The monoidal-categories tag has no usage guidance.

**34**

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741 views

### Enriched Categories: Ideals/Submodules and algebraic geometry

While working through Atiyah/MacDonald for my final exams I realized the following:
The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed ...

**15**

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182 views

### Are there small examples of non-pivotal finite tensor categories?

I'm looking for small concrete examples of non-pivotal finite tensor categories to do some calculations with.
Here a finite tensor category is, according to Etingof-Ostrik, a rigid monoidal category ...

**15**

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723 views

### Is the category of smooth manifolds equivalent to the opposite category of the category of commutative monoids of some additive symmetric monoidal category?

This is a followup to my previous question, which asked whether
the category of commutative or noncommutative C*-algebras or von Neumann algebras
is equivalent to the category of commutative or ...

**14**

votes

**0**answers

126 views

### Specific cases of the tangle hypothesis in terms of “classical” n-categories

As is well known, the tangle hypothesis of Baez and Dolan proposes that, for suitable definitions, the $n$-category of framed $n$-tangles in $n+k$ dimensions is the free $k$-tuply monoidal $n$-...

**9**

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183 views

### Grothendieck Construction, Categories of Operators and Opposites

Given a symmetric monoidal category $C$, we can construct its endomorphism operad (or multicategory) $End(C)$ whose objects are the objects of $C$, and for which the multimorphisms from $\{c_1,\ldots,...

**9**

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102 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 ...

**9**

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274 views

### Twisted duality in a symmetric monoidal category

I would like to know if the following definition is already known or even well-known. Is there a reference in the literature? Do you know prominent examples?
Definition. Let $\mathcal{C}$ be a ...

**9**

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184 views

### Are pivotal categories the algebras for a cartesian monad?

It seems to be "known" but not written down that the following are more-or-less equivalent:
...

**7**

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**0**answers

118 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 ...

**7**

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150 views

### When is Rep(U_q(g)) invariant under q -> -q and why?

Since this is a question about quantum groups, let me first fix notation. In this question I use the conventions from this paper of Sawin. (That particular paper conveniently lists in the ...

**6**

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118 views

### Is there a finite test for isomorphisms of rigid monoidal abelian categories, part II

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 an additive tensor functor $Rep_G ...

**6**

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250 views

### Enriched Categories: Metric Spaces, Monoidal Endofunctors and Lipschitz-Continuous Maps.

In the introduction to the reprint of "Metric spaces, generalized logic and closed categories" Lawvere talks about the following situation:
Let $\mathbb R_+$ denote $\mathbb R_{\geq 0}^\infty$. Every ...

**5**

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**0**answers

288 views

### Deligne's theorem on the characterisation of Tannakian categories

I am trying to understand the proof of the Theorem 7.1 from Catégories Tannakiennes by Pierre Deligne https://publications.ias.edu/sites/default/files/60_categoriestanna.pdf.
Essentially, it is ...

**4**

votes

**0**answers

47 views

### Temporal semantics for string diagrams

Suppose I have a string diagram $D$ which involves a set of strings $S$ and atomic processes $A$. Formally, we should think of this as a canonically chosen map in the free symmetric monoidal category (...

**4**

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57 views

### Are there any dominant pivotal functors such that the regular representation is not mapped onto a multiple of the regular representation?

This question is related to Pivotal functors of that are substantially different from finite group homomorphisms.
A tensor functor $F: \mathcal{C} \to \mathcal{D}$ is called dominant (sometimes ...

**4**

votes

**0**answers

181 views

### Schwede-Shipley theorem for monoidal categories?

The Schwede-Shipley theorem gives a criterion for a presentable stable $\infty$-category to be the category of modules over an $\mathcal{E}_1$-algebra. Is there any similar criterion for a monoidal ...

**4**

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**0**answers

156 views

### When does the canonical model structure on $\mathcal V$-$\mathbf{Cat}$ give a structure of monoidal model category?

Let $\mathcal V$ be a closed symmetric monoidal model category. It is well known that the category $\mathcal V$-$\mathbf{Cat}$ of $\mathcal V$-enriched categories is itself a closed symmetric monoidal ...

**4**

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106 views

### Ribbon Algebras and Co-(dual)-quasi-triangular Hopf Algebras

As is well-known, one can use the coquasi-triangular structure $\cal R$ of $U_q(\frak{g})$ to give it's category of (right) modules $\cal{M}_{U_q(\frak{g})}$ the structure of a braided monoidal ...

**4**

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**0**answers

375 views

### Tensor product of quivers

As a special case of a general construction I have constructed "accidentally" a tensor product of quivers aka directed multigraphs (aka directed graphs for category theorists). Probably this is ...

**4**

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260 views

### Reshetikhin-Turaev and links with a distinguished component

Hi,
This question came up to me when reading the paper of Cartier on Vassiliev invariants, but it can probably be turned into a more general question.
Let $T$ be the category whose objects are ...

**4**

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277 views

### When is the cofibrant replacement of a product the product of the cofibrant replacements?

I'm in a situation where I'd like to prove $Q(E\otimes E) \simeq QE \otimes QE$ for a monoid $E$ in a symmetric monoidal model category. I know it's not true in general that $Q(E\otimes F)\simeq QE \...

**3**

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96 views

### About the definition of lax.functor between tricategories

SUMMARY: Observing that monoids in a monoidal category are identified with lax.functors (with domain 1), I tried to generalize this argument wanting to get a skew-Monoidal-category as (tri)lax....

**3**

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243 views

### What is the right categorical framework for diagonal approximations, cup and cap products and identities between them?

I wanted to know what is the right categorical analog for the diagonal approximation for the singular chain complex $\Delta: C\rightarrow C\otimes C$ as a morphism in the homotopy category of chain ...

**3**

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122 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 ...

**3**

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**0**answers

175 views

### The multiplicative system in a symmetric monoidal category

Let $\mathcal{C}$ be a symmetric monoidal category. In the 1973 paper "Note on monoidal localisation" by Brian Day, the multiplicative system of morphism in $\mathcal{C}$ has been discussed. See also ...

**3**

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129 views

### Show that duality functor is anti-monoidal

Let $\mathcal{C}$ be a right rigid (not strict) monoidal category with associativity constraint $\Phi$. Let $J_{UV}: U^*\otimes V^*\to (V\otimes U)^*$ the canonical isomorphism for every objects $U,V\...

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194 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 ...

**3**

votes

**0**answers

168 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 ...

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38 views

### Analogy between Mac Lane's coherence theorem and independence of bracketing

It is an elementary fact that if a binary operation on a set is associative then any multi-term product can be written unambiguously without bracketing. Superficially the coherence theorem for ...

**2**

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44 views

### Branching behavior in string diagrams/monoidal categories?

I am currently working through Peter Selinger's paper "Towards a Quantum Programming Language", and trying to connect it with what I already know about monoidal categories and string diagrams.
...

**2**

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67 views

### Deligne tensor product of semisimple tensor categories

Let $T_1, T_2$ be two semisimple tensor categories over a field $k$ (i.e. symmetric rigid monoidal abelian $k$-linear). Then is the Deligne product, $T_1\otimes T_2$ also a $k$-tensor category?
...

**2**

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109 views

### How to compute (co)limits of enriched categories?

I've asked this question on math.stackexchange some time ago (http://math.stackexchange.com/questions/1380176/how-to-compute-colimits-of-enriched-categories) and I received no complete answers, so I'm ...

**2**

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73 views

### Involution of unital based ring (Grothendieck ring of a fusion category)

Let $A$ be a unital based ring in the sense of [Ostrik, arXiv:math/0111139]. As part of the data we have a base $B = \{b_i\}_{i\in I}$, and an involution $i \mapsto \bar i$ of $I$ whose induced map $\...

**2**

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81 views

### Notions of/References for freely generated (symmetric) monoidal categories

We often describe a category by giving a (directed, multi-)graph and freely generating a category of paths. I would like to know to what degree this intuition generalizes to monoidal categories, and ...

**2**

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147 views

### Pseudomodules, “general coherence theorem”

A pseudomonoid is defined within a monoidal bicategory. It is like a monoid in a monoidal category except that the usual axioms hold up to coherent invertible 2-cells. Pseudomonoid is like a monoidal ...

**2**

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80 views

### A question about braiding represented as pseudofunctors

I just asked a very similar question here (About symmetry, braids, and pseudo-functors.), still now I dont get a answere. I hope to express myself more clearly and correctly this time.
Let $(\...

**2**

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**0**answers

164 views

### Compatibility between strength and costrength of a monoidal monad

Let $C$ be a closed monoidal category, and let $T : C \to C$ be a monad on the underlying category. Let $\sigma$ be a tensorial strength of $T$ and let $\sigma'$ be a cotensorial strength of $T$. A ...

**2**

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217 views

### Braidings and Isomorphism Classes in a Monoidal Category

Let $X$ be an object in a monoidal category $({\cal C}, \otimes)$, and $\gamma:X \otimes X \to X \otimes X$ a braiding (that is to say a morphism in ${\cal C}$ from $X \otimes X$ to itself that ...

**1**

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68 views

### Non-trivial but simple concrete examples for some categories related to Tensor/Fusion categories

I'm writing a note on Tensor and Fusion categories, the readers of which are physicists rather than mathematicians. So instead of giving abstract definitions I have to give examples to inspire each ...

**1**

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**0**answers

281 views

### Does the Boardman-Vogt tensor product of operads commute with their W-construction

I have absolutely no idea whether this is true or not but it could well be useful for me in the future if it is. If we have topological operads $\mathcal{P}$ and $\mathcal{Q}$ and we let $W$ denote ...

**1**

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117 views

### How does one compute a colimit of monoidal categories?

The question is in the title. I'm also happy to get answers about (your favorite adjective) monoidal categories.
Here's a guess:
In order to compute a colimit of monoids we can push everything down ...

**1**

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191 views

### The bar construction or the quotient for monoidal category action on a category

Given a monoid $C$ acting (from the right) on a set $M$, there is a bar construction giving a simplicial set or equivalently a translation/action category,
$$
N_\bullet (M\rtimes C)= \cdots M\times C^...

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132 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 $F : \text{Mod}(R) \to C$, ...

**0**

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45 views

### Differential categories vs McBride's notion of derivative

Has anyone done an analysis to see if Blute, Cockett, and Seely's differential categories suffice to provide a notion of 1-hole context in the symmetric monoidal setting?