# nLab interval object in chain complexes

## Theorems

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## 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, $\left[0,1\right]$, with the evident cell decomposition.

## Definition

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

###### Definition

The standard interval object in chain complexes

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

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

${I}_{•}≔{N}_{•}\left(C\left(\Delta \left[1\right]\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$I_\bullet \coloneqq N_\bullet(C(\Delta[1])) \,.

In components this means that

${I}_{•}=\left[\cdots \to 0\to 0\to R\stackrel{\left(\mathrm{id},-\mathrm{id}\right)}{\to }R\oplus R\right]\phantom{\rule{thinmathspace}{0ex}}.$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.

Revised on September 3, 2012 13:21:36 by Tim Porter (95.147.237.93)