Questions tagged [profunctors]
The profunctors tag has no usage guidance.
25
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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 ...
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14
votes
3
answers
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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 ...
- 12.7k
13
votes
1
answer
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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) \...
- 1,862
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0
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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 ...
- 7,889
8
votes
1
answer
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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) ...
- 3,303
7
votes
3
answers
436
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 ...
- 7,889
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votes
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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 ...
- 161
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2
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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) ...
- 3,123
5
votes
1
answer
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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] \...
- 407
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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 ...
- 3,123
5
votes
1
answer
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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 ...
- 431
5
votes
1
answer
286
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
$$
\L : \...
- 12.7k
5
votes
2
answers
703
views
In what sense do the categorical trace and coend count fixed points?
According to the nlab, the categorical trace of a 1-endomorphism $F:C\to C$ in a 2-category is the set hom$(1_C, F)$ of global elements of $F$. If $F$ is a functor in the 2-category Cat, the ...
- 1,532
5
votes
1
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Strictness of two operations on proarrow equipments
There are several equivalent definitions of a profunctor between categories $C$ and $D$. I'm interested in the following two:
A functor $C\times D^o \to \text{Set}$
A co-continuous functor between ...
- 563
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Adjunctions with respect to profunctors
Let $P : W° \times Y \to \mathbf{Set}$ and $Q : X° \times V \to \mathbf{Set}$ be profunctors, and let $L : X \to W$ and $R : Y \to V$ be functors. Suppose that $$P(Lx, y) \cong Q(x, Ry)$$ natural in $...
- 7,889
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votes
1
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366
views
Does $\bf Prof$ admit all pseudolimits?
Does the bicategory $\bf Prof$ of categories, profunctors and natural transformations admit all pseudolimits?
By [Kel89, Prop. 5.1] it is enough to show that $\bf Prof$ admits products, cotensors, ...
- 12.7k
4
votes
0
answers
170
views
Promonoidal categories as $S$-algebras
A promonoidal category is a pseudomonoid in the monoidal bicategory $\bf Prof$, where the monoidal structure is given (on objects) by the product of categories.
I would like to show that promonoidal ...
- 12.7k
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votes
0
answers
163
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Isbell duality for profunctors
$\def\L{\mathfrak{L}}\def\Prof{\mathsf{Prof}}$Let $A,B$ be two $\cal V$-categories, and define the functor
$$
\L : \Prof(A,B) \to \Prof(B,A)
$$
sending $K : A^° \times B \to \cal V$ into $\L(K) : (b,a)...
- 12.7k
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votes
2
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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 \...
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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 ...
- 64.1k
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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 ...
- 1,532
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votes
0
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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 $...
- 2,039
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1
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298
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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 ...
- 12.7k
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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 ...
- 474
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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^{...
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