RE: Further to my question on adjoints

["Stephen Lack" <S.Lack@uws.edu.au>]

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
> 
> 
>