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

Edit this sidebar

Idea

The cohomology of a category CC is often defined to be the groupoid cohomology of the ∞-groupoid that is presented by the nerve of the category.

Hence for A\mathbf{A} some coeffiecient ∞-groupoid – typically, but not necessarily, an Eilenberg-MacLane object A=K(A,n)=B nA\mathbf{A} = K(A,n) = \mathbf{B}^n A – regarded as a Kan complex, the cohomology of CC in this sense is

H n(C,A):=π 0Grpd(F(N(C)),A), 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

=Ho SSet(N(C),A). \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 ThomasonQuillenSSet QuillenCat_{Thomason} \stackrel{Quillen}{\simeq} SSet_{Quillen} we have for α\alpha any category whose groupoidification is equivalent to A\mathbf{A}, i.e. any cateory such that F(N(α))AF(N(\alpha)) \simeq \mathbf{A} in ∞Grpd, we have

=Ho Cat Thomason(C,α). \cdots = Ho_{Cat_{Thomason}}(C,\alpha) \,.
Created on January 11, 2010 20:56:35 by Urs Schreiber (87.212.203.135)