categories: Re: Looking for adjoints

[Tom Leinster <T.Leinster@dpmms.cam.ac.uk>]

Here's an adjunction from which various basic results in category theory can
be read off.  (Useful, but somewhat inward-looking...)

Fix a small category C, and consider the forgetful functor 

U: [C^op, Set] ---> [ob C, Set].  

This has a left adjoint F, which can easily be written down explicitly (and
whose existence is also guaranteed because it's a Kan extension).  Hence U
preserves limits - and this is part of what's meant by the statement that
limits are computed pointwise in a presheaf category.  Moreover, the
adjunction is monadic, from which it follows that

(a) U creates limits (which is the rest of what's meant by the "computed
pointwise" slogan), and

(b) every presheaf is the colimit of representables (using the fact that
every algebra for a monad is a coequalizer of free algebras).

Dually, U has a right adjoint, so the dual results also hold. 

Tom


> From: Jean-Pierre Marquis <Jean-Pierre.Marquis@UMontreal.CA>
> To: <categories@mta.ca>
> 
> I would like to have a large pool of examples of adjoint functors in as many
> different fields of mathematics as possible.  I am looking for the "nicest",
> in whatever sense you can think of this expression (e.g. unexpected, their
> existence is equivalent to a classical theorem, etc), cases in various
> fields.
> 
> References or examples anyone?  (Besides the standard ones found in Mac
> Lane, etc.)
> 
> Thank you, 
> Jean-Pierre Marquis
> 
> 
>