On Sun, Mar 15, 2015 at 05:01:58PM +0000, Christopher King wrote:
David Leduc <david.leduc6 <at> googlemail.com> writes:
Hi,
A partial functor from C to D is given by a subcategory S of C and a functor from S to D. What is the appropriate notion of natural transformation between partial functors that would allow to turn small categories, partial functors and those "natural transformations" into a bicategory? The difficulty is that two partial functors from C to D might not have the same definition domain.
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I know this is late, but I find a quite obvious notion for it. Why not turn your partial functor into a regular functor from C->D+1 (1 and + are the terminal object and coproduct in the category of categories.) Now you can just use regular natural transformations.
If your idea is to mimic the construction used for modelling partial function as (total) function in Kleisli category for the monad (- ??? 1) in Set then this does not work in Cat. The reason is that a functor P : C ??? D ??? 1 in order to correspond to a partial functor P' : S ??? C ??? D should send the category S in D and al the other stuff in 1, nonetheless is ?? : s ??? c is a morphism from an object of S to an object in C ??? S there is no way to map ?? in a morphism in D ??? 1 (= D + 1 in your notation), because the two subcategories D and 1 in D ??? 1 are disjoint/disconnected and s should be mapped in D while c should be mapped in 1. Best regards Giorgio [For admin and other information see: http://www.mta.ca/~cat-dist/ ]