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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