Dear Michael, I do not remember your original question, but here is an answer to this. Let C be Cat^op and E be the arrow category 2. It's easier to work in Cat itself. Then we are interested in the full subcategory consisting of all categories X which admit a presentation I.2 --> J.2 --> X --> where I and J are sets, and "." is cotensor: e.g. J.2 denotes the coproduct of J copies of 2. But a category admits such a presentation if and only if it is free on a graph, and the free categories are of course not closed under coequalizers. Steve.
-----Original Message----- From: cat-dist@mta.ca [mailto:cat-dist@mta.ca] On Behalf Of Michael Barr Sent: Monday, May 12, 2008 10:34 PM To: Categories list Subject: categories: Further to my question on adjoints
In March I asked a question on adjoints, to which I have received no correct response. Rather than ask it again, I will pose what seems to be a simpler and maybe more manageable question. Suppose C is a complete category and E is an object. Form the full subcategory of C whose objects are equalizers of two arrows between powers of E. Is that category closed in C under equalizers? (Not, to be clear, the somewhat different question whether it is internally complete.)
In that form, it seems almost impossible to believe that it is, but it is surprisingly hard to find an example. When E is injective, the result is relatively easy, but when I look at examples, it has turned out to be true for other reasons. Probably there is someone out there who already knows an example.
Michael