nLab
cohomology of a category
**
cohomology **
*
cocycle ,
coboundary ,
coefficient
*
homology
*
chain ,
cycle ,
boundary
*
characteristic class
*
universal characteristic class
*
secondary characteristic class
*
differential characteristic class
*
fiber sequence /
long exact sequence in cohomology
*
fiber ∞-bundle ,
principal ∞-bundle ,
associated ∞-bundle ,
twisted ∞-bundle
*
∞-group extension
*
obstruction
### Special and general types ###
*
cochain cohomology
*
ordinary cohomology ,
singular cohomology
*
group cohomology ,
nonabelian group cohomology ,
Lie group cohomology
*
Galois cohomology
*
groupoid cohomology ,
nonabelian groupoid cohomology
*
generalized (Eilenberg-Steenrod) cohomology
*
cobordism cohomology theory
*
integral cohomology
*
K-theory
*
elliptic cohomology ,
tmf
*
taf
*
abelian sheaf cohomology
*
Deligne cohomology
*
de Rham cohomology
*
Dolbeault cohomology
*
etale cohomology
*
group of units ,
Picard group ,
Brauer group
*
crystalline cohomology
*
syntomic cohomology
*
motivic cohomology
*
cohomology of operads
*
Hochschild cohomology ,
cyclic cohomology
*
string topology
*
nonabelian cohomology
*
principal ∞-bundle
*
universal principal ∞-bundle ,
groupal model for universal principal ∞-bundles
*
principal bundle ,
Atiyah Lie groupoid
*
principal 2-bundle /
gerbe
*
covering ∞-bundle /
local system
*
(∞,1)-vector bundle /
(∞,n)-vector bundle
*
quantum anomaly
*
orientation ,
Spin structure ,
Spin^c structure ,
String structure ,
Fivebrane structure
*
cohomology with constant coefficients /
with a local system of coefficients
*
∞-Lie algebra cohomology
*
Lie algebra cohomology ,
nonabelian Lie algebra cohomology ,
Lie algebra extensions ,
Gelfand-Fuks cohomology ,
*
bialgebra cohomology
### Special notions
*
Čech cohomology
*
hypercohomology
### Variants ###
*
equivariant cohomology
*
equivariant homotopy theory
*
Bredon cohomology
*
twisted cohomology
*
twisted bundle
*
twisted K-theory ,
twisted spin structure ,
twisted spin^c structure
*
twisted differential c-structures
*
twisted differential string structure ,
twisted differential fivebrane structure
* differential cohomology
*
differential generalized (Eilenberg-Steenrod) cohomology
*
differential cobordism cohomology
*
Deligne cohomology
*
differential K-theory
*
differential elliptic cohomology
*
differential cohomology in a cohesive topos
*
Chern-Weil theory
*
∞-Chern-Weil theory
*
relative cohomology
### Extra structure
*
Hodge structure
*
orientation ,
in generalized cohomology
### Operations ###
*
cohomology operations
*
cup product
*
connecting homomorphism ,
Bockstein homomorphism
*
fiber integration ,
transgression
*
cohomology localization
### Theorems
*
universal coefficient theorem
*
Künneth theorem
*
de Rham theorem ,
Poincare lemma ,
Stokes theorem
*
Hodge theory ,
Hodge theorem
nonabelian Hodge theory ,
noncommutative Hodge theory
*
Brown representability theorem
*
hypercovering theorem
*
Eckmann-Hilton-Fuks duality
Idea
The cohomology of a category $C$ is often defined to be the groupoid cohomology of the ∞-groupoid that is presented by the nerve of the category.
Hence for $\mathbf{A}$ some coefficient ∞-groupoid – typically, but not necessarily, an Eilenberg-MacLane object $\mathbf{A} = K(A,n) = \mathbf{B}^n A$ – regarded as a Kan complex , the cohomology of $C$ in this sense is
$H^n(C,A) := \pi_0 \infty Grpd( F(N(C)), \mathbf{A} )
\,,$
where
Using the standard model structure on simplicial sets this is the same as the hom-set in the homotopy category of SSet
$\cdots = Ho_{SSet}(N(C), \mathbf{A})
\,.$
One can also use the Thomason model structure on Cat to say the same: due to the Quillen equivalence $Cat_{Thomason} \stackrel{Quillen}{\simeq} SSet_{Quillen}$ we have for $\alpha$ any category whose groupoidification is equivalent to $\mathbf{A}$ , i.e. any cateory such that $F(N(\alpha)) \simeq \mathbf{A}$ in ∞Grpd , we have
$\cdots = Ho_{Cat_{Thomason}}(C,\alpha)
\,.$