# Questions tagged [monoidal-categories]

A monoidal category, also called a tensor category, is a category $\mathcal{C}$ equipped with a bifunctor $\otimes\colon \mathcal{C}\times\mathcal{C}\to \mathcal{C}$ which is associative up to a natural isomorphism, and an object $\mathbb{1}$ which is both a left and right identity for $\otimes$ up to a natural isomorphism.

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### Monoidal categories are one-0-cell bicategories

I just saw this definition/interpretation of monoidal categories and I have two questions about it. The idea is to consider the objects of the monoidal category as 1-cells and the tensor product as 2-...
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### Reference request for monoidal structure on reflective subcategory

I am fairly sure the following lemma is folklore, but for some weird reason I am unable to track it down in the literature. Suppose $i:\mathcal{C}\hookrightarrow \mathcal{D}$ is a full reflective ...
1 vote
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### Characterization of monoid-realizable symmetric monoidal categories

Given any symmetric monoidal category $\mathbf{C}$, $\mathrm{Mon}(\mathbf{C})$ (the category of monoids, not necessarily commutative, in $\mathbf{C}$) is also a symmetric monoidal category. Now, is ...
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### Definition of monoidal category without set theory and without assuming small categories

I am interested in finding a definition of monoidal category that ultimately relies only on category theory elements such as objects, arrows, and categories. For instance, it should never assume we ...
1 vote
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### Left adjoint to the inclusion of semicocartesian symmetric monoidal categories in symmetric monoidal categories

Let $\mathbf{SMC}$ be the $2$-category of symmetric monoidal categories and strong symmetric monoidal functors. Also, let $\mathbf{SMC}_{0}$ be the full sub-$2$-category of $\mathbf{SMC}$ on the ...
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### What do you call monoidal categories where you can only delete on the left?

Cartesian monoidal categories come with projection morphisms that let you go from $A \otimes B$ to $A$ or from $A \otimes B$ to $B.$ I am wondering if there are kinds of monoidal categories which just ...
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### Freyd's algebraic real analysis, scale and linear order

I have been studying Freyd's algebraic theory of the reals in the past week. I have problems understanding his linear representation theorem (Theorem 8.1) that every scale can be embedded in a product ...
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### Construction of the monoidal structure on Set

The definition of a monoidal structure on $\textbf{Set}$ takes as product structure the direct (categorical) product of sets $X \times Y$. But this product is only defined up to unique isomorphism - ...
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### The functor category $[\mathbf{G},\textbf{Set}]$ is Cartesian closed. What is the explict description of the closed monoidal structure of it?

As the title says. We have the category of functors from a group $\mathbf{G}$ to $\mathbf{Set}$. The objects are the functors, the morphisms natural transformations between them. This is also another ...
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### Trouble justifying why the $S$-matrix of $\mathcal{C}(G,q)$ is $(b(g,h))_{g,h\in G}$

Let $G$ be a finite abelian group and $q: G \to k^\times$ be a quadratic form with $k$ a field of characteristic $0$. i.e. $(G,q)$ is a pre-metric group. It is known that such pre-metric groups ...
1 vote
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### Currying in the category of topological pairs

Let's denote by $\mathsf{Top}(2)$ the category of topological pairs. That is, let $\mathsf{Top}(2)$ be the category whose objects are all the pairs $(X,A)$ where $X$ is a topological space [*] ...
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### Verify evaluation and coevaluation in monoidal category $\mathsf{Vec}_G^\omega$ for a normalized cocycle $\omega$

$\newcommand{id}{\operatorname{id}}$ $\newcommand{ev}{\operatorname{ev}}$ $\newcommand{coev}{\operatorname{coev}}$ This question refers to Etingof, Gelaki, Nikshych, and Ostrik's book Tensor ...
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### Unit in strict monoidal category

Let $(\mathcal{C}, \otimes, 1)$ be a strict monoidal category. If there exists an isomorphism between $1$ and an object $X$, does it follow that $1=X$? (if not, can you give a counterexample?).
1 vote
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### Homotopy category of a symmetric monoidal $(\infty,1)$-category is symmetric monoidal

Let $\mathcal{C}$ be symmetric monoidal $(\infty,1)$-category, that is, a functor $\mathcal{C} \colon \Delta^{\text{op}} \times \Gamma^\text{op} \to \text{sSet}$, where $\Delta$ is the simplex ...
1 vote
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### Monoidal category with initial object

Let $(C, \otimes, 1)$ be monoidal category with initial object $0$. Is it true that $0 \otimes a \simeq 0$ for all $a \in C$? It is true in Set with cartesian product, $R-$modules with usual tensor ...
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### Hochschild homology of stable categories as topological chiral homology

Let $\mathscr{C}_0$ be a small idempotent complete stable category tensored over some symmetric monoidal category $\mathcal{E}$. Its Ind-completion $\mathscr{C} := \operatorname{Ind}(\mathscr{C}_0)$ ...
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### Construction of Free modules in a monoidal category

Let $(C,\otimes , 1^C , \alpha , \lambda , \rho)$ be a monoidal category and let $\mathsf{Mon}_C$ be the category of monoids in $C$. And let $A \in \mathsf{Mon}_C$. Denote the category of $A$-modules ...
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### Free object in the category of (not necessarily strict) monoidal category and lax (resp. oplax) monoidal functors.

Let $\mathsf{MonCat}_{\mathrm{lax}}$ (resp. $\mathsf{MonCat}_{\mathrm{oplax}}$) be the categories of (say small) not necessarily strict monoidal categories and lax (resp. oplax) monoidal functors. Let ...
1 vote
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### Underlying $\infty$-category of monoidal envelope

I'm learning Jacob Lurie's book Higher Algebra (HA for brief). HA 2.2.4. is about the monoidal envelope for fibration of $\infty$-operads. In HA 2.2.4.3. Lurie claimed that $\mathrm{Env}(\mathcal{C})$ ...
1 vote
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### Can a unit object in a strict monoidal category have any nontrivial endomorphisms?

I'm learning about strict monoidal categories and saw something interesting regarding the structure of units, which I hadn't seen before, and wanted to see if this was right. Suppose we have a strict ...
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1 vote
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### Do symmetric monoidal functors preserve reflexivity?

Suppose that $(C, \otimes_C, \mathcal{H}om_C, I_C), (D, \otimes_D, \mathcal{H}om_D, I_D)$ are closed symmetric monoidal categories. Let $$F: (C, \otimes_C, I_C) \rightarrow (D, \otimes_D, I_D)$$ ...
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### MacLane’s coherence theorem and free monoidal categories

Original question The nLab gives one formulation of the coherence theorem for monoidal categories: Every diagram in a free monoidal category made up of associators and unitors commutes. It seems to ...
1 vote
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I am looking for a reference to the following theorem. If $F:\mathcal{C}\to\mathcal{D}$ is a symmetric lax monoidal functor between symmetric monoidal categories, then it induces a well-defined ...
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### Can every (co)complete (in some sense) category be equipped with a monoidal structure?

Follow up to this question. Here the answers use the fact that not every category has an object whose endomorphism monoid is commutative. This can be fixed by for example only considering categories ...
By definition, a monoidal category is a category $\mathbf{C}$ equipped with a bifunctor $$\otimes :\mathbf{C} \times \mathbf{C} \rightarrow \mathbf{C}$$ and an object $I$ that is both a left and ...