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7
votes
1answer
848 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 ...
2
votes
3answers
437 views

Action on tensor power and “element notation” in monoidal categories

Let $C$ be a symmetric monoidal category. Fix an object $X$, let $S$ denote the symmetry $X \otimes X \to X \otimes X$. Also define $X^{\otimes n}$ by induction on $n$: $X^{\otimes 0} = 1$, ...
1
vote
0answers
124 views

A simple example of a tame tensor functor

Let $C$ be a $R$-linear cocomplete tame abelian tensor category (see here for definitions) with unit $1$, then there is a unique $R$-linear cocontinuous tensor functor $F : \text{Mod}(R) \to C$, ...
5
votes
1answer
288 views

representing tensor-C*-categories in BIM

Given a factor M (=von Neumann alg. with center ℂ), let us write BIM for the ⊗-C*-category of M-M-bimodules. Which ⊗-C*-categories can one faithfully embed into BIM? ⓵ Are ...
4
votes
1answer
356 views

Is the functor of divided powers a weakly monoidal functor?

Let $R$ be a commutative ring with identity $e$. For every $R$-module $M$ the algebra of divided powers $D(M)$ is defined as follows. The generators of $D(M)$ are the symbolls $m^{(k)}$ for every ...
3
votes
1answer
483 views

Tame abelian tensor categories

In the article "Tannaka duality for geometric stacks" (arxiv, see nlab for a summary) Jacob Lurie introduced the notion of a tame abelian tensor category. An abelian tensor category is called tame if ...
5
votes
1answer
227 views

Do plain functors out of monoidal categories factor into a nontrivial monoidal part and a plain part?

Given symmetric monoidal closed categories $C, D$, a symmetric monoidal (not closed!) functor $F:C\to D$ factors into two parts: the first is a symmetric monoidal closed functor from $C$ to a ...
0
votes
1answer
524 views

universal property of module categories internal to a tensor category

By a tensor category, I mean here a cocomplete $k$-linear symmetric tensor category, where $k$ is a fixed ground ring. Tensor functors are assumed to be $k$-linear and cocontinuous. I will denote the ...
3
votes
1answer
133 views

Epimorphisms between line objects

Let $\mathcal{A}$ be a $k$-linear abelian symmetric tensor category with unit $\mathcal{O}_A$; here $k$ is a comm. ring. By that I assume implicitly that $\otimes$ is finitely cocontinuous in each ...
5
votes
3answers
370 views

Does one of the hexagon identities imply the other one?

Suppose we have a monoidal category equipped with additional data that almost makes it a braided monoidal category except that only one of the hexagon identities is satisfied. Can we then prove the ...
8
votes
2answers
474 views

Gerbes and Z-graded symmetric monoidal categories

Let me first recall a known fact. Suppose $X$ is a complex algebraic variety, $\mathscr{L}$ is a line bundle on $X$ and $L^\times$ is the total space of $\mathscr{L}$ with the zero section removed. ...
5
votes
1answer
380 views

graded ring associated to a line bundle in a tensor category

Let $\mathcal{A}$ be an abelian tensor category with unit $\mathcal{O}$. An object $\mathcal{L}$ is called invertible or a line bundle if there is some $\mathcal{L}^{-1}$ such that $\mathcal{L} ...
10
votes
1answer
706 views

Alternative definition of monoidal categories

My question is about monoidal categories. To motivate it, let me first recall something about group objects. Assume you define a group object in a category $C$ with products by an object $G$ together ...
5
votes
2answers
288 views

Is the box product of morphisms associative?

Suppose $(C,\otimes)$ is a symmetric monoidal finitely-cocomplete category such that $\otimes$ preserves colimits. Given two morphisms $a:A_1\to A_2$ and $b:B_1\to B_2$, define $a\Box b$ to be the ...
3
votes
0answers
189 views

Symmetric monoidal structure on cosimplicial spaces

Is there a monoidal structure on the category $Spc^{\Delta}$ of cosimplicial spaces such that in the adjunction $$ \Delta^{\bullet}\otimes-\colon Spc\leftrightarrows Spc^{\Delta}\colon Tot(-) $$ the ...
2
votes
2answers
324 views

The condition End(1) = k in Tannakian Categories

A neutral Tannakian category over a field $k$ is a rigid $k$-linear abelian tensor category $\mathcal{C}$ whose unit $1$ satisfies $\mathrm{End}(1) \simeq k$, and is moreover equipped with an exact ...
3
votes
3answers
383 views

What is the proper name for “compact closed” multiplicative intuitionistic linear logic?

Multiplicative intuitionistic linear logic (MILL) has only multiplicative conjunction $\otimes$ and linear implication $\multimap$ as connectives. It has models in symmetric monoidal closed ...
7
votes
2answers
488 views

From tensor algebras in monoidal categories to (commutative?) monoids

Let $D$ be a monoidal category (without the structure of a symmetric monoidal category), with unit object $Id$, and let $L$ be an invertible object in $D$, so that $L$ is dualizable and the pairing ...
2
votes
2answers
202 views

A better way to compute the mapping spaces of the category of spans in an enriched tensored category?

Let X be a tensored and cotensored V-category, where V is a fixed complete, cocomplete, closed symmetric monoidal category. Define $C:=Span(X)$ to be the category of spans in X (this is the functor ...
5
votes
1answer
234 views

What is the right way to define the nerve of an unbiased monoidal category?

I've been toying around with unbiased composition in higher categorical structures on and off for a while now. In particular, I've been playing around with unbiased monoidal 2-categories. One ...
8
votes
3answers
601 views

Skeleton of a braided monoidal category

Does every (lax) braided monoidal category have a braided monoidal skeleton? That is, I want to define a (lax) braided monoidal structure on a skeleton so that it is braided monoidal equivalent to ...
1
vote
2answers
298 views

In which categories is every coalgebra a sum of its finite-dimensional subcoalgebras?

Let $\mathcal V$ be a reasonably nice category — I'm interested in the case when $\mathcal V$ is $\mathbb K$-linear for some field $\mathbb K$, abelian, and has all products and coproducts ...
6
votes
3answers
680 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 ...
4
votes
1answer
297 views

De-equivariantization by Rep(G)

I'm trying to understand Proposition 2.9 of this paper on weakly group theoretical fusion categories. First of all I have a problem with understanding the settings for de-equivariantization process. ...
19
votes
8answers
3k views

Resources for graphical languages / Penrose notation / Feynman diagrams / birdtracks?

There is an idea I've recently gotten interested in that doesn't seem to have a good agreed-upon name ("diagrammatic algebra?"). It centers around the use of two-dimensional diagrams of dots, ...
8
votes
4answers
2k views

What is the universal enveloping algebra?

Let ${\mathfrak g}$ be a Lie algebra in a symmetric monoidal category enriched over $K$-vector spaces, i.e., in particular, hom-s are $K$-vector spaces (where $K$ is a field of characteristic zero). ...
7
votes
3answers
676 views

Tannakian description of a semi-direct product

Let $H$ and $K$ be affine group schemes over a field $k$ of caracteristic zero. Let $\varphi:H\to Aut(K)$ be a group action. Then we can form the semi-direct product $G = K\ltimes H$. Problem: ...
4
votes
1answer
413 views

Is assigning the endomorphism object in some sense functorial?

Let $\mathcal V$ be a monoidal category and let $\mathcal C$ be a $\mathcal V$-category. Let's denote the $\mathcal V$-valued hom-functor $[-,-]$. Now for every object $X\in\mathcal C$ we have it's ...
3
votes
0answers
164 views

What is the underlying graphical calculus of the Interactions-Round-a-Face lattice model?

Background Let $\mathcal{L}$ be an $m \times n$ square lattice on a torus, and let $\Sigma$ be a finite set. We think of $\Sigma$ as the possible spin values that can be assigned to the points of the ...
7
votes
1answer
352 views

Category of copresheaves over commutative monoids

Let C be a symmetric monoidal category. Let Comm(C) be the category of commutative monoids in C. Consider the topos X = CoPSh(Comm(C)) of covariant functors from Comm(C) to the category Set of sets. ...
4
votes
2answers
453 views

Closed monoidal structure on the derived category of sheaves

Given a topological space X, i'd like to find Der X - the derived category of sheaves of abelian groups on X - to be a closed monoidal category. Hom should be cohomological and the internal-hom should ...
5
votes
5answers
428 views

What are the correct axioms for a “weakly associative monoidal functor”?

Definitions and the main question Recall that a category $\mathcal C$ is monoidal if it is equipped with the following data (two functors, three natural transformations, and some properties): a ...
5
votes
1answer
468 views

Is there a relative version of Tannakian reconstruction?

According to some form of Tannakian reconstruction, given a finite tensor category with a fiber functor to the category of vector spaces, one determines a Hopf algebra by considering tensor ...
16
votes
1answer
2k 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 ...
7
votes
2answers
368 views

Effects of “weak” vs. “strict” categories in Eckmann-Hilton arguments

A standard example for demonstrating the need for genuinely weak n-categories is that a weak 3-category with unique 0- and 1-cells amounts to the same thing as a braided monoidal category (by an ...
10
votes
4answers
832 views

Why is a monoid with closed symmetric monoidal module category commutative?

Given a symmetric monoidal category and a monoid object A in it, one can form the category of modules over this monoid object, i.e. objects are $A \otimes M \rightarrow M$ satisfying the natural ...
23
votes
4answers
1k 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 ...
15
votes
1answer
776 views

Semiadditivity and dualizability of 2

Short version: Let (C, ⊗, 1) be a locally presentable closed symmetric monoidal category with a zero object, and write 2 = 1 ∐ 1. Suppose the object 2 has a dual. Does it follow that C is a ...
2
votes
1answer
246 views

When does a certain natural construction on monoidal categories yield a Hopf algebra?

Let $\mathcal C = (\mathcal C_0,\mathcal C_1)$ be a (small) strict monoidal category. Pick a field $\mathbb K$, and let $\mathbb K[\mathcal C_1]$ be the vector space with basis the morphism of ...
12
votes
3answers
624 views

Model Structure/Homotopy Pushouts in topological monoids?

Let C be the category of topological monoids, that is, the category of monoids in (Top, $\times$). Can the model category structure on Top (Serre fibrations, cofibrations, weak homotopy ...
9
votes
2answers
3k 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 ...
11
votes
1answer
763 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, Module categories, weak ...
4
votes
1answer
257 views

Explicit description of a free braided monoidal groupoid with inverses

Let G be a braided monoidal groupoid: it does no harm to suppose that the monoidal product on G is strictly associative, so I'll do that. "With inverses" means that for every object $X$ of G, there ...
12
votes
6answers
998 views

What structure on a monoidal category would make its 2-category of module categories monoidal and braided?

So, many of us know the answer to "what kind of structure on an algebra would make its category of representations braided monoidal": your algebra should be a quasi-triangular Hopf algebra (maybe if ...
7
votes
4answers
509 views

What's the right object to categorify a braided tensor category?

The yoga of categorification has gained a lot of popularity in recent years, and some techniques for it have made a lot of progress. It's well-understood that, for example, a ring is probably ...
7
votes
3answers
456 views

Are supervector spaces the representations of a Hopf algebra?

Supervector spaces look a lot like the category of representations of $\mathbb{Z}/2\mathbb{Z}$ - the even part corresponds to the copies of the trivial representation and the odd part corresponds to ...
19
votes
4answers
3k 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, ...
11
votes
2answers
727 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 ...
9
votes
3answers
796 views

When does Tannakian theory work over affine schemes besides fields?

By 'work' I would like the correspondence between fiber functors (to finitely generated projective modules) and algebraic groups to be the same as in the field case. Specifically, if $A$ is an affine ...
11
votes
6answers
866 views

How do I describe a fusion category given a subfactor?

I felt like following up on Kate's question. There were some good motivational answers there. Given a pair of factors M < N, there is a standard way to construct a 2-category whose objects are M ...