Re: Dualizing comma categories

[Steve Lack <steve.lack@mq.edu.au>]

Dear David,

As usual,  constructing colimits in Cat (and other concrete categories or 2-categories)
is more difficult than constructing limits. 

As you suspected, a cocomma object over the initial object is just a coproduct.  This is
completely general, not just true in Cat. Seen as a general 2-categorical fact, it becomes
the same fact that you mentioned: a comma object over a terminal object is just a product. 

I won't try to describe the general cocomma object, but another special case of a cocomma 
object is the *collage* of an arrow f:A-->B. This is the universal diagram containing arrows 
i:A->C and j:B->C and a 2-cell jf->i. It can be seen as a cocomma object of f and the identity
1_A. This special case is easy to describe in Cat. The object-set of C is the disjoint 
union of the object-sets of A and of B. A morphism in C between objects of A is a 
morphism in A; a morphism in C between objects of B is a morphism of B. There is
a morphism fa->a for each a in A, and these are the only morphisms from objects
of B to objects of A; there are no morphisms from objects of A to objects of B. 
(Thus the morphisms can be described by the 2x2 matrix with entries A, f, 0, B; this can 
be made precise if you think of the underlying span of a category as a matrix.)

Steve Lack.

On 30/10/2011, at 11:34 PM, David Leduc wrote:

> Hi,
> 
> A comma category is a comma object in the 2-category Cat of categories
> and functors. And a comma object is defined by a universal property.
> Now, one can dualize the notion of comma object by turning around the
> 1-cells and/or 2-cells in its definition. My question is: when we
> instantiate those dualized definitions to Cat, what do we obtain? In
> other words, what is a "co-comma category"?
> 
> For example, since the product of two categories is a special case of
> comma category, I would expect that the coproduct of two categories is
> a special case of "co-comma category".
> 
> Thanks!
> 



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