nLab
interval object in chain complexes

Idea

The standard interval object in a category of chain complexes in $R$Mod is an “abelianization” of the standard simplicial interval, the 1-simplex and a model of the unit interval, $[0,1]$, with the evident cell decomposition.

Definition

Let $R$ be some ring and let $𝒜=R$Mod be the abelian category of $R$-modules. Write ${\mathrm{Ch}}_{•}(𝒜)$ for the corresponding category of chain complexes.

Definition

The standard interval object in chain complexes

$$I_{•}\in {\mathrm{Ch}}_{•}(𝒜)$$I_\bullet \in Ch_\bullet(\mathcal{A})

is the normalized chain complex of the simplicial chains on the simplicial 1-simplex:

$$I_{•}≔N_{•}(C(\Delta [1])).$$I_\bullet \coloneqq N_\bullet(C(\Delta[1])) \,.

In components this means that

$$I_{•}=[\cdots \to 0\to 0\to R\stackrel{(\mathrm{id},-\mathrm{id})}{\to }R\oplus R].$$I_\bullet = [ \cdots \to 0 \to 0 \to R \stackrel{(id,-id)}{\to} R \oplus R ] \,.

Properties

Homotopies

Proposition

A homotopy with respect to $I_{•}$ gives a chain homotopy and conversely.

See the entry on chain homotopy for more details.