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interval object in chain complexes

Context

Homotopy theory

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

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

Definition

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

Definition

The standard interval object in chain complexes

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

In components this means that

I =[00R(id,id)RR]. 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 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)