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 , and a domain and coefficient , the cocycle space is the “space”, or rather the the ∞-groupoid/homotopy type, whose
elements/objects are cocycles in -cohomology theory on with coefficients in (hence maps/morphisms );
edges/morphisms are coboundaries between these cocycles (i.e. homotopies/gauge transformations)
faces/2-morphisms are coboundaries between coboundaries;
…
n-cells/n-morphisms are th order coboundaries (i.h. higher homotopies/higher gauge transformations).
Precisely: For some (∞,1)-topos, and two objects, the cocycle space of cocycles on with coefficients in is the (∞,1)-categorical hom-space .
The actual cohomology set is the 0-truncation/connected components of the cocycle space:
Similarly, if is equipped with the structure of a pointed object , the cocycle space becomes canonically pointed by the constant morphism 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.
Last revised on February 16, 2020 at 07:11:21. See the history of this page for a list of all contributions to it.