From: kramsay@aol.commangled (Keith Ramsay)
Subject: Re: adjoint functors
Date: 21 Jan 2000 04:48:56 GMT
Newsgroups: sci.math
Summary: [missing]

In article <s5t3ds5zjft.fsf@goedel3.math.washington.edu>, John Palmieri
<palmieri@math.washington.edu> writes:
|I don't know about "simple", but there is Freyd's Adjoint Functor
|Theorem.  You can find it in many books on category theory and/or
|homological algebra (none of which I have in my office right now, or
|else I would provide the precise statement).

MacLane states it like this:

   Given a small-complete category A with small hom-sets, a functor
   G:A->X has a left adjoint if and only if it preserves all small
   limits and satisfies the following Solution Set Condition. For
   each object x \in X there is a small set I and an I-indexed family
   of arrows f_i:x->Ga_i such that every arrow x->Ga can be written
   as a composite h=Gt \circ f_i for some index i and some t:a_i->a.

Small-complete means all small diagrams have limits. "Small" is in the
sense in which one defines Set as the category of small sets, which is
itself a "large" object.

One application for example is to proving the existence of a free
group on a set of generators. The category of groups is small-complete
and the functor forgetting the group structure is limit-preserving.
This leaves you need just one piece of information which I find a bit
odd: given a (small) set S, there's a (small) *set* of groups such
that if we have a mapping from S to a group G, then it factors through
a mapping to one of the groups in our family. Mac Lane just says that
the subgroup generated by the image of S is bounded in cardinality,
hence there is a set of such groups, up to isomorphism.

Apparently the analogous criterion for representability of a functor
was used in Wiles' proof somewhere.

They're all variations on a criterion for a category having an initial
element: a small-complete category with small hom-sets has an initial
object iff there's a small family k_i of objects such that every
object has a mapping to one of the k_i.

Keith Ramsay
who got his copy of Mac Lane from a jazz pianist
a long long time ago
==============================================================================

From: kramsay@aol.commangled (Keith Ramsay)
Subject: Re: adjoint functors
Date: 23 Jan 2000 03:59:31 GMT
Newsgroups: sci.math

In article <20000120234856.01549.00000698@nso-fj.aol.com>,
kramsay@aol.commangled (Keith Ramsay) writes:
|They're all variations on a criterion for a category having an initial
|element: a small-complete category with small hom-sets has an initial
|object iff there's a small family k_i of objects such that every
|object has a mapping to one of the k_i.

Sorry; that should have been "from one of the k_i".

Keith Ramsay