Re: Dualizing comma categories

[Michael Shulman <mshulman@ucsd.edu>]

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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