Re: Partial functors ..

[henry@phare.normalesup.org]

I don't this this works: if (g .g') is in the domain of the partial
functor and g is not, then g will be sent to 0 by the extension of the
partial functor, hence g.g' will also be sent to 0 which shouldn't be the
case because it is in the domain of the functor.

Also it is not entirely clear to me if the original question wanted
functor defined on a full subcategory or on an arbitrary sub-category. But
in both case, my opinion would be that the only reasonable notion is to
talk about "partial natural transformations" which are defined on a
subcategory included in the domain of definition of both functor (and
hence are natural transformation between ordinary functor).


Best wishes,
Simon Henry.


> The previous suggestion of considering functors to D + 1 was a false start
> for reasons Fred and Uwe pointed out, but it suggests a better approach:
> consider functors to the category D~ formed from D by freely adjoining a
> zero object.  Arrows not in S now have somewhere to go (the zero arrow
> with the appropriate source and target).
>
> I think at the one-categorical level, taking Hom(C,D) to be the
> zero-preserving functors from C~ to D~, and letting C and D range over all
> small categories gives  a category isomorphic to that of small categories
> with partial  functors as arrows.
>
> Natural transformations between (zero-preserving) functors from C~ to D~
> would
> then give a reasonable notion of partial natural transformations.  It
> certainly captures some, at least, of the natural transformations "more
> partial"  than their source functor, since there will be a zero natural
> transformation between any two partial functors, corresponding to a
> "defined nowhere" partial natural transformation when zero-ness is
> interpreted as undefined as  it was in the correspondence between
> zero-preserving functors from C~ to D~ and partial functors from C to D.
>
> I'm not sure how this fits with the restrictions Robin points out.  It
> seems to allow more partial natural transformations than Robin's
> observation, since zero arrows can fill in whenever the image object under
> either the source or target functor is undefined, a partial natural
> transformation to be a natural transformation between the restrictions of
> the two partial functors to the intersections of their domain of
> definition (or a subcategory thereof).
>
> Best Thoughts,
> David Yetter

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