14
votes
Accepted
A multicategory is a ... with one object?
This has been called a "fc-multicategory" by Tom Leinster, for example here.
I think this as also been called a "Hypervirtual double category" here, but I don't remember if this is exactly the same ...
- 42.4k
11
votes
Prof and the completion of Cat under right adjoints
The idea that passing from $Cat$ to $Prof$ is a great way to give functors adjoints features prominently in Richard Wood's theory of proarrow equipments. Rosebrugh and Wood showed in Proarrows and ...
- 63.9k
10
votes
A multicategory is a ... with one object?
Another established structure very close to what you want is opetopic bicategories, which are equivalent to classical bicategories but formulated opetopically; see e.g. §3 of Cheng 2003, Opetopic ...
- 21.3k
8
votes
Accepted
The Kan construction, profunctors, and Kan extensions
I will try to answer the second question.
Prop 1. Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. Moreover $\text{lan}_if$ is ...
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6
votes
A multicategory is a ... with one object?
I think it worth mentioning that precisely the notion described in the question is given in Cockett–Koslowski–Seely's Morphisms and modules for poly-bicategories, where it is called a multi-bicategory....
- 10.6k
6
votes
Accepted
Relations with "for each" composition and its properties (coming from profunctors with end composition)
Your operation $\square$ defined with $\forall$ and $\land$ is associative, because both $a(R\square(S\square T))d$ and $a((R\square S)\square T)d$ expand to
$$ (\forall b_1.\ a R b_1)\land(\forall ...
- 8,481
5
votes
Relations with "for each" composition and its properties (coming from profunctors with end composition)
If you define the universal composition with respect to the disjunction, rather than the conjunction, then you do get another category structure as Paul mentioned.
In this case, the identities are ...
- 169
5
votes
Accepted
Comonoid homomorphisms in the bicategory of profunctors
It's helpful to consider the case of posets and $\mathbf{2}$-enriched profunctors between them, where $\mathbf{2}$ is the cartesian closed poset $(0 \leq 1)$. (This obviates the complications of ...
- 53.3k
4
votes
Accepted
Profunctors as a Kleisli bicategory
This beautiful story take place in $\mathsf{Cat}$, the bicategory of locally small categories. There, the construction of small presheaves induces a pseudomonad whose pseudoalgebrabas are cocomplete ...
- 9,111
4
votes
Profunctors and multicategories
Hyland's Elements of a theory of algebraic theories describes a precise connection between multicategories and $\mathbf{Prof}$ in Section 4.3. I shall briefly describe the intuition; a complete ...
- 10.6k
4
votes
Accepted
Ends and coends – analogues for higher arity – Horn Filling
This is exactly the subject of the paper Coends of higher arity by Loregian and de Oliveira Santos.
- 10.6k
3
votes
Accepted
The $\mathfrak L$ functor on $\textsf{Prof}$
This is the right Kan extension of $\hom_A: A \nrightarrow A$ along $K: A \nrightarrow B$ in the bicategory of profunctors. Which is to say that for every profunctor (aka bimodule) $L: B \nrightarrow ...
- 53.3k
3
votes
Accepted
Does $\bf Prof$ admit all pseudolimits?
There is another simple kind of peusdo-limits called inverters, in which one universally inverts a 2-morphism between two parallel 1-morphisms (such pseudo-limits can also be constructed in a simple ...
- 9,481
3
votes
Is there a name for this variant of the category of elements of a profunctor?
It might be enough to work out just the case of $P=\hom$!
I'll just leave an idea here since I don't have time to check it properly and I'll be glad if you took the fun away from me (you nerd-sniped ...
- 13.6k
3
votes
Accepted
Strictness of two operations on proarrow equipments
I believe the answer to (2) is yes.
First, apply the strictification theorem for bicategories twice, to make composition of arrows and proarrows both strictly associative. Thus, when our equipment is ...
- 66.7k
2
votes
Adjunctions with respect to profunctors
I remember I tried to work with this definition for a while when I still believed in the notion of relative category. Under some coherence assumptions, your notion is related to the classical notion ...
- 9,111
2
votes
Prof and the completion of Cat under right adjoints
I shall sketch out a proof that $\mathbf{Prof}$ is almost obtained from $\mathbf{Cat}$ by adjoining right adjoints to every 1-cell, following Roald Koudenburg's suggestions in the comments. The ...
- 10.6k
2
votes
Prof and the completion of Cat under right adjoints
I discovered a related characterisation in Betti's Formal theory of internal categories. For $\mathcal E$ a finitely complete category, Betti claims (in the theorem at the top of page 49) that $\...
- 10.6k
2
votes
Profunctors as a Kleisli bicategory
A nice account of this is given in Martin Hyland's Elements of a theory of algebraic theories.
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1
vote
The Kan construction, profunctors, and Kan extensions
What follows is kind of preliminary and complementary to Ivan's answer.
Let $A$ be a small category, $\mathcal B$ and $\mathcal C$ be locally presentable categories
and $i\colon A \to \mathcal B$ and ...
- 604
1
vote
The Kan construction, profunctors, and Kan extensions
Edit. The construction of $\eta$ below is not correct. See the comments.
Ad 2. I think that indeed $\mathrm{Lan}_G(F):\mathbf{C} \to \mathbf{B}$ is left adjoint to $\mathrm{Lan}_F(G):\mathbf{B} \to \...
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