nLab center of a category




The notion of center of a monoid has a horizontal categorification to a notion of center of a category.


For π’ž\mathcal{C} a category, its center Z(π’ž)Z(\mathcal{C}) is defined to be the commutative monoid

Z(π’ž)≔[π’ž,π’ž](Id π’ž,Id π’ž) Z(\mathcal{C}) \;\coloneqq\; [\mathcal{C},\mathcal{C}]\big( Id_\mathcal{C} ,\, Id_\mathcal{C} \big)

of endo-natural transformation of the identity functor Id C:C→CId_C \,\colon\, C \to C, i.e. the endomorphism monoid of Id CId_C in the functor category [C,C][C,C].

If π’ž\mathcal{C} carries extra structure this may be inhereted by its center. Notably the center of an additive category is not just a commutative monoid but a commutative ring: the endomorphism ring of the identity functor. For more on this see at center of an additive category.



For AA an ordinary monoid and BA\mathbf{B}A it delooping-category, the ordinary center of AA is naturally identified with the category theoretic center of BA\mathbf{B}A.


For a generator GG of a category π’ž\mathcal{C} there is an embedding of Z(π’ž)Z(\mathcal{C}) into the monoid Hom(G,G)Hom(G,G) given by η↦η G\eta\mapsto\eta _G. In particular, if Hom(G,G)Hom(G,G) or Z(Hom(G,G))Z(Hom(G,G)) is trivial, as happens e.g. for SetSet with G=*G=\ast, then so is Z(π’ž)Z(\mathcal{C}).

[Hoffmann (1975)]


For Cauchy complete π’ž\mathcal{C} the idempotent elements of Z(π’ž)Z(\mathcal{C}) correspond precisely to the quintessential localizations of π’ž\mathcal{C}.

[Johnstone (1996)]


See also:

Last revised on January 21, 2024 at 01:28:58. See the history of this page for a list of all contributions to it.