* Re: Defining composition via colimit [not found] ` <a31ae7ae-b5f5-4f86-9b4f-0b36adab1ea7-0@mqoutlook.onmicrosoft.com> @ 2023-11-03 10:29 ` JS Lemay 0 siblings, 0 replies; 3+ messages in thread From: JS Lemay @ 2023-11-03 10:29 UTC (permalink / raw) To: Categories mailing list [-- Attachment #1: Type: text/plain, Size: 2620 bytes --] [[The following message is sent on behalf of streicher@mathematik.tu-darmstadt.d -- for whatever reason it did not seem to have been sent/approved properly 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=EscalatedMessage&action=leave&GuestId=4d7e0354-0b41-4b99-be68-d7eebf3df1ae<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=4d7e0354-0b41-4b99-be68-d7eebf3df1ae> > [-- Attachment #2: Type: text/html, Size: 3722 bytes --] ^ permalink raw reply [flat|nested] 3+ messages in thread
* Signing off @ 2023-10-26 14:14 Bob Rosebrugh 2023-11-02 18:55 ` Defining composition via colimit Jamie Vicary 0 siblings, 1 reply; 3+ messages in thread From: Bob Rosebrugh @ 2023-10-26 14:14 UTC (permalink / raw) To: categories Dear Colleagues, From March 1990 until recently it was my privilege to moderate the categories mailing list. For a couple of years it has been my intention to pass the list to a new moderator, but I was too slow to act. Changes to IT service at my former employer, Mount Allison University, abruptly made it impossible for me to continue running the list. Those changes were unexpected for me, but I have been away from the campus for over three years and was not aware. The IT staff at Mount Allison have always been very supportive of the list and I thank them for their more than three decades of generous assistance. It is a reassuring pleasure to know that the list is now in the capable hands of JS Lemay who kindly accepted the invitation to revive it. I am very grateful. Moreover, there is no one better suited to the role and our community is fortunate to have him moderating. I am also very happy that the new home of the list is Macquarie. Best wishes to everyone. I am hoping to see many of you in the not distant future, Bob Rosebrugh ---------- 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=EscalatedMessage&action=leave&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b ^ permalink raw reply [flat|nested] 3+ messages in thread
* Defining composition via colimit 2023-10-26 14:14 Signing off Bob Rosebrugh @ 2023-11-02 18:55 ` Jamie Vicary 2023-11-02 23:23 ` Richard Garner 0 siblings, 1 reply; 3+ messages in thread From: Jamie Vicary @ 2023-11-02 18:55 UTC (permalink / raw) To: categories Dear all, 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=EscalatedMessage&action=leave&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b ^ permalink raw reply [flat|nested] 3+ messages in thread
* Re: Defining composition via colimit 2023-11-02 18:55 ` Defining composition via colimit Jamie Vicary @ 2023-11-02 23:23 ` Richard Garner 0 siblings, 0 replies; 3+ messages in thread From: Richard Garner @ 2023-11-02 23:23 UTC (permalink / raw) To: jamievicary; +Cc: categories Hi Jamie, That does sound interesting. Did you look at: - Kelly, G. M., & Le Creurer, I. J. On the monadicity over graphs of categories with limits. Cahiers de Topologie et G\'eom\'etrie Diff\'erentielle Cat\'egoriques, 1997(38), 179–191. This paper, among other things, shows that categories with limits indexed by finite acyclic directed graphs are monadic over directed graphs. It seems like you could be describing an algebra structure for this monad (which is a fortiori a category). The paper gives a fairly explicit presentation of this monad by generators and relations, so you should be able to check fairly easily if that is indeed what is going on. Richard Jamie Vicary <jamievicary@gmail.com> writes: > Dear all, > > 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 ^ permalink raw reply [flat|nested] 3+ messages in thread
end of thread, other threads:[~2023-11-03 10:30 UTC | newest] Thread overview: 3+ messages (download: mbox.gz / follow: Atom feed) -- links below jump to the message on this page -- [not found] <f93fc7025b48c018f4d644ae95d2f11e.squirrel@webmail.mathematik.tu-darmstadt.de> [not found] ` <a31ae7ae-b5f5-4f86-9b4f-0b36adab1ea7-0@mqoutlook.onmicrosoft.com> 2023-11-03 10:29 ` Defining composition via colimit JS Lemay 2023-10-26 14:14 Signing off Bob Rosebrugh 2023-11-02 18:55 ` Defining composition via colimit Jamie Vicary 2023-11-02 23:23 ` Richard Garner
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