nLab cocycle space

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

• elements/objects are cocycles in $\mathbf{H}$-cohomology theory on $X$ with coefficients in $A$ (hence maps/morphisms $X \to A$);

• edges/morphisms are coboundaries between these cocycles (i.e. homotopies/gauge transformations)

• faces/2-morphisms are coboundaries between coboundaries;

• …

• n-cells/n-morphisms are $n$th order coboundaries (i.h. higher homotopies/higher gauge transformations).

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.

Related concepts

• hom-space, space of maps

• Cohomotopy charge map

	homotopy	cohomology	homology
	$[S^n,-]$	$[-,A]$	$(-) \otimes A$
category theory	covariant hom	contravariant hom	tensor product
homological algebra	Ext	Ext	Tor
enriched category theory	end	end	coend
homotopy theory	derived hom space $\mathbb{R}Hom(S^n,-)$	cocycles $\mathbb{R}Hom(-,A)$	derived tensor product $(-) \otimes^{\mathbb{L}} A$