On 2015-03-15 18:01, 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.
I think this construction will not work since the set-theoretic difference C\S is not a subcategory of C while 1 is a subcategory of D+1. The collection of all partial functors from C to D is a partial ordering due to the inclusion of definition domains. For each subcategory S of C you have the functor category [S->C] and each inclusion functor In_S,S':S->S' gives you a functor from [S'->D] into [S->D]. Combining both structures (via an appropriate variant of the Grothendieck construction] you should get a category with objects all partial functors (S,F:S->D) and morphisms (In_S,S', \alpha:F => In_S,S';F'):(S,F)->(S',F'). Composition of partial functors is given by pullback (inverse image) construction. I don't know if this gives a bicategory put maybe it helps to have a look in the paper of Barry Jay "Partial Functions, Ordered Categories, Limits and Cartesian Closure (1993) " http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.48.6433 Uwe [For admin and other information see: http://www.mta.ca/~cat-dist/ ]