nLab interval object in chain complexes

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

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

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.

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