nLab
cocycle space

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

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

Mapping space

Contents

Idea

Given any kind of generalized cohomology theory H\mathbf{H}, and a domain XX and coefficient AA, the cocycle space H(X,A)\mathbf{H}(X,A) is the “space”, or rather the the ∞-groupoid/homotopy type, whose

Definition

Precisely: For H\mathbf{H} some (∞,1)-topos, and X,AHX,A \in \mathbf{H} two objects, the cocycle space of cocycles on XX with coefficients in AA is the (∞,1)-categorical hom-space H(X,A)\mathbf{H}(X,A).

Truncation to cohomoloy sets

The actual cohomology set H(X,A)H(X,A) is the 0-truncation/connected components of the cocycle space:

H(X,A)=π 0(H(X,A)). H(X,A) \;=\; \pi_0\big( \mathbf{H}(X,A) \big) \,.

Similarly, if AA is equipped with the structure of a pointed object *a 0A\ast \overset{a_0}{\to} A, the cocycle space H(X,A)\mathbf{H}(X,A) becomes canonically pointed by the constant morphism const a 0:X*a 0Aconst_{a_0} \colon X \to \ast \overset{a_0}{\to} A and the 0-truncation/connected components of the corresponding based loop space of the cocucle space is the cohomoloy set in one degree lower:

H(X,ΩA)π 1(H(X,A),const a 0)π 0(Ω const a 0H(X,A)). H(X,\Omega A) \;\simeq\; \pi_1 \big( \mathbf{H}(X,A), const_{a_0} \big) \;\simeq\; \pi_0 \big( \Omega_{const_{a_0}} \mathbf{H}(X,A) \big) \,.

Etc.

homotopycohomologyhomology
[S n,][S^n,-][,A][-,A]()A(-) \otimes A
category theorycovariant homcontravariant homtensor product
homological algebraExtExtTor
enriched category theoryendendcoend
homotopy theoryderived hom space Hom(S n,)\mathbb{R}Hom(S^n,-)cocycles Hom(,A)\mathbb{R}Hom(-,A)derived tensor product () 𝕃A(-) \otimes^{\mathbb{L}} A

Last revised on October 18, 2019 at 07:39:04. See the history of this page for a list of all contributions to it.