The monoidal-categories tag has no usage guidance.

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### 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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174 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 ...

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708 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**

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111 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 ...

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180 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:
...

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263 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 ...

**7**

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143 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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110 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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247 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 ...

**4**

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54 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**

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169 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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148 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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99 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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361 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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258 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 ...

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266 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 ...

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238 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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119 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

171 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 ...

**3**

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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**

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165 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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80 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 ...

**2**

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98 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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67 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 ...

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76 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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129 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 ...

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158 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 ...

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214 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 ...

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51 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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204 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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110 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 ...

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185 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 ...

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

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78 views

### the category of right comodule of coalgebra is a monoidal category , why?

the category of right comodule of coalgebra is a monoidal category according the following
the associativity constraint is defined as
a_{U,V,W}((u\otimes v)\otimes w)=u_0\otimes (v_0\otimes ...

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47 views

### Monoidal product over database table entries

We have seen from Spivak that database schemas are categories, having tables as objects and relationships as morphisms. I am wondering if we can have a monoidal product over objects, that is, over ...