Looking for adjoints

Looking for adjoints

[Jean-Pierre Marquis]

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

Re: Looking for adjoints

[Paul Taylor]

> 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

Re: Looking for adjoints

[Derek Ross]

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
>

Re: Looking for adjoints

[Michael Barr]

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.

Re: Looking for adjoints

[Michael Batanin]

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.
> 

Re: Looking for adjoints

[Tom Leinster]

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
> 
> 
>