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cohomology of a category

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cohomology

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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 coefficient ∞-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) \,.

Last revised on August 12, 2016 at 08:53:32. See the history of this page for a list of all contributions to it.