Re: equality is beautiful

[Michael Shulman <shulman@uchicago.edu>]

That's a good point.  However, if C is a non-strict category, then
while you can define products over its preset of objects, such a
product is no longer necessarily a particular case of a limit, since
the preset may not have any "discrete" category structure.  So while
you can construct limits over arbitrary (non-strict) categories via
"products" and equalizers if you generalize the notion of "product" in
this way, the converse now fails -- having all limits doesn't seem to
guarantee that you have all "products" in this generalized sense.

Mike

On Tue, Mar 16, 2010 at 3:03 AM, Richard Garner <rhgg2@hermes.cam.ac.uk> wrote:
>> To rephrase what Toby said: the construction of limits via products and
>> equalizers only works for limits over a domain category which has a
>> set(oid) of objects (what Toby calls a "strict category"), whether that
>> set is large or small.
>
> Is this really the case? Given any type (=preset) A and any
> term A --> ob C (for C a non-strict category), one can define
> what it means to be a product of this family of objects in C.
> Now given a non-strict category J and a functor F:J->C, one
> may construct the limit of F as an equaliser of two morphisms
> between products in the usual way. I don't see where equality
> on objects is necessary, or even useful.
>
> Richard
>
>
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