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