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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?
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?
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.
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:
Sorry I don’t have time to write more.
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
I don’t understand, are you asking me another question?
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.
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.
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