internalization and categorical algebra
algebra object (associative, Lie, β¦)
The notion of center of a monoid has a horizontal categorification to a notion of center of a category.
For a category, its center is defined to be the commutative monoid
of endo-natural transformation of the identity functor , i.e. the endomorphism monoid of in the functor category .
If 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 an ordinary monoid and it delooping-category, the ordinary center of is naturally identified with the category theoretic center of .
For a generator of a category there is an embedding of into the monoid given by . In particular, if or is trivial, as happens e.g. for with , then so is .
For Cauchy complete the idempotent elements of correspond precisely to the quintessential localizations of .
Rudolf-E. Hoffmann, Γber das Zentrum einer Kategorie, Math. Nachr. 68 (1975) 299-306 [doi:10.1002/mana.19750680122]
Peter Johnstone, Remarks on Quintessential and Persistent Localizations, TAC 2 8 (1996) 90-99. [tac:2-08, pdf]
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.