# Questions tagged [profunctors]

The profunctors tag has no usage guidance.

27
questions

6
votes

0
answers

71
views

### A distributor between categories induces a distributor between their categories of presheaves

Let $P$ be a distributor/profunctor from a small category $A$ to a small category $B$, i.e. a functor $P : B^\circ \times A \to \mathrm{Set}$.
We may then define a distributor from $[A^\circ, \mathrm{...

6
votes

2
answers

206
views

### Relations with "for each" composition and its properties (coming from profunctors with end composition)

$\newcommand{\sq}{\mathbin{\square}}$The usual composition $S\diamond R$ of two relations $R\colon A⇸B$ and $S\colon B⇸C$ is defined as follows:
For each $(a,c)\in A\times C$, we declare $a\sim_{S\...

7
votes

1
answer

170
views

### Comonoid homomorphisms in the bicategory of profunctors

Cartesian bicategories axiomatize the intuitively evident but mathematically elusive "cartesian" product on bicategories such as Rel, Span, and Prof. An important concept for cartesian ...

5
votes

1
answer

196
views

### Is there a name for this variant of the category of elements of a profunctor?

Let $\mathsf{C}$ be a category and let $P : \mathsf{C}^{\text{op}} \times \mathsf{C} \to \mathsf{Set}$ be a functor.
Let $\mathsf{E}$ be the category whose:
objects are pairs $(X,x)$, where $X$ is an ...

6
votes

1
answer

144
views

### Strictness of two operations on proarrow equipments

There are several equivalent definitions of a profunctor between categories $C$ and $D$. I'm interested in the following two:
A functor $C\times D^o \to \text{Set}$
A co-continuous functor between ...

6
votes

1
answer

318
views

### Adjunctions with respect to profunctors

Let $P : W° \times Y \to \mathbf{Set}$ and $Q : X° \times V \to \mathbf{Set}$ be profunctors, and let $L : X \to W$ and $R : Y \to V$ be functors. Suppose that $$P(Lx, y) \cong Q(x, Ry)$$ natural in $...

9
votes

0
answers

176
views

### Michel Thiébaud's thesis ("Self-Dual Structure-Semantics and Algebraic Categories")

I am looking for a copy of Michel Thiébaud's 1971 thesis Self-Dual Structure-Semantics and Algebraic Categories, which appears to be an early reference for the relationship between the Kleisli ...

7
votes

3
answers

450
views

### Prof and the completion of Cat under right adjoints

In Bénabou's Les distributeurs, in which the bicategory of profunctors is introduced, Bénabou remarks (page 17, quoted below) that $\mathbf{Prof}$ may be viewed as the construction of a bicategory ...

4
votes

0
answers

182
views

### Promonoidal categories as $S$-algebras

A promonoidal category is a pseudomonoid in the monoidal bicategory $\bf Prof$, where the monoidal structure is given (on objects) by the product of categories.
I would like to show that promonoidal ...

3
votes

2
answers

493
views

### Profunctors as a Kleisli bicategory

There is some discussion on the nLab on seeing the free cocompletion $\mathbf{Psh}(\mathbf{A}) = [\mathbf{A}^{op}, \mathbf{Sets}]$, as a pseudomonad. The Yoneda embedding $よ \colon \mathbf{A} \to \...

18
votes

3
answers

1k
views

### A multicategory is a ... with one object?

We all know that
A monoidal category is a bicategory with one object.
How do we fill in the blank in the following sentence?
A multicategory is a ... with one object.
The answer is fairly ...

1
vote

0
answers

180
views

### Is the category of profunctors $Prof(A,B)$ equivalent to $Prof(B,A)^{op}$?

$\def\Prof{\mathsf{Prof}}\def\Set{\mathsf{Set}}\def\tobar{\mathrel{\mkern3mu \vcenter{\hbox{$\scriptscriptstyle+$}}\mkern-12mu{\to}}}$Let $A$ and $B$ be categories. Define a profunctor $A\tobar B$ to ...

3
votes

0
answers

161
views

### Exponentials of profunctors

Suppose $f:B\to A$ is an exponentiable functor, so that pullback $f^\ast$ has a right adjoint $\Pi_f$. Then $f\times \mathbf{2} : B\times \mathbf{2} \to A\times \mathbf{2}$ is also exponentiable, and ...

5
votes

1
answer

292
views

### The $\mathfrak L$ functor on $\textsf{Prof}$

$\def\L{\mathfrak{L}}\def\Prof{\mathsf{Prof}}$ Recall that Isbell duality $\text{Spec}\dashv {\cal O} : {\cal V}^{A^°} \leftrightarrows \big({\cal V}^A\big)^°$ allows us to define the functor
$$
\L : \...

4
votes

0
answers

171
views

### Isbell duality for profunctors

$\def\L{\mathfrak{L}}\def\Prof{\mathsf{Prof}}$Let $A,B$ be two $\cal V$-categories, and define the functor
$$
\L : \Prof(A,B) \to \Prof(B,A)
$$
sending $K : A^° \times B \to \cal V$ into $\L(K) : (b,a)...

4
votes

1
answer

420
views

### Does $\bf Prof$ admit all pseudolimits?

Does the bicategory $\bf Prof$ of categories, profunctors and natural transformations admit all pseudolimits?
By [Kel89, Prop. 5.1] it is enough to show that $\bf Prof$ admits products, cotensors, ...

1
vote

1
answer

305
views

### Profunctors and multicategories

I've been told that there is a way to link profunctors and multicategories, probably obtaining a multicategory from $\bf Prof$; I feel I didn't understand the meaning of this claim.
Can you provide ...

14
votes

3
answers

2k
views

### The Kan construction, profunctors, and Kan extensions

It's been a long time since I tried to understand the deep meaning of the "Kan construction", or "nerve-realization" adjunction
$$
\text{Lan}_y F \dashv N_F = \hom(F,1)
$$
that exists among the left ...

5
votes

1
answer

327
views

### Enriched Cauchy completions and underlying categories

The ordinary Cauchy completion $\overline{C}$ of a small category $C$ satisfies a number of conditions: Every idempotent in $\overline{C}$ splits, there's an equivalence of categories $[C^{op}, Set] \...

2
votes

0
answers

58
views

### Comparing right and left quasi-representable bimodules

Let $\mathcal V$ be your favourite (closed, symmetric) monoidal model category. To fix ideas, set $\mathcal V = \mathrm{Ch}(k)$, the category of chain complexes over a fixed commutative ring. Given a $...

6
votes

1
answer

326
views

### Ends and coends – analogues for higher arity – Horn Filling

Consider the setting of categories enriched over a suitable monoidal category $\mathbb V$.
We define $$\mathrm{Dist}(X,Y):=\mathbb V−\mathrm{Cat}(X^ \mathrm{op}⊗Y,\mathbb V).$$
Recall the definition ...

5
votes

2
answers

682
views

### grothendieck construction for profunctors

Given categories $X$ and $Y$ and a strong functor
$$D:X^{op}\times Y\to Cat$$
we can of course build the oplax colimit
$$\mathrm{colim}^{oplax}_{X^{op}\times Y}D$$
via the usual (covariant) ...

1
vote

0
answers

192
views

### Composition of Cat-valued distributors - compatible with grothendieck construction?

Let $C$ be a category and $F\in[C^{op}, Cat]$ be a strong functor.
(1) There are functors
$$hom_C(c',c)\times F(c)\to F(c').$$
(2) The grothendieck construction gives a 2-equvalence
$$\int_C: [C^{...

8
votes

1
answer

438
views

### Ends as a "cotrace" operation on profunctors

As mentioned here, there is a trace operation on the monoidal category of profunctors given by taking coends: for any profunctor $F : A\times X \nrightarrow B \times X$, there is a profunctor $Tr^X(F) ...

2
votes

1
answer

292
views

### References to using profunctors in program analysis?

Profunctors from a category to itself seem like they'd be useful in representing the result of a program analysis; I can imagine a profunctor that given some information about a function it tells you ...

5
votes

2
answers

721
views

### In what sense do the categorical trace and coend count fixed points?

According to the nlab, the categorical trace of a 1-endomorphism $F:C\to C$ in a 2-category is the set hom$(1_C, F)$ of global elements of $F$. If $F$ is a functor in the 2-category Cat, the ...

13
votes

1
answer

1k
views

### Co-ends as a trace operation on profunctors

The n-lab site on profunctors (http://ncatlab.org/nlab/show/profunctor) describes profunctor composition as using a co-end to "trace out" the connecting variable:
$F\circ G := \int^{d\in D} F(-, d) \...