From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8564 Path: news.gmane.org!not-for-mail From: Robin Cockett Newsgroups: gmane.science.mathematics.categories Subject: Re: Partial functors .. Date: Tue, 17 Mar 2015 14:31:50 -0600 Message-ID: References: Reply-To: Robin Cockett NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: ger.gmane.org 1426681547 18942 80.91.229.3 (18 Mar 2015 12:25:47 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 18 Mar 2015 12:25:47 +0000 (UTC) Cc: Categories list To: David Yetter Original-X-From: majordomo@mlist.mta.ca Wed Mar 18 13:25:34 2015 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.127]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1YYD2T-0006O4-KT for gsmc-categories@m.gmane.org; Wed, 18 Mar 2015 13:25:33 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:43996) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1YYD1c-0003Bb-K4; Wed, 18 Mar 2015 09:24:40 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1YYD1a-0006KM-K7 for categories-list@mlist.mta.ca; Wed, 18 Mar 2015 09:24:38 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8564 Archived-At: Ah I did say that this is certainly not the only alternative!!!! David's suggestion also works and is interesting in its own right. Adding a zero to categories is a monad on the category of (small) categories which is (as far as I can see) is a perfectly good example of a partial map classifier. However it does not classify ALL subcategories: once f is taken to the zero map then all composites gfh must be taken to the zero map. This means that the subcategory of 4 (i.e. the linear order 0 -> 1 -> 2 -> 3 ) which consists of the identity maps and the map 0 -> 3 cannot be classified (as the middle map is taken to zero). The subcategories which CAN be classified are exactly those which are factor closed (i.e. fg in \X' \subseteq \X => f,g \in \X'). BTW: The original suggestion seen in this light also works: adding a disconnected object, \X +1, is also a partial map classifier. However, the subcategories classified must be connection closed (i.e. a disjoint sum component of the category). This may not have been quite the original objective .... Of course, Cat does not have a partial map classifier for ALL subcategories ... however, this does not stop one from building partial functor categories using all partial functors which have perfectly happy natural transformations :-) ... but there are still some choices to make along the natural transformation road. All choices lead to interesting alternatives ... some more interesting than others! -robin On Tue, Mar 17, 2015 at 9:04 AM, David Yetter wrote: > 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 > Sent: Monday, March 16, 2015 6:12 PM > To: Categories list > Subject: categories: Partial functors .. > > David Leduc 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 > > > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] > > --001a11c133443cbef6051181db4e Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
Ah I did say that this is certainly not the only alternati= ve!!!!

David's suggestion also works and is interest= ing in its own right. =C2=A0

Adding a zero to cate= gories is a monad on the category of (small) categories which is (as far as= I can see) is a perfectly good example of a partial map classifier.=C2=A0 = However it does not classify ALL subcategories: once f is taken to the zero= map then all composites gfh must be taken to the zero map. =C2=A0 This mea= ns that the subcategory of 4 (i.e. the linear order 0 -> 1 -> 2 ->= 3 ) which consists of the identity maps and the map 0 -> 3 cannot be cl= assified (as the middle map is taken to zero).=C2=A0 The subcategories whic= h CAN be classified are exactly those which are factor closed (i.e. fg in \= X' \subseteq \X =3D> f,g \in \X'). =C2=A0=C2=A0

BTW: The original suggestion seen in this light also works: adding= a disconnected object, \X +1, is also a partial map classifier.=C2=A0 Howe= ver, the subcategories classified must be connection closed =C2=A0(i.e. a d= isjoint sum component of the category).=C2=A0 This may not have been quite = the original objective ....

Of course, Cat does no= t have a partial map classifier for ALL subcategories ... however, this doe= s not stop one from building partial functor categories using all partial f= unctors which have perfectly happy natural transformations :-) ... but ther= e are still some choices to make along the natural transformation road.=C2= =A0 All choices lead to interesting alternatives ... some more interesting = than others!

-robin

On Tue, Mar 17, 2015 at 9:04 AM, Dav= id Yetter <dyetter@ksu.edu> wrote:
The previous suggestion of considering functors to D + 1 was a fal= se start for reasons Fred and Uwe pointed out, but it suggests a better app= roach:=C2=A0 consider functors to the category D~ formed from D by freely a= djoining a zero object.=C2=A0 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-preser= ving functors from C~ to D~, and letting C and D range over all small categ= ories gives=C2=A0 a category isomorphic to that of small categories with pa= rtial=C2=A0 functors as arrows.

Natural transformations between (zero-preserving) functors from C~ to D~ wo= uld
then give a reasonable notion of partial natural transformations.=C2=A0 It = certainly captures some, at least, of the natural transformations "mor= e partial"=C2=A0 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=C2=A0 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.=C2= =A0 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 transf= ormation 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&q= uot; into
> a bicategory? The difficulty is that two partial functors from C to D<= br> > 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=C2=A0 F =3D> G iff F is less defined than G and on th= eir common
domain (which is just the domain of F) there is a natural transformation from F =3D> \rst{F} G.=C2=A0 =C2=A0Partial functors, of course, form a r= estriction
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.=C2=A0 =C2=A0Here one does have to be a bit carefu= l: 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).=C2=A0 This then works too ....

I hope this helps.

-robin



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


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