group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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
hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
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).
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 $\mathbf{H}(X,A)$.
The actual cohomology set $H(X,A)$ is the 0-truncation/connected components of the cocycle space:
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:
Etc.
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 $(-) \otimes^{\mathbb{L}} A$ |
Last revised on February 16, 2020 at 07:11:21. See the history of this page for a list of all contributions to it.