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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]