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
Excerpts from that book may also be found online at: http://www.dcs.qmw.ac.uk/~pt/Practical_Foundations/html/c7.html Derek.
I would like to have a large pool of examples of adjoint functors
See "Practical Foundations of Mathematics" (Cambridge University Press, 1999), especially Section 7.1.
Paul
Here are a few interesting examples. One of the earliest, although not called an adjoint, was the Bohr compactification of an abelian group and the construction was similar to that of the original adjoint functor theorem. The earliest that was labelled such was Kan's Ex^\infty, which was a left adjoint to the inclusion of (what are now called) Kan simplicial sets into simplicial sets. All Galois connections are contravariant adjunctions. The contravariant power set functor is adjoint to itself on the left. Hom(A,PB) = Hom(B,PA). Valid in any topos. Also valid in toposes, for any f: A --> B, the induced inverse image function f*: Sub B --> Sub A has both a left adjoint (the familiar direct image) and a right adjoint (not familiar), considering the subobject lattices as categories. And many, many more.
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
This is just a small correction to Michael Barr message. The Kan's Ex^{\infty} is not a left adjoint to the inclusion of Kan simplicial sets into simplicial sets. In the original paper by Kan "On c.s.s. complexes" there is no mention about it being adjoint. Yet, the following is true Kan(Ex^{\infty}X,Y) is homotopy equivalent to Ssets(X,Y). So it is some sort of homotopy adjunction. Michael Batanin.
on 2/4/01 9:16 PM, Michael Barr at barr@barrs.org wrote:
The earliest that was labelled such was Kan's Ex^\infty, which was a left adjoint to the inclusion of (what are now called) Kan simplicial sets into simplicial sets.
participants (6)
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Derek Ross -
Jean-Pierre Marquis -
Michael Barr -
Michael Batanin -
Paul Taylor -
Tom Leinster