Questions tagged [profunctors]

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9
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0answers
134 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
1answer
253 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 ...
2
votes
0answers
89 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
2answers
312 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
3answers
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
0answers
120 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
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0answers
147 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 ...
4
votes
1answer
226 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
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0answers
122 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)...
3
votes
1answer
281 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
1answer
244 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 ...
12
votes
3answers
1k 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
1answer
242 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
0answers
54 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 $...
3
votes
0answers
230 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
2answers
562 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
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0answers
172 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
1answer
396 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
1answer
274 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 ...
4
votes
2answers
609 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 ...
12
votes
1answer
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) \...