Re: Partial functor

[Giorgio Mossa <mossa@poisson.phc.unipi.it>]

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.
>>
>> [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.
>

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



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