Questions tagged [monoidal-categories]
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84
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Reference for "lax monoidal functors" = "monoids under Day convolution"
Suppose $A$ and $C$ two symmetric monoidal categories. Let's say that $A$ is small and $C$ is locally presentable, and let's assume also that the tensor product on $C$ preserves colimits separately in ...
30
votes
1
answer
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Can one explain Tannaka-Krein duality for a finite-group to ... a computer ? (How to make input for reconstruction to be finite datum?)
Consider a finite group. Tannaka-Krein duality allows to reconstruct the group from the
category of its representations and additional structures on it (tensor structure + fiber functor). Somehow ...
5
votes
1
answer
290
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Existence of a multiplication bifunctor for the category of groups
For $\mathsf{Grp}$ the category of groups, a bifunctor $M: \mathsf{Grp} \times \mathsf{Grp}\to \mathsf{Grp}$ is a multiplication bifunctor if:
$M(C_n,C_m) \simeq C_{nm}$,
$M(C_1,G) \simeq M(G,C_1) \...
4
votes
1
answer
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Existence of an addition bifunctor for the category of groups
Let $\mathsf{Grp}$ be the category of groups. A bifunctor $A: \mathsf{Grp} \times \mathsf{Grp}\to \mathsf{Grp}$ is an addition bifunctor if:
$A(C_n,C_m) \simeq C_{n+m}$,
$A(C_0,G) \simeq A(G,C_0) \...
40
votes
4
answers
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Tannakian Formalism
The Tannakian formalism says you can recover a complex algebraic group from its category of finite dimensional representations, the tensor structure, and the forgetful functor to Vect. Intuitively, ...
18
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2
answers
1k
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Monoidal categories whose tensor has a left adjoint
Is there a name for monoidal categories $(\mathscr V, \otimes, I)$ such that $\otimes$ has a left adjoint $(\ell, r) : \mathscr V \to \mathscr V^2$? Have they been studied anywhere? What are some ...
16
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2
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How are the Walker-Wang TQFT and the Crane-Yetter TQFT related?
Mathematical physicists in solid state physics and topological insulators talk a lot about Walker-Wang models, which are a family of Hamiltonians defined on a 3d lattice. Unfortunately, the original ...
14
votes
2
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800
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Examples of rings in monoidal categories
Ring objects are usually defined on Cartesian monoidal categories, but one can define them more generally on non-Cartesian symmetric monoidal categories as follows:
Let $(\mathcal{C},\otimes,\mathbf{...
13
votes
3
answers
967
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Model Structure/Homotopy Pushouts in topological monoids?
Let $\mathsf C$ be the category of topological monoids, that is, the category of monoids in $(\textsf{Top}, \times)$.
Can the model category structure on $\textsf{Top}$ (Serre fibrations, ...
12
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4
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Kan extensions in the $2$-category of monoidal categories
Kan extensions make sense in any $2$-category. But so far I have only really seen them in the case of the $2$-category of categories, functors, natural transformations and the $2$-category of $k$-...
10
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1
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Hovey's unit axiom in monoidal model categories
Let $\mathcal{C}$ be a monoidal model category in the sense of Hovey's book. He assumes the following unit axiom not considered in other references (e.g. Schwede-Shipley): given a cofibrant ...
10
votes
1
answer
422
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Is there a monoidal analogue of equalizers?
There are three different kinds of finite limits in categories: terminal objects, binary products, and equalizers. In a category $C$, these define functors $1_{C},\times,\mathrm{Eq}\colon\mathrm{Fun}(...
8
votes
1
answer
335
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What's the (monoidal) image of a monoidal functor?
For an ordinary functor $F\colon \mathcal{C} \to \mathcal{D}$ of categories, there is a construction $\operatorname{im} F$, the image of $F$, which is again a category, and $F$ factors through that ...
6
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2
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A general theory of quasi-functors, generalizing from dg-categories to $\mathcal V$-categories, with $\mathcal V$ monoidal model category
I employ the vast majority of the post to develop the notion of quasi-functor between dg-categories: I think it is important to get the idea.
Let $k$ be a field, and let $\mathcal V =\mathbf C(k)$ ...
6
votes
2
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432
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Does the Day convolution induce the structure of a bimonoidal category on $Fun(C,D)$?
Let $\mathcal{C}$ and $\mathcal{D}$ be symmetric monoidal categories and assume that the symmetric monoidal product $\otimes_{\mathcal{D}}$ on $\mathcal{D}$ preserves colimits in both variables.
Then ...
5
votes
1
answer
818
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When is homotopy orbit space weakly equivalent to orbit space, other than situation of free action?
Let $M$ be a closed symmetric monoidal model category. Let $X$ be a cofibrant object (it can also be fibrant if you like) and let $\Sigma_n$ act on $X^{\otimes n}$ by permuting the factors (note that ...
3
votes
1
answer
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Corepresentability of involutory objects in monoidal $\infty$-categories
The group $\mathbb{Z}/2$ corepresents the functor $\mathrm{Inv}\colon\mathsf{Mon}\to\mathsf{Sets}$ sending a monoid $A$ to its set of involutory elements (those satisfying $a^2=1_A$).
A similar story ...
3
votes
0
answers
216
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symmetric monoidal dagger endofunctor categories
Take an arbitrary category $C$, and consider $End(C)$, any endofunctor category consisting of some or all endofunctors of $C$.
I have several related questions:
What restrictions must we impose on ...
41
votes
4
answers
4k
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Understanding the definition of the Lefschetz (pure effective) motive
For all those who are unlikely to have answers to my questions, I provide some
Background:
In some sense, pure motives are generalisations of smooth projective varieties. Every Weil cohomology ...
23
votes
5
answers
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Do all 3D TQFTs come from Reshetikhin-Turaev?
The Reshetikhin-Turaev construction take as input a Modular Tensor Category (MTC) and spits out a 3D TQFT. I've been told that the other main construction of 3D TQFTs, the Turaev-Viro State sum ...
21
votes
2
answers
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Modular Tensor Categories: Reasoning behind the axioms
(Sorry for the length of the question, I'm trying to communicate what is bothering me as thoroughly as possible)
In the construction of modular tensor categories (MTC) from ground zero, we put ...
20
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1
answer
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Calculating 6j-symbols (aka Racah-Wigner coefficients) for quantum groups
Which $6j$-symbols for quantised enveloping algebras are known explicitly?
The quantum $6j$-symbols for $sl(2)$ are well-known. The references are
Masbaum and Vogel and
Frenkel and Khovanov.
What ...
19
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3
answers
3k
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Free symmetric monoidal category on a monoidal category
Consider the $2$-categories
$\mathsf{MonCat}$ of monoidal categories, with strong monoidal functors and monoidal transformations,
$\mathsf{SymMonCat}$ of symmetric monoidal categories, with strong ...
17
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4
answers
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What is the monoidal equivalent of a locally cartesian closed category?
If a closed monoidal category is the monoidal equivalent of a Cartesian closed category, is there an analogous equivalent for locally cartesian closed categories? Is there a standard terminology or ...
16
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0
answers
2k
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What is known about module categories over general monoidal categories?
All of the literature I have seen on module categories over monoidal categories has been in the rigid $k$-linear semisimple case, more or less in the spirit of Ostrik's paper,
Ostrik, V. Module ...
16
votes
2
answers
1k
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The symmetric monoidal category of finite sets
It is well-known that the (augmented) simplex category is the universal monoidal category with a monoid object. What about a commutative analogue? Consider the category $\mathsf{FinSet}$ of finite ...
15
votes
3
answers
823
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Rectifying the definition of a closed category
The definition of a closed category I'm using is here.
Suppose $V$ is a closed category and that for each object $b\in V$, $[b,-]$ has a left adjoint $- \otimes b$. The result is nearly a monoidal ...
15
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2
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657
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Monoidal functors $\mathcal C \to [\mathcal D,\mathcal V]$ are monoidal functors $\mathcal C \otimes \mathcal D \to \mathcal V$?
It is well known (e.g., Reference for "lax monoidal functors" = "monoids under Day convolution" ) that if $\mathcal C$ is a monoidal $\mathcal V$-enriched category, then a monoid ...
15
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2
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Categorifying the Reals via von Neumann Algebras?
So one way to categorify the natural numbers is to replace them with vector spaces. Then the dimension of the vector space reproduces the natural number. More generally you can categorify integers to ...
14
votes
2
answers
973
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Recovering a monoidal category from its category of monoids
What kind of additional properties and/or structures one needs to impose on the category
of (commutative or noncommutative) monoids of some monoidal category
so that one can recover the original ...
14
votes
1
answer
819
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Cartesian envelope of a symmetric monoidal category
Is there a left adjoint to the forgetful functor from cartesian monoidal categories to symmetric monoidal categories? And if so: how can it be described explicitely.
In more detail: Given a symmetric ...
13
votes
1
answer
585
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A cohomology theory for fusion categories
It is well known that for a finite group $G$, the associator of the fusion category of $G$-graded $k$-vector spaces is given by an element of $H^3(G,k^*)$, up to equivalence of categories. ($k^*$ is ...
13
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2
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Is there a meaningful difference between biased and unbiased composition?
In higher category theory, there are notions of biased and unbiased definitions of composition of $n$-morphisms (or, as a special case, tensor products of objects). In the biased framework, we define ...
12
votes
0
answers
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Biased vs unbiased lax monoidal categories
There are two principal ways to define a monoidal category:
The biased definition includes a unit object $I$, a binary tensor product $A\otimes B$, and a ternary associativity isomorphism $(A\otimes ...
12
votes
2
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Abelian categories that are not monoidal
Forgive me if this turns out to be a naive question. I'm quite convinced that not all abelian categories admit (symmetric?) monoidal structure (of course, with the tensor product being additive, ...
12
votes
1
answer
252
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Is there a finite test for isomorphisms of rigid monoidal abelian categories?
Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.)
Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with pair of exact tensor functors $...
12
votes
1
answer
553
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Uniqueness of dualizing objects
One definition of (symmetric) star-autonomous category is as a closed symmetric monoidal category $(C,\otimes,I,\multimap)$ equipped with an object $\bot$ such that all double-dualization maps $A \to (...
11
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1
answer
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When is the adjoint to a monoidal functor monoidal?
Let $\mathcal C,\mathcal D$ be monoidal categories. Recall that a functor $F : \mathcal C \to \mathcal D$ is lax monoidal if it is equipped with maps $1_{\mathcal D} \to F(1_{\mathcal C})$ and $F(X) \...
10
votes
4
answers
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180˚ vs 360˚ Twists in String Diagrams for Ribbon Categories
Ribbon categories are braided monoidal categories with a twist or balance, $\theta_B:B\to B$, which is a natural transformation from the identity functor to itself. In the string diagram calculus for ...
10
votes
2
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What are TQFTs that are multiplicative under connected sums? Do bordisms with connected sum as monoidal product exist?
In general, one extracts a manifold invariant from a TQFT by interpreting the closed manifold as a bordism from the empty set to the empty set. The TQFT sends this bordism to a homomorphism of the ...
10
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2
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671
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On functors preserving monoid objects
If $C$ is a monoidal category, we can define the category $Mon(C)$ of monoids in $C$; call $U_C : Mon(C) \to C$ the forgetful functor. I'm interested in functors between categories of monoids:
...
10
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1
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Reference for a generalization of Γ-spaces to monoidal model categories
Γ-spaces were introduced by Segal in 1969 as models for what can be now described
as commutative ∞-monoids and ∞-groups in cartesian symmetric monoidal ∞-categories, e.g., E_∞-spaces and connective ...
10
votes
1
answer
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Exterior powers in tensor categories
Let $\mathcal{C}$ be a cocomplete $R$-linear tensor category. Many notions of commutative algebra can be internalized to $\mathcal{C}$. For example a commutative algebra is an object $A$ in $\mathcal{...
9
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2
answers
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What is the free monoidal category generated by a monoid?
In several places in a segment on cohomology (for example, here (PDF)) in John Baez's online lecture notes for a course in 2007 on quantum gravity, much is made of the fact that the simplex category $...
9
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4
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The tensor product of two monoidal categories
Given two monoidal categories $\mathcal{M}$ and $\mathcal{N}$, can one form their tensor product in a canonical way?
The motivation I am thinking of is two categories that are representation ...
9
votes
1
answer
230
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Elementary equivalence of monoidal categories =?
Recall that, in model theory, two models $M_1$ and $M_2$ of the same signature are elementary equivalent if $ M_1 \models \phi \Leftrightarrow M_2 \models \phi $ for every first order formula $\phi$ ...
9
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0
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t-structures on the tensor product of stable $\infty$-categories
It is a matter of checking definitions that the tensor product of presentable $\infty$-categories restrict to a tensor product between stable presentable $\infty$-categories; I was wondering if there ...
9
votes
1
answer
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Image, kernel, quotient and first isomorphism theorem, in a category of monoid objects
Let $\mathcal{C}$ be a monoidal category and Mon$_{\mathcal{C}}$ the category of monoids (also called algebra objects) on $\mathcal{C}$.
Questions: are there definitions of image and kernel for a ...
9
votes
1
answer
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Are differential rings monoids in a monoidal category?
$\newcommand{\pt}{\mathrm{pt}}\newcommand{\N}{\mathbb{N}}\newcommand{\Z}{\mathbb{Z}}$A number of algebraic structures can be defined as monoids in some appropriate monoidal category:
A monoid ...
9
votes
1
answer
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Saavedra's Definition of Tannakian Category
I have been reading Deligne-Milne's Tannakian Categories and got to the point at the end of part 3 where they discuss what went wrong with Saavedra's definition. Motivated by their counterexample, ...