Dualizing comma categories

Dualizing comma categories

[David Leduc]

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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Re: Dualizing comma categories

[Steve Lack]

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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Re: Dualizing comma categories

[Michael Shulman]

On Sun, Oct 30, 2011 at 3:29 PM, Steve Lack <steve.lack@mq.edu.au> wrote:
> another special case of a cocomma
> object is the *collage* of an arrow f:A-->B.

More generally, if H: A --|--> B is a
profunctor/distributor/module/relator/correspondence/etc., then it has
a collage, whose objects are the disjoint union of those of A and B,
and whose morphisms are built out of those in A, B, and the elements
of the image of H, as a "matrix" like Steve describes:
http://nlab.mathforge.org/nlab/show/cograph+of+a+profunctor
This gives a cospan A --> coll(H) <-- B.  The coproduct of two
categories is the special case of the collage of the empty profunctor.

On the other hand, by the generalized Grothendieck construction, H
also gives rise to a span A <-- fib(H) --> B which is a discrete
two-sided fibration in the sense of Street:
http://nlab.mathforge.org/nlab/show/two-sided+fibration
As Street also pointed out in his paper "Fibrations in bicategories",
the collages of profunctors are exactly the COdiscrete two-sided
COfibrations.

The reason I mention this in the context of limits and colimits is
that one of the nice "exactness" properties of Cat is that coll(H) is
the cocomma object of the span with vertex fib(H), and fib(H) is the
comma object of the cospan with vertex coll(H).  Moreover, the
codiscrete cofibrations and discrete fibrations from A to B are the
fixed objects for an idempotent adjunction between spans and cospans.
So you can think of the cocomma object of an arbitrary span as "the
collage of the profunctor generated by that span."

I see you also asked about reversing the 2-cells in a comma object.
This doesn't give you anything new in terms of limits: the "op-comma
object" of two morphisms f and g is just the comma object of g and f
(in the other order).

Mike


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