From: t Subject: Re: Kan extensions Date: Tue, 05 Sep 2000 14:33:35 GMT Newsgroups: sci.math Summary: [missing] Fergal Daly wrote: > Hi, > I'm trying to figure what the right Kan extension of F along X means > for two functors X:A->C and F:A->B. I can see why, if B is the category > with 1 object and no arrows then Ran_F X just produces the limit of X, > when X is considered as a diagram in C. > I think that this is covered in the book by Bousfield and Kan ("Homotopy limits, completions and localizations". If I go near a university this week, I'll find out the spot exactly. > > Also if B is any discrete category with objects {1, 2, ...} then F > divides A into disconnected subcategories A_1, A_2 (objects are in the > same subcat in A if they map to the same object in B) so we can think of > X as made up of X_1:A_1->C, X_2->A_2->C,... and Ran_F X will map 1 to > the limit of X_1, 2 to the limit of X_2,... > > Can anyone tell me what other interesting things you can do with Kan > extensions (left or right) and point me towards a reference. For one thing, it's necessary for the construction of K-theory. See book above, book by Quillen on homotopy (before his fundamental K-theory paper) (can't remember name, but part of the springer LNM series). For a particularly intereting view, see the work of Jardine on simplicial homotopy theory (and construction of K-theory from that point of view). hope this helps some, t ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Kan extensions Date: 13 Sep 2000 06:13:27 GMT Newsgroups: sci.math Fergal Daly wrote: > Can anyone tell me what other interesting things you can do with Kan > extensions (left or right) and point me towards a reference. The geometric realization of a simplicial set is a nice example of a large class of constructions that can be slickly described as Kan extensions. Once you can turn the simplices into topological spaces, Kan extension automatically does the rest. Have you read the end of MacLane's "Categories for the Working Mathematician"? It makes the (half-joking) claim that *all* concepts in category theory are examples of Kan extensions. And it gives a bunch of examples.