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/ ]
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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (7)
-
Carchedi, D.J. (Dave) -
Christopher King -
David Leduc -
Fred E.J. Linton -
Giorgio Mossa -
Steve Lack -
Uwe Egbert Wolter