Partial functor

Partial functor

[David Leduc]

Hi,

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.


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RE: Partial functor

[Carchedi, D.J. (Dave)]

David,

My guess would be to start with the tricategory of spans of categories:

The objects are categories, and a 1-morphism from C to D is category E with  two functors, F:E \to C and G:E \to D, and composition is given by computing 2-categorical pullbacks of categories (so is associative only up to equivalence), 2-morphisms between a span (F,G):E \to C \times D and a span (F',G'):E' \to C \times D are given by a functor H:E \to E' together with natural isomorphisms making everything commute, and 3-morphisms between two such  2-morphisms H,H':E \to E' (I'm suppressing the natural isomorphisms in my notation, but they're still there) are natural transformations compatible with the natural isomorphisms associated to H and H'.

Now a partial functor from C to D is a particular case of a span, F:E \to C  and G:E \to D, but where F is required to be full and faithful. Since Span(Cat) is a tricategory, Hom_Span(Cat)(C,D) is a bicategory. Take the full sub-bicategory of Hom_Span(Cat)(C,D) on those spans (F,G) where F is full and faithful. This is a bicategory of partial functors.

-Dave 

________________________________________
From: David Leduc [david.leduc6@googlemail.com]
Sent: Monday, November 07, 2011 1:55 PM
To: categories
Subject: categories: Partial functor

Hi,

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.


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Re: Partial functor

[Steve Lack]

Dear David,

There are many possible meanings of partial morphism between categories,
depending on  what meaning you attach to "subcategory". Different meanings
will be appropriate depending on the applications in question.

One possibility, which I believe was first considered by Lawvere, is to take "subcategory" 
of C to be a discrete opfibration over C. The resulting 2-category is described in 
detail in the appendix of 

Stephen Lack and Ross Street, The formal theory of monads II, JPAA 175:243-265, 2002.

where it is also shown that these partial maps are classified, in a suitable sense,
by the Fam construction. 

Regards,

Steve Lack.


On 07/11/2011, at 11:55 PM, david leduc wrote:

> Hi,
> 
> 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.
> 

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Re: Partial functor

[Christopher King]

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

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

I know this is late, but I find a quite obvious notion for it. Why not turn
your partial functor into a regular functor from C->D+1 (1 and + are the
terminal object and coproduct in the category of categories.) Now you can just
use regular natural transformations.



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Re: Re: Partial functor

[Uwe Egbert Wolter]

On 2015-03-15 18:01, Christopher King wrote:
> David Leduc <david.leduc6 <at> googlemail.com> writes:
>
>> Hi,
>>
>> 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.
>>
>> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>>
>>
> I know this is late, but I find a quite obvious notion for it. Why not turn
> your partial functor into a regular functor from C->D+1 (1 and + are the
> terminal object and coproduct in the category of categories.) Now you can just
> use regular natural transformations.
>

I think this construction will not work since the set-theoretic
difference C\S is not a subcategory of C while 1 is a subcategory of D+1.

The collection of all partial functors from C to D is a partial ordering
due to the inclusion of definition domains. For each subcategory S of C
you have the functor category [S->C] and each inclusion functor
In_S,S':S->S' gives you a functor from [S'->D] into [S->D]. Combining
both structures (via an appropriate variant of the Grothendieck
construction] you should get a category with objects all partial
functors (S,F:S->D) and morphisms (In_S,S', \alpha:F =>
In_S,S';F'):(S,F)->(S',F'). Composition of partial functors is given by
pullback (inverse image) construction.

I don't know if this gives a bicategory put maybe it helps to have a
look in the paper of Barry Jay "Partial Functions, Ordered Categories,
Limits and Cartesian Closure (1993) "

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.48.6433

Uwe


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Re: Partial functor

[Giorgio Mossa]

On Sun, Mar 15, 2015 at 05:01:58PM +0000, Christopher King wrote:
> David Leduc <david.leduc6 <at> googlemail.com> writes:
>
>>
>> Hi,
>>
>> 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.
>>
>> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>>
>>
>
> I know this is late, but I find a quite obvious notion for it. Why not turn
> your partial functor into a regular functor from C->D+1 (1 and + are the
> terminal object and coproduct in the category of categories.) Now you can just
> use regular natural transformations.
>

If your idea is to mimic the construction used for modelling partial function
as (total) function in Kleisli category for the monad (- ??? 1) in Set then this does
not work in Cat.
The reason is that a functor P : C ??? D ??? 1 in order to correspond to a partial
functor P' : S ??? C ??? D should send the category S in D and al the other stuff in 1,
nonetheless is ?? : s ??? c is a morphism from an object of S to an object in C ??? S
there is no way to map ?? in a morphism in D ??? 1 (= D + 1 in your notation),
because the two subcategories D and 1 in D ??? 1 are disjoint/disconnected
and s should be mapped in D while c should be mapped in 1.

Best regards

Giorgio



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Re: Partial functor

[Fred E.J. Linton]

An issue with Christopher King's proposal, below,
is what to do for a map between an object of S and 
an object of C not in S.

Cheers, -- Fred 

---

------ Original Message ------
Received: Mon, 16 Mar 2015 08:59:05 AM EDT
From: Christopher King <G.nius.ck@gmail.com>
To: <categories@mta.ca>
Subject: categories: Re: Partial functor

> David Leduc <david.leduc6 <at> googlemail.com> writes:
> 
>>
>> Hi,
>>
>> 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.
>>
>> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>>
>>
> 
> I know this is late, but I find a quite obvious notion for it. Why not turn
> your partial functor into a regular functor from C->D+1 (1 and + are the
> terminal object and coproduct in the category of categories.) Now you can
just
> use regular natural transformations.




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