From: t <t@sympatico.ca>
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.