Questions tagged [monoidal-categories]

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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 ...
Clark Barwick's user avatar
30 votes
1 answer
2k views

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 ...
Alexander Chervov's user avatar
5 votes
1 answer
290 views

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) \...
Sebastien Palcoux's user avatar
4 votes
1 answer
235 views

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) \...
Sebastien Palcoux's user avatar
40 votes
4 answers
7k views

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, ...
Dinakar Muthiah's user avatar
18 votes
2 answers
1k views

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 ...
varkor's user avatar
  • 8,675
16 votes
2 answers
1k views

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 ...
Manuel Bärenz's user avatar
14 votes
2 answers
800 views

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{...
Emily's user avatar
  • 10.3k
13 votes
3 answers
967 views

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, ...
Joey Hirsh's user avatar
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12 votes
4 answers
1k views

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 $k$-...
Martin Brandenburg's user avatar
10 votes
1 answer
685 views

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 ...
Fernando Muro's user avatar
10 votes
1 answer
422 views

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}(...
Emily's user avatar
  • 10.3k
8 votes
1 answer
335 views

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 ...
Manuel Bärenz's user avatar
6 votes
2 answers
1k views

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 $\mathcal V =\mathbf C(k)$ ...
Francesco Genovese's user avatar
6 votes
2 answers
432 views

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 ...
Jan Steinebrunner's user avatar
5 votes
1 answer
818 views

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 ...
David White's user avatar
  • 29.4k
3 votes
1 answer
243 views

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 ...
Emily's user avatar
  • 10.3k
3 votes
0 answers
216 views

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 ...
Ben Sprott's user avatar
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41 votes
4 answers
4k views

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 ...
Konrad Voelkel's user avatar
23 votes
5 answers
3k views

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 ...
Chris Schommer-Pries's user avatar
21 votes
2 answers
2k views

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 ...
Hamed's user avatar
  • 593
20 votes
1 answer
3k views

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 ...
Bruce Westbury's user avatar
19 votes
3 answers
3k views

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 ...
Martin Brandenburg's user avatar
17 votes
4 answers
1k views

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 ...
pnips's user avatar
  • 171
16 votes
0 answers
2k views

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 ...
Evan Jenkins's user avatar
  • 7,107
16 votes
2 answers
1k views

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 ...
Martin Brandenburg's user avatar
15 votes
3 answers
823 views

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 ...
SCappella's user avatar
  • 474
15 votes
2 answers
657 views

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 ...
John Gowers's user avatar
15 votes
2 answers
1k views

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 ...
Chris Schommer-Pries's user avatar
14 votes
2 answers
973 views

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 ...
Dmitri Pavlov's user avatar
14 votes
1 answer
819 views

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 ...
Gerrit Begher's user avatar
13 votes
1 answer
585 views

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 ...
Manuel Bärenz's user avatar
13 votes
2 answers
4k views

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 ...
Evan Jenkins's user avatar
  • 7,107
12 votes
0 answers
410 views

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 ...
Mike Shulman's user avatar
12 votes
2 answers
1k views

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, ...
xuq01's user avatar
  • 1,054
12 votes
1 answer
252 views

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 $...
Will Sawin's user avatar
  • 135k
12 votes
1 answer
553 views

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 (...
Mike Shulman's user avatar
11 votes
1 answer
1k views

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) \...
Theo Johnson-Freyd's user avatar
10 votes
4 answers
1k views

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 ...
supercooldave's user avatar
10 votes
2 answers
583 views

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 ...
Manuel Bärenz's user avatar
10 votes
2 answers
671 views

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: ...
LorenzoPerticone's user avatar
10 votes
1 answer
431 views

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 ...
Dmitri Pavlov's user avatar
10 votes
1 answer
2k views

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{...
Martin Brandenburg's user avatar
9 votes
2 answers
2k views

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 $...
ziggurism's user avatar
  • 1,436
9 votes
4 answers
2k views

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 ...
Nadia SUSY's user avatar
9 votes
1 answer
230 views

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 $\phi$ ...
André Henriques's user avatar
9 votes
0 answers
254 views

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 ...
fosco's user avatar
  • 13k
9 votes
1 answer
2k views

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 ...
Sebastien Palcoux's user avatar
9 votes
1 answer
317 views

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 ...
Emily's user avatar
  • 10.3k
9 votes
1 answer
940 views

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, ...
Bugs Bunny's user avatar
  • 12.1k