (non-commutative) diagram: $$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} \newcommand{\la}[1]{\!\!\!\!\!\!\!\xleftarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\ua}[1]{\left\uparrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{ccccccc} ... & \ra{} & S_{m+1} & \ra{\partial_{m+1}^S} & S_{m} & \ra{\partial_{m}^S} & S_{m-1} & \ra{\partial_{m-1}^S} & S_{m-2} & \ra{} & ...\\ & \swarrow ^{} & \da{f_{m+1}} \da{g_{m+1}}& \swarrow ^{h_{m}} & \da{f_{m}} \da{g_{m}} & \swarrow ^{h_{m-1}} & \da{f_{m-1}} \da{g_{m-1}}& \swarrow ^{h_{m-2}} & \da{f_{m-2}} \da{g_{m-2}}& \swarrow ^{} & \\ ... & \ra{} & T_{m+1} & \ra{\partial_{m+1}^T} & T_{m} & \ra{\partial_{m}^T} & T_{m-1} & \ra{\partial_{m-1}^T} & T_{m-2} & \ra{} & ...\\ \end{array} $$ The left-hand side of the equation is the difference of two adjacent vertical arrows and the right-hand side is the sum of the two triangles adjacent to them.

All these definitions apply to a cochain complex which is a (reversed) sequence of abelian groups and their homomorphisms: $$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} \newcommand{\la}[1]{\!\!\!\!\!\!\!\xleftarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\ua}[1]{\left\uparrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{ccccccc} ... & \la{} & S^{m+2} & \la{d^{m+1}} & S^{m+1} & \la{d^{m}} & S^{m} & \la{d^{m-1}} & S^{m-1} & \la{} & ...\\ \end{array} $$ that satisfies: $$d^{m+1} d^{m} = 0.$$