The monoidal-categories tag has no usage guidance.

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### Balanced Tensor Product of Module Categories

(Moved from MSE)
Let C be a k-linear (Vectk-enriched) monoidal category and consider the 2-category Mod_C of k-linear (C,C)-bimodule categories in the sense of Ostrik (https://arxiv.org/abs/math/...

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**1**answer

114 views

### When is a Module category monoidal?

Let $\mathcal{C}$ be a monoidal category and $M$ a left module category over $\mathcal{C}$. That is, a category equipped with an exact bifunctor $F:\mathcal{C}\otimes M\rightarrow M$ satisfying some ...

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121 views

### Completeness of 2-category of Monoidal Categories

Is the 2-category of monoidal categories complete? If not, can any conditions be imposed to satisfy completeness?

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**1**answer

193 views

### Coherence theorem for symmetric lax monoidal functors

Let $V$ and $W$ be symmetric monoidal categories. Let $F:V\to W$ be a lax symmetric monoidal functor with multiplication $\nabla:FA\otimes FB \to F(A\otimes B)$. Consider the following statements:
1) ...

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189 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,...

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40 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 ...

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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?

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316 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 ...

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**1**answer

90 views

### When are Morita classes represented by certain structured algebra objects?

Let $\mathcal{C}$ be a monoidal category. There is a notion of Morita equivalence of algebra objects internal to $\mathcal{C}$. Does each Morita class have a symmetric Frobenius representative? A Hopf ...

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**1**answer

96 views

### Self-enrichment of reflective subcategories of self-enriched categories

I'll go straight to the point of my question:
Say $\mathscr{A}$ is a reflexive subcategory of $\mathscr{B}$, meaning the inclusion functor $i: \mathscr{A} \to \mathscr{B}$ is fully faithful and ...

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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 (...

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174 views

### Twisted Day convolution

Has anyone studied a version of Day convolution for an enriched presheaf category $V^{A^{\mathrm{op}}}$ where the monoidal structure of $V$ is "twisted" on one side by an action of $A$? I'm thinking ...

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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.
...

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132 views

### Is the biproduct of dualizable objects itself dualizable

In a monoidal category with biproducts, let $A$ and $B$ be objects with right duals. Then does $A \oplus B$ have a right dual?
The question is a bit subtle. Suppose I already know that $A \oplus B$ ...

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99 views

### When modular tensor categories are equivalent?

I asked this question at math stack exchange math stack exchange but I haven't got any answer yet there.
I would like to know when we say that two modular tensor categories are equivalent.
Is it ...

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68 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?
...

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121 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 ...

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113 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 ...

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**1**answer

142 views

### Seeking more information regarding the “rigoidal category” of $\mathbb{N}$-graded sets

Definitions.
By an $\mathbb{N}$-graded set, I mean a set $X$ together with a function $|\Box|_X:\mathbb{N} \leftarrow X,$ called the grading. These will simply be called graded sets hereafter.
If $Y$...

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119 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 ...

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190 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 $...

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69 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 ...

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156 views

### Mac Lane strictness theorem and categorifiability of fusion rings

The Mac Lane strictness theorem states that any monoidal category is monoidally equivalent to a strict monoidal category (see here section 2.8).
Q1: Is it true that any fusion category is monoidally ...

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443 views

### What categorical property of monoidal categories picks out the ones with duals?

Recall that a monoidal category $\mathcal C$ is rigid if every object $X\in \mathcal C$ has both left and right duals, i.e. objects $X^l$ and $X^r$ with maps $X^l \otimes X \to \mathbf 1 \to X \otimes ...

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128 views

### The definition of unitary fusion category

I just come across a definition of the unitary fusion category:
A fusion category $\mathcal{C}$ over the complex number is said to be unitary if we have:
We have a Hilbert space structure on each ...

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158 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 $n$-...

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218 views

### Enriching categories and equivalences

Let $\mathcal{C}$ and $\mathcal{D}$ be two equivalent categories. Furthermore, assume $\mathcal{C}$ is enriched over a monoidal category $(\mathcal{M}, \otimes)$. Can one use the equivalence to ...

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259 views

### Conditions on the fusion data of symmetric fusion category

We know that every symmetric fusion category (SFC) gives rise to data
$N^{ij}_k$ that describe the fusion of simple objects:
$i\times j = N^{ij}_k k$, and the data $\theta_i =\pm 1$ that describe the ...

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98 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....

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### 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 ...

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113 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 ...

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182 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 ...

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### 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 $\...

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81 views

### Semisimple monoidal category with duals

We say that an abelian category is semisimple if every object is a semisimple object, which is to say, a direct sum of finitely many simple objects.
Let $({\cal C},\otimes,*)$ be a semisimple ...

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144 views

### About a closed strucure on profunctors

Let $Prof$ the bicategory with profunctors (on small categories), arrows are like $D: \mathscr{A} \dashrightarrow \mathscr{B}$ and this means that $D: \mathscr{A}^{op} \times \mathscr{B}\to Set$.
...

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112 views

### Adjointable Abelian Monoidal Functor

Given two abelian monoidal categories ${\cal C,D}$ (where the monoidal operation is bilinear) and an additive monoidal functor $F:{\cal C} \to {\cal D}$. Will $F$ always admit an adjoint?

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239 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) \...

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368 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 ...

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687 views

### Does projective imply flat?

Let $\mathcal C$ be an abelian category equipped with a closed symmetric monoidal structure. This implies in particular that the monoidal structure $\otimes$ is right exact in each variable. I care ...

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237 views

### Categorical definition of infinite symmetric product

Let $(C,\otimes,I)$ be a symmetric monoidal category with coequalizers and directed colimits.
Fix some object $X$ and morphism $\tau\colon I\to X.$
Using $\tau$ one can construct a sequence of ...

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452 views

### A cosmos where coproduct injections are not monic

The injections (coprojections) of a coproduct in a category are very often monomorphisms. For instance, this happens in any extensive category (essentially by definition) and also in any category ...

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### Reference for an unbiased definition of a symmetric monoidal category

In the references I know, a symmetric monoidal category is defined as a monoidal category equipped with some additional data, so in particular the monoidal product is a functor
$$
\mathcal{C}\times \...

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271 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 ...

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282 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 ...

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381 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 $k$-...

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### 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 ...

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82 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 ...

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### Proofs in monoidal categories [closed]

I have to do some pretty ugly proofs in monoidal categories. Basically, I have some long identities that I would like to prove. A random example:
$$(a\otimes b)\circ (c\otimes d) \circ q = q $$
Are ...

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360 views

### Computation of extreme rays of rational polyhedral cones - Hemmecke's project and lift algorithm

I am working on an implementation of Raymond Hemmecke's algorithm for finding generating sets of cones: http://arxiv.org/abs/math/0203105
Unfortunately I am struggling to make the algorithm work on ...

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### 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 ...