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• CommentRowNumber1.
• CommentAuthorFosco
• CommentTimeOct 29th 2013

I have a couple of lines to write about the question I made on MO some times ago. What would you advice me to do?

1. Create a stub-page here in nlab, pointing to join of categories
2. Expand the same page adding the “profunctorial” POV
3. Open another question on MO, linking the original topic and explaining what I would like to prove.
4. Other?

Here is what I want to prove: the join of categories can be neatly described to be the cograph of the terminal profunctor, and as such it enjoys a number of properties: it is a bifunctor $Cat\times Cat\to Cat$ (in fact, to $Cat/I$ where $I$ is the walking arrow”), and as such it can be characterized as the right adjoint to the functor $i^*$ induced by the inclusione $\partial I\subset I$ (many of you will recognize Joyal’s touch in this definition!). Moreover, any $A\star-$ has a right adjoint, and the same is true for $-\star B$, for categories $A,B\in Cat$.

Now I would like to prove the same results in the more general setting of the “collage” of categories along a generic profunctor. But I’m rather unsatisfied since in that setting it seems I must specify not only the functor $F\colon C\to C'$ to induce $C\star_W D\to C'\star_{W'}D$ (notice that $W\neq W'$!), but also a morphism of profunctors $W\Rightarrow W'\circ F^\uparrow$, where $F^\uparrow$ is the profunctor$C'(F-,=)$. Not to mention the fact that a priori there’s would be no counterpart for $i$.

Maybe the right thing to do would be adding a paragraph to the existing page join of categories, and then open another thread on MO pointing to that page?

• CommentRowNumber2.
• CommentAuthorFosco
• CommentTimeOct 29th 2013
• (edited Oct 29th 2013)

I got it! I pray for the attention of mr. Shulman, as I found out that what I need to formalize the situation is the proarrow equipment $(\Cat, \Dist)$.

I explained in full detail what I have in mind here.

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeOct 29th 2013

Yes, equipments and double categories are a good way to think about the universal property of the collage of a profunctor. Some references you may be interested in:

• Ross Street, “Cauchy characterization of enriched categories” TAC reprint
• Grandis-Pare, “Limits in double categories”
• Garner-Shulman, “Enriched categories as a free cocompletion”, 2013
• Recent papers of Susan Niefield

Sorry I don’t have time to write more.

• CommentRowNumber4.
• CommentAuthorFosco
• CommentTimeOct 30th 2013

I tried to skim over Wood’s “Abstract proarrows”, but it seems quite hard-to-read; no problem if you are busy, just tell me if according to you my questions are well-suited for some other people. :)

Many thanks!

Fosco

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeOct 30th 2013

I don’t understand, are you asking me another question?

• CommentRowNumber6.
• CommentAuthorFosco
• CommentTimeOct 30th 2013

In some sense yes. I managed to state in a precise form my questions about the link between (weighted) limits and weighted joins, i.e. collages along profunctors. But I don’t want to bother you! I simply would appreciate an help, since I feel like a beginner, and I am alone here. I was wondering whether there is somebody among the authors you quoted me (Street, Pare’, Grandis, Garner, Niefeld, others?) that I can contact and discuss with.

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeOct 30th 2013

I think that reading the papers I mentioned would help answer your questions, but yes, some of them are difficult to read. You don’t need to feel bad about bothering me; on the contrary, I feel bad that I don’t have the time right now to help more. Maybe someone else here, or at MO, would be able to help; or you could try asking on the categories email list, perhaps.