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Questions tagged [profunctors]

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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{...
varkor's user avatar
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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\...
Emily's user avatar
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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 ...
Evan Patterson's user avatar
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 ...
diracdeltafunk's user avatar
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 ...
Max New's user avatar
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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 $...
varkor's user avatar
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9 votes
0 answers
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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 ...
varkor's user avatar
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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 ...
varkor's user avatar
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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 ...
fosco's user avatar
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3 votes
2 answers
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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 \...
Mario Román's user avatar
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 ...
John Gowers's user avatar
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 ...
SCappella's user avatar
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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 ...
Mike Shulman's user avatar
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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 : \...
fosco's user avatar
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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)...
fosco's user avatar
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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, ...
fosco's user avatar
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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 ...
fosco's user avatar
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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 ...
fosco's user avatar
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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] \...
Richard Jennings's user avatar
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 $...
Francesco Genovese's user avatar
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 ...
Gerrit Begher's user avatar
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) ...
Gerrit Begher's user avatar
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^{...
Gerrit Begher's user avatar
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) ...
Noam Zeilberger's user avatar
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 ...
Mike Stay's user avatar
  • 1,532
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 ...
Mike Stay's user avatar
  • 1,532
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) \...
Aleks Kissinger's user avatar