The monoidal-categories tag has no wiki summary.

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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 ...

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### 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 ...

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### 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 ...

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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 ...

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**1**answer

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### 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 ...

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### 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 ...

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### 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 ...

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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 ...

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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, Module categories, weak ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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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, ...

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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 ...

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### 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 ...

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### 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 ...

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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 ...

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### When do PROP-morphisms induce adjunctions?

If (C,tensor,1) is a symmetric monoidal category and f:A-->B is a morphism of PROPs (or monoidal cats = colored PROPs), one gets a forgetful functor f^*:B-Alg(C)-->A-Alg(C) (where ...

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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 ...

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### What is a formula for the “group-like Drinfeld element”?

Any quantized universal enveloping algebra (in fact, any toplogically quasi-triangular Hopf algebra) has an (in its completion) an element u called the Drinfeld element which gives an isomorphism from ...

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### Are there interesting monoidal structures on representations of quantum affine algebras?

Is there a good monoidal structure on a category of integrable representations of a quantum affine algebra? In the ordinary affine Kac-Moody case, there is the usual tensor product (symmetric, adds ...

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### Are abelian nondegenerate tensor categories semisimple?

A pivotal monoidal category is called non-degenerate if the inner product (x,y) = Tr(xy*) (where y* is the dual map) is non-degenerate. As a rule of thumb non-degenerate is closely related to ...