# nLab cocycle space

Contents

cohomology

### Theorems

#### Mapping space

internal hom/mapping space

# Contents

## Idea

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

## Definition

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

## Truncation to cohomology sets

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

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

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

$H(X,\Omega A) \;\simeq\; \pi_1 \big( \mathbf{H}(X,A), const_{a_0} \big) \;\simeq\; \pi_0 \big( \Omega_{const_{a_0}} \mathbf{H}(X,A) \big) \,.$

Etc.

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

Last revised on February 16, 2020 at 02:11:21. See the history of this page for a list of all contributions to it.