Re: equality is beautiful

[Toby Bartels <toby+categories@ugcs.caltech.edu>]

David Leduc wrote:

>What about the characterization of limits in terms of products and
>equalizers? It states that the limit of a functor F:J->C is
>constructed by products indexed by the set(oid) of objects and the
>set(oid) of arrows of J. But if you don't allow equality on objects in
>J, you only have a preset of objects, not a set(oid).

Consider the analogy between small and strict categories.
(A category is strict if it is equipped with a notion of equality of objects.
Logically, this is a structure rather than a property like smallness.)

Often when speaking of small categories, one speaks relative to a universe
which is a collection of set(oid)s or a collection of set(oid) cardinalities,
so every small preset automatically comes equipped with a notion of equality.
In this case, every small category is strict.  Conversely,
every strict category is small relative to some universe,
if you accept an axiom such as Grothendieck's axiom of universes.
So these are very closely related concepts.

As is well known, we are most interested in the limit of F: J -> C for small J.
Less well known, but also true, we are most interested in it when J is strict.
In that case, there is no problem; the arrows of J form a set(oid),
and we can consider products indexed by that set(oid).
In principle, J doesn't have to be strict, any more than it has to be small,
but if you have some reason to believe that the limit exists,
then you can examine that reason to see what product is relevant.
Most of the time, you can just assume that J is small and strict.


--Toby


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