Re: How to visualize limits in category theory

From: Todd Trimble (trimble1_at_optonline.net)
Date: 12/18/04 

Date: Sat, 18 Dec 2004 03:21:28 +0000 (UTC)

On 17 Dec 2004, Lee Rudolph wrote:
>"Jesse F. Hughes" <jesse@phiwumbda.org> writes:
>
>>barr@barrs.org (Michael Barr) writes:
>>
>>> When I think about limits, I think about subobjects of products.
>>
>>That's *much* clearer than what I wrote, when I said a category is
>>complete just in case it has equalizers and products. My answer is
>>relevant to completeness, but not a good answer to how to visualize
>>limits.
>
>I was pleased to see Michael Barr rework the verb being used
>into "think about", and I'm distressed to see you return to
>"visualize". Nothing I've read in this thread, from the
>beginning on, has seemed to *me* to have anything at all to
>do with "visualization" (except, maybe, the occasional post
>referring to pentagons). Probably the original poster didn't
>mean what I mean by "visualization" either, but if he did, he
>hasn't been replied to.
>
>Lee Rudolph

Perhaps you're right, but there are ways of "visualizing" limits
and colimits of functors of the form F: B --> Set [B small].

The colimit of F: B --> Set can be visualized as the set of
path components of the category of elements of F, el(F). Recall
that the objects of el(F) are pairs (x, b) where b is an
object of B and x is an element of Fb. Morphisms in el(F)
of the form (x, b) --> (y, c) are morphisms f: b --> c
such that F(f)(x) = y. You can visualize E = el(F) as fibered
over B, where the fiber over an object b is the set Fb, and
given an arrow f: j --> k in the base, there is a unique
arrow in E over f whose source is a given element x over b.

Let E_1 denote the collection of arrows of E, and let E_0
denote the set of objects. The domain/source and codomain/target
give a parallel pair of functions
 
                  -->
              E_1 --> E_0

and the coequalizer of this pair, denoted \pi_0(E), is the
colimit of F. This is just the set of path components of the
simplicial set given by the nerve of E.

There is a way of dualizing all this to get limits (which I
could explain if pressed), but to make a long story short, the
limit of F is the set of sections of the simplicial projection
map p between the nerves of E and B, part of which looks like

                 -->
             E_1 --> E_0
              | |
         p_1 | | p_0
              v v
             B_1 --> B_0 .
                 -->

Each section s of p is uniquely determined by its 0-dimensional
part s_0 (because of the unique-arrow lifting property of p
mentioned above), but not all sections s_0 of p_0 extend to
sections s of p. In short, sections s are matching families:
tuples <x_b in Fb>_(b), indexed by objects b of B, which satisfy
the matching condition:

           F(f)(x_b) = x_c

whenever f: b --> c is an arrow of B.

How satisfactory all this is from a "visual" standpoint I leave
to the individual reader to decide.

Todd Trimble