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Nov 2023 10:29:05.3737 (UTC) X-MS-Exchange-CrossTenant-fromentityheader: Hosted X-MS-Exchange-CrossTenant-id: 82c514c1-a717-4087-be06-d40d2070ad52 X-MS-Exchange-CrossTenant-mailboxtype: HOSTED X-MS-Exchange-CrossTenant-userprincipalname: QuFFxI48J/mfHPHVvuW8uQM0hAPonNYEigKuq9oH2gDm6xzssxzUWFasP1G1JNWxIO3LBOlAeFt9y18dbnDtfg== X-MS-Exchange-Transport-CrossTenantHeadersStamped: SYZPR01MB7627 X-Mimecast-Spam-Score: 0 X-Mimecast-Originator: mq.edu.au Content-Language: en-US Content-Type: multipart/alternative; boundary="_000_ME2PR01MB31229E56D7A2AC04E426C8CFCDA5AME2PR01MB3122ausp_" --_000_ME2PR01MB31229E56D7A2AC04E426C8CFCDA5AME2PR01MB3122ausp_ Content-Type: text/plain; charset=WINDOWS-1252 Content-Transfer-Encoding: quoted-printable [[The following message is sent on behalf of streicher@mathematik.tu-darmst= adt.d -- for whatever reason it did not seem to have been sent/approved pro= perly to the mailing list, apologies if you receive multiple copies]] Composition of distributors or profunctors would be an example. But composition is only defined up to isomorphism and so one gets a bicategory. This was done in the second half of the sixties by Benabou. But the situation can be rectified when redefining distributors from A to B as cocontinuous functors from Psh(A) to Psh(B). Thomas PS I take the opportunity to thank Bob for organising and moderating the categories list for such a long time. And thanks to the people who have taken over! I missed it for quite some time already! > In a current project we have the following situation. For a category > we are attempting to define, we know what the objects are, and also > the morphisms. Unfortunately we do not have an obvious composition > operation. What we do have is a "colimit" operation, which operates on > a directed graph labelled by our objects and morphisms, and returns a > putative colimit object equipped with a family of morphisms in the > usual way (or fails.) > > We then define the composite of morphisms A->B, B->C to be the > colimit of the diagram A->B->C. We then check that this composition > operation satisfies the axioms of a category, and that our earlier > colimit construction is indeed an actual colimit with respect to the > compositional structure. It seems that everything works fine, and we > are happy. > > My question is whether this has any precedent in the literature. The > situation as I have described it is a bit simplified, in reality there > is some higher categorical stuff going on. Personally I'm sure I've > read similar things in the literature in the past but I can't track > them down now that I actually need them. The nLab article on > "composition" has some stuff about this with regard to transfinite > composition, but we're not trying to do anything transfinite here. > > Best wishes, > Jamie > > > > ---------- > > You're receiving this message because you're a member of the Categories > mailing list group from Macquarie University. > > Leave group: > https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.= ashx?source=3DEscalatedMessage&action=3Dleave&GuestId=3D4d7e0354-0b41-4b99-= be68-d7eebf3df1ae > --_000_ME2PR01MB31229E56D7A2AC04E426C8CFCDA5AME2PR01MB3122ausp_ Content-Type: text/html; charset=WINDOWS-1252 Content-Transfer-Encoding: quoted-printable
[[The following message is sent on behalf of streicher@mathematik.tu-darmstadt.d -- for whatever reason it did no= t seem to have been sent/approved properly to the mailing list, apologies i= f you receive multiple copies]] 


Composition of distributors or profunctors would be an example. But composition is only defined up to isomorphism and so one gets a
bicategory. This was done in the second half of the sixties by Benabou.

But the situation can be rectified when redefining distributors from A to B=
as cocontinuous functors from Psh(A) to Psh(B).

Thomas

PS I take the opportunity to thank Bob for organising and moderating the categories list for such a long time.
And thanks to the people who have taken over! I missed it for quite some time already!

> In a current project we have the following situation. For a category > we are attempting to define, we know what the objects are, and also > the morphisms. Unfortunately we do not have an obvious composition
> operation. What we do have is a "colimit" operation, which o= perates on
> a directed graph labelled by our objects and morphisms, and returns a<= br> > putative colimit object equipped with a family of morphisms in the
> usual way (or fails.)
>
> We then define the composite of morphisms A->B, B->C to be the > colimit of the diagram A->B->C. We then check that this composit= ion
> operation satisfies the axioms of a category, and that our earlier
> colimit construction is indeed an actual colimit with respect to the > compositional structure. It seems that everything works fine, and we > are happy.
>
> My question is whether this has any precedent in the literature. The > situation as I have described it is a bit simplified, in reality there=
> is some higher categorical stuff going on. Personally I'm sure I've > read similar things in the literature in the past but I can't track > them down now that I actually need them. The nLab article on
> "composition" has some stuff about this with regard to trans= finite
> composition, but we're not trying to do anything transfinite here.
>
> Best wishes,
> Jamie
>
>
>
> ----------
>
> You're receiving this message because you're a member of the Categorie= s
> mailing list group from Macquarie University.
>
> Leave group:
> https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.as= hx?source=3DEscalatedMessage&action=3Dleave&GuestId=3D4d7e0354-0b41= -4b99-be68-d7eebf3df1ae
>

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