interval object in chain complexes

**homotopy theory, (∞,1)-category theory, homotopy type theory**

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also **algebraic topology**

**Introductions**

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**Homotopy groups**

**Basic facts**

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(also nonabelian homological algebra)

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.

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

The standard **interval object in chain complexes**

$I_\bullet \in Ch_\bullet(\mathcal{A})$

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

$I_\bullet \coloneqq N_\bullet(C(\Delta[1]))
\,.$

In components this means that

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

A homotopy with respect to $I_\bullet$ 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)