Partial functors ..

[Robin Cockett <robin@ucalgary.ca>]

David Leduc <david.leduc6 <at> googlemail.com> writes:

> 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.


Here is a basic and quite natural interpretation (if someone has not
already pointed this out):

One can have a n.t  F => G iff F is less defined than G and on their common
domain (which is just the domain of F) there is a natural transformation
from F => \rst{F} G.   Partial functors, of course, form a restriction
category so they are naturally partial order enriched (by restriction).
This 2-cell structure must simply respect this partial order ...

This is certainly not the only possibility, unfortunately ... for example
why not also allow partial natural transformations ... which are less
defined than the functor.   Here one does have to be a bit careful: a
natural transformation must "know" the subcategory it is working with ...
thus defining the natural transformation as a function on arrows (rather
than just objects) is worthwhile adjustment (see MacLane page 19, Excercise
5).  This then works too ....

I hope this helps.

-robin


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