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 ...
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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 ...
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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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Does the currying of a closed abelian monoidal category preserves addition?
Let $V$ be a left-closed abelian monoidal category. That is, $V$ is a left-closed monoidal category which is also an abelian category. Let $\Phi:\hom(y\otimes x,z)\to\hom(x,[y,z])$ be the currying of $...
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different notions of functor products
There are two constructions which deserve to be called product of functors:
The first is the morphism part of the product functor $\times: \mathcal{Cat} \to \mathcal{Cat}$, sending a pair $F: A \to A'$...
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Is there a classification of symmetric monoidal closed structures on $\mathbf{Ab}$?
How much is known/has been studied about symmetric monoidal closed structures on the category of abelian groups $\mathbf{Ab}$ up to equivalence? Is there any nice characterization of the usual tensor ...
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Statement and Proof of Deligne's Theorem
I am interested in Deligne's Theorem regarding how all tensor categories satisfying some "nice" properties are equivalent to $\textrm{Rep}(G, \epsilon)$ where $G$ is a supergroup. I have two ...
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Intuition behind large diagrams in category theory
I am attempting to read "Tensor Categories" by Pavel Etingof et. al.
The following pentagon axiom is a part of the definition of a monoidal category:
The following diagram is part of the ...
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General category with finite products as a cartesian monoidal category
According to Wikipedia:
Any category with finite products (a "finite product category") can be thought of as a cartesian monoidal category.
I cannot see this for a general category other ...
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Categories which are not monoidal
I am reading about monoidal categories and I am not able to think of categories which are non-monoidal. Am I thinking in the wrong direction?
Is being monoidal, an additional property like topology?
...
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Why are PROPS and their morphisms strict?
According to both nlab and wikipedia, PROPS are defined as special kinds of strict symmetric monoidal categories and morphisms between them are defined as strict symmetric monoidal functors. Why is ...
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Internalizing cyclic groups?
The nLab-article on internalization describes how to define certain algebraic structures internally to a category-with-structure. This gives, for example, the notion of a group object in a category ...
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Opposite of braided monoidal category
I am currently learning about the theory of monoidal categories, and something is bothering me. I want to define the opposite category $C^{op}$ of a (braided) monoidal category. My guess would be that ...
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A question about monoidal functors
For two monoidal categories $(M,\otimes)$ and $(N,\circledast)$, let $F:M \to N$ be a monoidal functor, and denote the structure natural isomorphism of $F$ by $J_{m,n}: F(m) \circledast F(n) \to F(m \...
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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 ...
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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?).
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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 ...
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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 ...
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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 $\mathrm{Env}(\mathcal{C})$ ...
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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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Deligne’s tensor product of algebra module categories
$A$ and $B$ are finite dimensional $\mathbb{k}$-algebras. $\textrm{Mod}_A$ is the category of finite dimensional $A$-modules.
In Proposition 1.46.2. of the note，it is claimed that $\textrm{Mod}_{A\...
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Morphism can be pulled through braid crossings
I stumbled upon this statement in the context of monoidal braided categories. How do I see, as pointed out in the text, that this follows from functoriality? Here is the source
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How does one show that coev is independent of the choice of basis?
Given $\operatorname{coev} \colon k \rightarrow V \otimes V^{*}$ with the mapping $1 \mapsto \sum_{i} b_{i} \otimes b^{*}_{i}$ for left dual of $V \in \mathbf{vect}_{k}$.
How can I conclude from this ...
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Monoid structures induced on hom objects
If $a$ is a set and $b$ is some algebraic structure, then the set of functions $a\to b$ also has the obvious algebraic structure. Is there a popular categorification of this?
More abstractly, let $V$ ...
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The category of graded $\mathbb S$-modules form a monoidal category
I am reading paragraph 6.2 in Algebraic Operads by Jean-Louis Loday and Bruno Vallette. Proposition 6.2.2 states:
The category of graded $\mathbb S$-modules, with
the (composite) product $\circ$ and ...
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In the coherence theorem for braided monoidal categories, why does it suffice to show the result for strict monoidal categories only?
$\newcommand{\M}{\mathcal{M}}\newcommand{\B}{\mathfrak{B}}\newcommand{\hom}{\operatorname{Hom}}\newcommand{\BM}{\mathsf{BM}}\newcommand{\SBM}{\mathsf{SBM}}\newcommand{\S}{\mathcal{S}}$I refer to the ...
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About Rigid monoidal subcategory
I have just sturted to study about monoidal categories and I am trying to solve this question...
Suppose $A=\mathbb{C}/(x^3)$ and $A_c$ be additive closure of $A$ and $A\otimes_{\mathbb{C}}A$ in $A-...
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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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Does the product always exist in a monoidal category?
Certainly there are categories where the product does not in general exist. Say for instance we had a category with the 3 objects $1, A$ and $B$ where $\otimes$ is defined as $1\otimes A=A=A\otimes1, ...
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Dual objects where only one zig-zag identity holds?
Recall the definition of a (left) dual object in a monoidal category.
If one requires that both the evaluation and the coevaluation are isomorphisms, one zig-zag-identity implies the other (see here).
...
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Does every closed symmetric monoidal category admit a faithful monoidal functor to Sets?
Is there an example of a closed symmetric monoidal category which does not admit a faithful, monoidal functor to $\mathrm{Sets}$? (Here $\mathrm{Sets}$ has the usual cartesian closed structure.)
The ...
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Adjoint functor theorem applied to a forgetful functor
Let $\mathbf{Cat}$ denote the category of small categories and $\mathbf{MCat}$ the category of small monoidal categories with monoidal functors.
Consider the forgetful functor $\operatorname{U}:\...
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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 ...
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Symmetric lax monoidal functor takes operads to operads
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 ...
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Can every category be equipped with a monoidal structure?
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 ...
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Isomorphism between $\operatorname{End}(F\otimes F)$ and $\operatorname{End}(F)\otimes \operatorname{End}(F)$, where F is an exact faithful functor.
Let $\mathcal{C}$ be a finite $k$-linear abelian category, and $\operatorname{Vec}$ be the category of finite dimensional vector spaces over $k$. Let $F_1,\ F_2:\ \mathcal{C}\rightarrow \operatorname{...
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Topological order at finite temperature
For the past few decades the study of so-called topological orders has remained an active area of research. By definition (following the motivation section of this recent invitation to the subject, ...
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About "6j symbol": How to understand a vertor space tensor with an object in a tensor category?
Let $\mathcal{C}$ be a semisimple (multi)tensor categroy over field $k$, with simple objects $\{V_i\}_{i\in I}$. We define
$$
H_{i, j}^{\ell}=\operatorname{Hom}_\mathcal{C}\left(V_{\ell}, V_{i} \...