Re: Partial functors ..

["Sergei Soloviev" <Sergei.Soloviev@irit.fr>]

Dear David,

of recent papers, there are for exemple

K. Dosen, Z. Petric
Coherence in substructural categories, Studia Logica, vol.
> 70 (2002), pp. 271-296 (available at: http://arXiv.org)
>
> where the diagonal case (Relevant categories) is treated separately. The
> cartesian case is treated in
>
> K. Dosen, Z. Petric, The maximality of cartesian categories, Mathematical
> logic Quarterly, vol. 47 (2001), pp. 137-144 (available at:
> http://arXiv.org)
>
> and also in our book Proof-Theoretical Coherence, Chapter 9. (available at
> http://www.mi.sanu.ac.rs/~kosta/coh.pdf).
>
There is also a recent paper
>
K. Dosen, On Sets of Premises  (available at: http://arXiv.org)

As to the historical aspect, there was a theorem in an old paper by Grigori Mints,
published in Russian in 1980
(reprinted in "Selected papers in Proof Theory", Bibliopolis, 1992), but I have
to find the paper and exact formulation of his theorem.

regards,

Sergei Soloviev



Le Mardi 17 Mars 2015 16:04 CET, David Yetter <dyetter@ksu.edu> a écrit:

> 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
> ________________________________________
> From: Robin Cockett <robin@ucalgary.ca>
> Sent: Monday, March 16, 2015 6:12 PM
> To: Categories list
> Subject: categories: Partial functors ..
>
> 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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