Re: Re: Partial functor

[Uwe Egbert Wolter <Uwe.Wolter@uib.no>]

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