Subject: categories: Re: Looking for adjoints

From: Michael Barr <[email-masked]>
Date: Mon, 2 Apr 2001 07:16:50 -0400 (EDT)

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.

---

From: Michael Batanin <[email-masked]>
Date: Thu, 05 Apr 2001 12:59:11 +1000

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 [email-masked] wrote: