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1. This is basically a reference request.

What is a good source which describes the theory of colimits in the 2-category Cat? Here I mean colimits in the 2-categorical pseudo snese, but not necessarily the lax sense.

In usually category theory we know that if you have all coequalizers and all coproducts, then you have all colimits. Is there a similar statement here? Is there an explicit way to understand the colimit of a diagram of categories, maybe for certain fundamental sorts of colimits? What if I restrict myself to diagrams which are just ordinary 1-categories?

I am really interested in the case where we replace Cat be the 2-category of Abelian categories and additive functors. Where should I look to find these sorts of colimits?

• CommentRowNumber2.
• CommentAuthorDavidRoberts
• CommentTimeApr 12th 2012

Presumably there are ’co-’ analogues of PIE-limits and an analogous theorem about pseudocolimits. I don’t know the situation for either of the examples you mention, though.

• CommentRowNumber3.
• CommentAuthorTim_Porter
• CommentTimeApr 12th 2012
• (edited Apr 12th 2012)

Chris, have you looked at Chapter 4 of Tom Fiore’s book. Memoirs of the American Mathematical Society 182 (2006), no. 860.

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeApr 12th 2012

The nLab has a section on 2-colimits in Cat constructed via the Grothendieck construction, which presumably can be identified with Tom’s version.

For the construction of (co)limits out of (co)products and (co)equalizers in an arbitrary bicategory, see the Correction to Ross Street’s paper “Fibrations in bicategories”. (The paper itself gave an incorrect construction; the correction fixed it.)

2. Thanks everyone. This is quite helpful and will at least get me looking in the right places. Thanks again.