The monoidal-categories tag has no usage guidance.

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### String diagrams for bimodules over noncommutative algebras?

I'm trying to do some calculations with bimodules (over Azumaya algebras, as it happens), and I need a string diagram notation that mixes the tensor product over the base ring (a symmetric monoidal ...

**2**

votes

**1**answer

221 views

### Localization of symmetric monoidal category

Let $\mathcal M$ be a symmetric monoidal category, $S\subset \mathcal M$ a collection of objects and morphisms. I would like to construct the localization $\mathcal M \mathop{\longrightarrow}^T ...

**2**

votes

**1**answer

139 views

### Does the category PCM (partial commutative monoids) have a closed symmetric monoidal product?

A partial commutative monoid (PCM) is, roughly speaking, a set with a partially defined binary operation that is as associative as it can be (given that not all products are defined) and commutative. ...

**5**

votes

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222 views

### How to construct a free 2-group on a groupoid?

Let G
be a groupoid. I'm wondering how to construct the free 2-group on G.
By the free 2-group I mean a 2-group $\mathcal{F}\left(G\right)$
equipped with a functor ...

**3**

votes

**1**answer

145 views

### Strictifying strong monoidal functors

Let $C_1$ and $C_2$ be monoidal categories (not necessarily symmetric or strict) and let $\Psi : C_1 \rightarrow C_2$ be a strong monoidal functor. Is it possibly to construct a strict monoidal ...

**11**

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**1**answer

397 views

### Categorical interpretation of disjoint cycle notation for tracing permutations

For a natural number $n\in\mathbb{N}$, let $\underline{n}$ denote the finite set $$\underline{n}:=\{1,2,\ldots,n\}.$$
A permutation $\sigma\in Aut(\underline{n})$ can be uniquely written (up to order) ...

**2**

votes

**1**answer

242 views

### Inducing a Monoidal Structure using an Equivalence of Categories [closed]

Given an equivalence of categories $C \equiv D$, such that $C$ has a monoidal structure, is it clear that we can use the equivalence to induce a monoidal structure on $D$. Is there a standard ...

**11**

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**1**answer

194 views

### Understanding the computation of the center of Tambara-Yamagami fusion categories when realized as C* categories

Recall that the Tambara-Yamagami categories are those with fusion ring $\mathbb{Z}[A \cup m]$ where $A$ is an abelian group and $m$ is a non-invertible (simple) object such that $ma = am = m$ for all ...

**2**

votes

**1**answer

189 views

### When is an Eilenberg-Moore category or Kleisli category braided monoidal? When semisimple?

I have a braided monoidal, semisimple linear category $\mathcal{C}$. (Imagine representations of a semisimple quasitriangular Hopf algebra.) I also have a monad $(T,\mu,\eta)$ on it, however, $T$ is ...

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votes

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178 views

### symmetric monoidal double categories?

Let me preface this by saying that I don't know much category theory.
I am running into a situation where I have a double category and additionally there is a multiplication. Moreover, choosing ...

**5**

votes

**2**answers

182 views

### Why is a braided left autonomous category also right autonomous?

In Selinger, P. A survey of graphical languages for monoidal categories (New Structures for Physics, Springer, 2011, 813, 289-233), it is stated that:
Lemma 4.17 ([23, Prop. 7.2]). A braided ...

**6**

votes

**1**answer

211 views

### Making additive envelopes of monoidal categories monoidal

I have a monoidal category $(\mathcal{C},\otimes)$ enriched over abelian groups, for which I want to take the additive envelope $\mathcal{M}at\,\mathcal{C}$.
(This is defined as the category with ...

**2**

votes

**0**answers

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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votes

**3**answers

297 views

### Rank vanishing in tensor categories

Let $\mathcal{C}$ be a cocomplete $\mathbb{Q}$-linear symmetric monoidal category (if convenient, assume that it is locally finitely presentable and/or abelian). Recall that the rank (or dimension) of ...

**12**

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**1**answer

276 views

### Does every commutative variety of algebras have a cogenerator?

By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes.
Equivalently (i) every interpretation of ...

**3**

votes

**3**answers

458 views

### When does a monoidal functor between ribbon categories preserve cups and caps, but not necessarily braidings?

Imagine a monoidal functor between ribbon categories (i.e. monoidal, with a braiding, a twist and compatible left and right duals). An important example would be the restriction functor from the ...

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votes

**1**answer

551 views

### Can one explain Tannaka-Krein duality for a finite-group to … a computer ? (How to make input for reconstruction to be finite datum?)

Consider a finite group. Tannaka-Krein duality allows to reconstruct the group from the
category of its representations and additional structures on it (tensor structure + fiber functor). Somehow ...

**4**

votes

**1**answer

109 views

### Does trace handle composition in a traced symmetric monoidal category?

Suppose that $(C,\otimes,I)$ is a traced symmetric monoidal category (TSMC) with symmetrizor $\sigma$ and trace $Tr$. Given two morphisms $f\colon A\to B$ and $g\colon B\to C$, I can tensor them to ...

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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 ...

**12**

votes

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2k views

### History and motivation for Tannaka, Krein, Grothendieck, Deligne et al. works on Tannaka-Krein theory?

I am trying to wrap my mind around Tannaka-Krein duality and it seems quite mysterious for me, as well, as its history. So let me ask:
Question: What was the motivation and historical context for ...

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votes

**1**answer

274 views

### Reference for a generalization of Γ-spaces to monoidal model categories

Γ-spaces were introduced by Segal in 1969 as models for what can be now described
as commutative ∞-monoids and ∞-groups in cartesian symmetric monoidal ∞-categories, e.g., E_∞-spaces and connective ...

**12**

votes

**1**answer

322 views

### Associators, Grothendieck-Teichmüller group and monoidal categories

The standard definition of an associator seems to be that it a a grouplike power series in two variables $x$ and $ y $ satisfying some pentagon and hexagon relations.
In other words, denoting by $ ...

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votes

**2**answers

420 views

### It looks so coKleisli, but it's not. What is it?

Fix a symmetric monoidal category $(M,\otimes,I)$ and a small discrete monoidal subcategory $M'\subseteq M$. Define a new symmetric monoidal category $C:=CoKl(M,M')$ as follows: $Ob(C):=Ob(M)$, and ...

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331 views

### Is “being a modular category” a universal or categorical/algebraic property?

A semisimple braided category with duals is called modular when a certain matrix $S$ is invertible. The components $S_{AB}$ are indexed by (isomorphism classes of) simple objects of the category and ...

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votes

**1**answer

252 views

### Functors with an epi-mono factorization property

This is a simple question about terminology and a request for any related references. Specifically, what would you call a functor $F:\mathbf{D}\rightarrow\mathbf{C}$ with the following property?
...

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votes

**1**answer

262 views

### Two definitions of modules in monoidal category

The standard definition of a (left) simplicial module $V$ over some simplicial algebra $A$ is the map of simplicial vector spaces $A\otimes V\to V$ that gives the usual modules component-wise. Here ...

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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 ...

**6**

votes

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270 views

### About an embedding of abelian categories into categories of modules

Let $k$ be a field. Let $C$ be an abelian $k$-linear category with a symmetric tensor product $\otimes$ and internal homomorphisms, such that $\mathrm{End}(1)=k$. Let $M$ be another $k$-linear abelian ...

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174 views

### Monoidal transformations are isomorphisms at dualizable objects

Here is a cute observation: Let $F,G : \mathcal{C} \to \mathcal{D}$ be a symmetric monoidal functors between symmetric monoidal categories, and let $\eta : F \to G$ be a monoidal transformation. Then ...

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213 views

### Definitions and coherence in “rigid” monoidal categories

In "Catégories Tannakiennes" by Savedra Rivano (under A. Grothendieck supervision) at pag.78 he define a rigid category $\mathscr{C}$ as a monoidal symmetrical closed such that the natural morphisms ...

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314 views

### Generalization of analytic functors

It's been a long time since I posted the following question on stackexchange.
Now I think it's better to adress it to you, in the hope I will reach the right audience: Martin's comment, albeit ...

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238 views

### Coherence in pseudo.monoids

In the article of A. Joyal and R. Street:
Braided tensor categories, (Adv. Math. N.102 (1993), no. 1, 20-78)
they define a tensor object (or pseudo.monoids) in a monoidal 2-category $\mathcal{C}$ ...

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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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445 views

### Cartesian envelope of a symmetric monoidal category

Is there a left adjoint to the forgetful functor from cartesian monoidal categories to symmetric monoidal categories? And if so: how can it be described explicitely.
In more detail: Given a symmetric ...

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votes

**1**answer

571 views

### Reference for “lax monoidal functors” = “monoids under Day convolution”

Suppose $A$ and $C$ two symmetric monoidal categories. Let's say that $A$ is small and $C$ is locally presentable, and let's assume also that the tensor product on $C$ preserves colimits separately in ...

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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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votes

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401 views

### “Wrong” strictification of symmetric monoidal categories

It is well-known that any symmetric monoidal category is equivalent to a strict symmetric monoidal category. The construction of this strict monoidal category is rather technical and it appears to me ...

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421 views

### K-theory of monoidal categories

I am novice in the algebraic K- theory and don' t know if this is the right place for the following questions. So some people might consider them as basic questions.
Consider an exact monoidal ...

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votes

**1**answer

725 views

### Are Lurie's operads special SMCs?

In Higher Operads, Higher Categories, Leinster nicely characterizes operads among monoidal categories (as PROs). Roughly speaking, a monoidal category comes from an operad if its set of objects is ...

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votes

**1**answer

413 views

### What about schemes built up out of graded rings?

Toen-Vaquié construct a category of schemes relative to some complete cocomplete closed symmetric monoidal category $C$. Affine schemes correspond by definition 1:1 to commutative monoid objects in ...

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votes

**1**answer

287 views

### Unitalization internal to monoidal categories

Let $C$ be a monoidal category, not assumed to be symmetric. Assume that the underlying category of $C$ is nice enough, for example cocomplete, perhaps even presentable. A semigroup object in $C$ is a ...

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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 ...

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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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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 ...

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**1**answer

269 views

### Example of a non-closed cocomplete symmetric monoidal category

Background
By a cocomplete symmetric monoidal category $C$ I mean a symmetric monoidal category whose underlying category is cocomplete and such that $- \otimes X : C \to C$ is cocontinuous for all ...

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523 views

### The symmetric monoidal category of finite sets

It is well-known that the (augmented) simplex category is the universal monoidal category with a monoid object. What about a commutative analogue? Consider the category $\mathsf{FinSet}$ of finite ...

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322 views

### Not-so-symmetric monoidal categories

Is the notion of a commutative monoidal category, where every product is isomorphic to its opposite, but not necessarily functorially, known not to be useful? I have not been able to find any ...

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785 views

### Free symmetric monoidal category on a monoidal category

Consider the $2$-categories
$\mathsf{MonCat}$ of monoidal categories, with strong monoidal functors and monoidal transformations,
$\mathsf{SymMonCat}$ of symmetric monoidal categories, with strong ...

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177 views

### Seems like Reader monad composed with a strong monad produces a monad, am I right?

Take a Cartesian (or monoidal) closed category; define Reader monad for a given object $E$ as
$X \mapsto X^E$; and take a strong monad $M$ (strong means preserves product or tensor product).
Now the ...

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687 views

### What is the free monoidal category generated by a monoid?

In several places in a segment on cohomology (for example, here (PDF)) in John Baez's online lecture notes for a course in 2007 on quantum gravity, much is made of the fact that the simplex category ...