nLab interval object in chain complexes

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Contents

Context

Homotopy theory

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

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Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

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diagram chasing

Schanuel's lemma

Homology theories

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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.

Last revised on September 3, 2012 at 13:21:36. See the history of this page for a list of all contributions to it.