Re: Partial functors ..

[Robin Cockett <robin@ucalgary.ca>]

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 <dyetter@ksu.edu> 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 <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
>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>
>

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<div dir=3D"ltr">Ah I did say that this is certainly not the only alternati=
ve!!!!<div><br></div><div>David&#39;s suggestion also works and is interest=
ing in its own right. =C2=A0</div><div><br></div><div>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 -&gt; 1 -&gt; 2 -&gt;=
  3 ) which consists of the identity maps and the map 0 -&gt; 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&#39; \subseteq \X =3D&gt; f,g \in \X&#39;). =C2=A0=C2=A0</div><div><br></=
div><div>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 ....</div><div><br></div><div>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!</div><div><br></div><div>-robin</div></div><div class=3D"gmail=
_extra"><br><div class=3D"gmail_quote">On Tue, Mar 17, 2015 at 9:04 AM, Dav=
id Yetter <span dir=3D"ltr">&lt;<a href=3D"mailto:dyetter@ksu.edu" target=
=3D"_blank">dyetter@ksu.edu</a>&gt;</span> wrote:<br><blockquote class=3D"g=
mail_quote" style=3D"margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-l=
eft:1ex">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).<br>
<br>
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.<br>
<br>
Natural transformations between (zero-preserving) functors from C~ to D~ wo=
uld<br>
then give a reasonable notion of partial natural transformations.=C2=A0 It =
certainly captures some, at least, of the natural transformations &quot;mor=
e partial&quot;=C2=A0 than their source functor, since there will be a zero=
  natural transformation between any two partial functors, corresponding to =
a &quot;defined nowhere&quot; 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.<br=
>
<br>
I&#39;m not sure how this fits with the restrictions Robin points out.=C2=
=A0 It seems to allow more partial natural transformations than Robin&#39;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).<br>
<br>
Best Thoughts,<br>
David Yetter<br>
________________________________________<br>
From: Robin Cockett &lt;<a href=3D"mailto:robin@ucalgary.ca">robin@ucalgary=
.ca</a>&gt;<br>
Sent: Monday, March 16, 2015 6:12 PM<br>
To: Categories list<br>
Subject: categories: Partial functors ..<br>
<br>
David Leduc &lt;david.leduc6 &lt;at&gt; <a href=3D"http://googlemail.com" t=
arget=3D"_blank">googlemail.com</a>&gt; writes:<br>
<br>
&gt; A partial functor from C to D is given by a subcategory S of C and a<b=
r>
&gt; functor from S to D. What is the appropriate notion of natural<br>
&gt; transformation between partial functors that would allow to turn small=
<br>
&gt; categories, partial functors and those &quot;natural transformations&q=
uot; into<br>
&gt; a bicategory? The difficulty is that two partial functors from C to D<=
br>
&gt; might not have the same definition domain.<br>
<br>
<br>
Here is a basic and quite natural interpretation (if someone has not<br>
already pointed this out):<br>
<br>
One can have a n.t=C2=A0 F =3D&gt; G iff F is less defined than G and on th=
eir common<br>
domain (which is just the domain of F) there is a natural transformation<br=
>
from F =3D&gt; \rst{F} G.=C2=A0 =C2=A0Partial functors, of course, form a r=
estriction<br>
category so they are naturally partial order enriched (by restriction).<br>
This 2-cell structure must simply respect this partial order ...<br>
<br>
This is certainly not the only possibility, unfortunately ... for example<b=
r>
why not also allow partial natural transformations ... which are less<br>
defined than the functor.=C2=A0 =C2=A0Here one does have to be a bit carefu=
l: a<br>
natural transformation must &quot;know&quot; the subcategory it is working =
with ...<br>
thus defining the natural transformation as a function on arrows (rather<br=
>
than just objects) is worthwhile adjustment (see MacLane page 19, Excercise=
<br>
5).=C2=A0 This then works too ....<br>
<br>
I hope this helps.<br>
<br>
-robin<br>
<br>
<br>
<br>
[For admin and other information see: <a href=3D"http://www.mta.ca/~cat-dis=
t/" target=3D"_blank">http://www.mta.ca/~cat-dist/</a> ]<br>
<br>
</blockquote></div><br></div>

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