Questions tagged [monoidal-categories]
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Braided monoidal category, example
Let $H$ be a cocommutative hopf algebra.
Let $M$ be the category of $H$-bimodules.
Does the category $M$ form a braided monoidal category with tensor product $\otimes_{H}$ ?
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Abelianization of monoids in arbitrary (symmetric) monoidal categories
Under which 'minimal' conditions on a symmetric monoidal category does an abelization functor from its monoids to its commutative monoids exist?
More precisely: let $(\mathcal{C},\otimes,1)$ be a ...
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Deligne Tensor Product of Categories, Explicit Equivalence of $A\otimes_\mathbb{C} B\text{-Mod} \cong A\text{-Mod}\boxtimes B\text{-Mod}$
$\newcommand\Mod[1]{#1\text{-Mod}}$Does any one have a reference on a explicit equivalence between
$$\Mod{A\otimes_\mathbb{C} B} \cong \Mod A\boxtimes \Mod B?$$
The proof in "Tensor Categories ...
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Are there non-semisimple complex "non-unital special Frobenius algebras"?
I'm interested in "non-unital special Frobenius algebras", consisting of two linear maps (morphisms in the symmetric monoidal category of finite-dimensional complex vector spaces)
$$\mu: V\...
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Constructing the inverse of a braiding in a braided pivotal category
Assume we have a braided pivotal monoidal category. This means we assume the braiding $c$ to be a natural isomorphism. But looking at the corresponding string diagram, it seems to me as if we could ...
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Drinfeld center of $\mathrm{Mod}_R$
Let $R$ be a commutative ring and let $\mathrm{Mod}_R$ be the category of (left) $R$-modules.
Question: Is it true that the categories $\mathcal{Z}(\mathrm{Mod}_R)$ and $\mathrm{Mod}_R$ are ...
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Recovering the center of a monoid from the Drinfeld center
The Drinfeld center construction is intended to be a categorification of the center of a monoid. It seems to be folklore (eg this answer or this one) that when the Drinfeld center is taken over a ...
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Coherence for closed bicategories
A right-closed bicategory [1] is a bicategory that has all right extensions (i.e. right adjoints to precomposition with a fixed 1-cell). A one-object right-closed bicategory is therefore a right-...
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Obstruction to delooping
Let $G$ be a finite group. It can be think of as a $1$-category with one object and $|G|$ many morphisms. If $A$ happens to be abelian, then one can think of it to an $n$-category. Conversly, this ...
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"Boundaries" in Free Simplicial Monoids
I suspect that this has been addressed somewhere already, but I cannot find anything. Let $\hat{\Delta}$ denote simplicial sets, and $Mon\hat{\Delta}$ simplicial monoids. There is a forgetful functor $...
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Self-enrichments of rigid monoidal categories
A right closed monoidal category $\mathcal{C}$ is one for which the functor $X \otimes -$ admits a right adjoint, for every $X \in \mathcal{C}$. As explained here
Enrichments vs Internal homs
this ...
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Frobenius algebras and traces of modules
$\newcommand{\Hom}{\mathscr{Hom}}$
Let $C$ be a cocomplete closed symmetric monoidal category, and the tensor product preserves colimits in each variable;
Let $A$ be a commutative algebra in $C$, ...
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Being (co)cartesian as a property (rather than structure) of a plain monoidal category
It's well-known that being cocartesian is a property of a symmetric monoidal category: $(\mathcal C,I,\otimes,\sigma)$ is cocartesian if and only if the following conditions are satisfied:
Every ...
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Proof that a Cartesian category is monoidal
If $\mathcal C$ is a category with products and a terminal object, then $\mathcal C$ is monoidal.
This seems obvious, but wherever I look for a proof or a reference it simply states that the proof is &...
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Derived category of an abelian monoidal category
For any abelian category $\mathcal{A}$, we can consider its derived category $\mathcal{D(A)}$, which is naturally triangulated. If $\mathcal{A}$ is endowed with a monoidal structure (bilinear with ...
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Example of a sub-bialgebra of a Hopf algebra that is not a Hopf subalgebra
It's all in the question! What is an example of a sub-bialgebra of a Hopf algebra that is not a Hopf subalgebra?
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Monoid objects constructed from duals
Let $(M,\otimes)$ be a rigid monoidal category, for which left and right duals coincide. For any object $X \in M$, we can define a monoid structure on $X \otimes X^*$: Multiplication is defined by ...
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Monoidal categories whose tensor has a left adjoint
Is there a name for monoidal categories $(\mathscr V, \otimes, I)$ such that $\otimes$ has a left adjoint $(\ell, r) : \mathscr V \to \mathscr V^2$? Have they been studied anywhere? What are some ...
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Function spaces satisfying $\mathcal{F}(M\times N)\simeq\mathcal{F}(M)\otimes\mathcal{F}(N)$
Let $M \mapsto\mathcal{F}(M)$ be a map associating topological vector spaces of some type (that I will call "function spaces") to geometric spaces $M$ of some type.
For $M$, I'm mostly ...
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Rank of a finite group and its representations
$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\rank{rank}$Let $G$ be a finite group, and $C=\Rep(G)$ be the monoidal category of complex finite-dimensional representations of $G$. As $C$ is finite ...
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Categorical presentation of direct sums of vector spaces, versus tensor products
My apologies in advance if this question is to vague, but here goes.... In the category of vector spaces, products are given by direct sums. In general category theory, the existence of products is a ...
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What is the difference between the Yang--Baxter equation and the quantum Yang--Baxter equation?
For a vector space $V$ and a linear operator $R:V \otimes V \to V \otimes V$, we say that $R$ satisfies the Yang--Baxter equation if
$$(R\otimes id)(id\otimes R)(R\otimes id)
= (id\otimes R)(R\otimes ...
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Yoneda lemma for monoidal categories
I am looking at the Yoneda lemma trying to see where the assumption of "locally small" really comes in. Obviously in order to define a functor to the category sets using $Hom$-spaces we need ...
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Comodule Morita equivalence for Hopf algebras
Let $A$ and $B$ be two Hopf algebras, and denote by $\mathcal{M}^A$ and $\mathcal{M}^B$ their respective categories of right comodules. If we have a monoidal equivalence between $\mathcal{M}^A$ and $\...
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Are cofibrations in topological monoids preserved by forming the product with the identity?
Consider the category $\mathrm{Mon}(\mathbf{Top})$ of topological monoids, together with the model structure transferred along the adjunction $F:\mathbf{Top}\rightleftarrows \mathrm{Mon}(\mathbf{Top}):...
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In a rigid monoidal category, why is $W^*\otimes V^*$ a left dual of $V \otimes W$? [closed]
Question: Let $\mathcal{C}$ be a monoidal category, $V,W$ in $\mathcal{C}$ are objects. Show that if $V, W$ have left duals $V^*, W^*$, respectively, then $V\otimes W$ has a left dual $W^* \otimes V^*$...
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When is an object determined by the number of maps from the other objects?
Let $C$ be a category with finite hom-sets.
Suppose that $X$ and $Y$ are objects in $C$ such that $C(Z,X)\cong C(Z,Y)$ for any Z (with no naturality condition).
For which categories $C$ does it follow ...
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Does the functor $\mathcal{C} \to \mathcal{Z}(\mathcal{C})$ have adjoints?
Let $\mathcal{C}$ be a braided monoidal category. We have a canonical functor $\mathcal{C} \to \mathcal{Z}(\mathcal{C})$ from $\mathcal{C}$ to the Drinfeld center $\mathcal{Z}(\mathcal{C})$ sending an ...
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References about "monoidal fibrations" in $\infty$-category theory
$\newcommand{\cat}{\mathsf} \newcommand{\fun}{\mathrm{Fun}} \newcommand{\calg}{\mathrm{CAlg}}$
Let $\cat C^\otimes,\cat D^\otimes, \cat E^\otimes$ be symmetric monoidal $\infty$-categories, and $p^\...
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Subobject classifiers with a quantale structure
Quantales were kind of popular a few decades ago, as some sort of "quantum" version of the much more widespread concept of locale (Mulvey introduced them to study $C^*$ algebras). I was ...
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Infinite coproducts and monoidal structure
It is a well-known fact that on categories admitting all finite coproducts one can define a monoidal structure where the monoidal product is exactly the coproduct and the monoidal unit is the initial ...
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monoidal (∞,1)-categories from weakly monoidal model categories
In Higher Algebra section 4.1.7, Jacob Lurie proves that the underlying $(\infty,1)$-category of a monoidal model category is a monoidal $(\infty,1)$-category.
Dominic Verity and Yuki Maehara have (...
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Non-counital coalgebras
For any unital algebra $A$, we have an associated dual coalgebra $A^{\circ}$. (Recall that it is defined to be the largest subalgebra of the $\mathbf{C}$-linear dual of $A$ such that the coproduct $\...
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Representation theory in braided monoidal categories
The crux of what I wish to know is what results from representation theory, a subject usually framed within the category $\text{Vect}_\mathbb{k}$, follow in more general braided monoidal categories? I ...
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Tensor product of unit and co-unit in a closed compact category
Consider a compact closed category, i.e., a symmetric monoidal category with a unit $\eta$ and co-unit $\epsilon$. It seems natural to demand that the tensor product of two units (for different ...
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A question on cokernel
I was reading the proof of Theorem 7.12.11 in the book "Tensor categories" by Etingof et al. Let $\mathcal{C}$ be a finite multi-tensor category, and $A$ be an algebra in $\mathcal{C}$. Let $...
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Left Kan extensions of "strong" monoidal functors
Consider the 2-category $\mathsf{MonCat}$ where objects are monoidal categories,
1-cells are strong monoidal functors, and 2-cells are monoidal natural transformations.
Given arrows $f: \mathsf{C} \to ...
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Braided category inside braided 2-category
Let $\mathcal{C}$ be a semistrict braided monoidal $2$-category in the sense of [BN] (so in particular a strict $2$-category). Let $\mathcal{C}_1$ be the category of $1$-morphisms (objects) and $2$-...
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Every monoidal category is equivalent to a strict monoidal category
I'm reading the book "Quantum groups" by Kassel. I'm reading the proof that every tensor category is tensor equivalent with a strict tensor category. Here is the proof given. Below is more ...
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Strictifying closed monoidal categories?
Let $C$ be a cartesian closed category. It's well known that $C$ is equivalent to a category where the product is strict monoidal; i.e. where there are equalities of the functors given by the ...
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Automated rewriting of string diagrams in symmetric monoidal categories
Many algebraic structures, such as Frobenius algebras, or quasi-triangular Hopf algebras, can be formulated in an arbitrary symmetric monoidal category. They are given by a collection of morphisms, ...
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Postnikov invariants of the Brauer 3-group
Given a commutative ring $k$ there is a bicategory with
algebras over $k$ as objects,
bimodules as morphisms,
bimodule homomorphisms as 2-morphisms.
This is a monoidal bicategory, since we can ...
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Is the center of an abelian rigid monoidal category, abelian?
Is the Drinfeld-Majid center of an abelian rigid monoidal category, abelian?
[stated in 1J of On the center of fusion categories" by Bruguières and Virelizier (link at Virelizier's page)]
In ...
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Oplax monoidal functors of $\infty$-categories
In Higher Algebra, a notion of lax symmetric monoidal functors (in what follows, I'll remove the adjective "symmetric", but I'm mainly interested in the symmetric situation) is defined : if you have ...
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Infinitesimal categories and left duality
I have been reading Kassel's Quantum groups and there is something I can not understand.
In Section 4 of chapter $XX$, he introduces the notion of a Infinitesimal symmetric category, that is
a ...
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Does Cantor Bernstein hold in a Closed Symmetric Monoidal Category?
In a closed symmetric monoidal category with $[I,X] \cong X$ for all $X$ is it true that having monomorphisms $m :A \rightarrow B$ and $m: B \rightarrow A$ is enough to imply $A \cong B$ ?
I tried to ...
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A question about subobjects of the unit in a rigid abelian tensor category
I have a question about Proposition 1.17 in Deligne and Milne, Tannakian Categories (see here), in the last 4 lines of the proof.
I don't know how it follows from $U\otimes U\simeq U$ that $T=\ker(...
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Examples of strict monoidal categories and monoidal categories with nontrivial associators
What are some "natural" motivating examples of the following:
i) A strict monoidal category,
ii) A monoidal with non-trivial associatots?
For i) the only examples I know are categories which ...
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Example of a projective bimodule with isomorphic left and right duals
What is an example of a non-free finitely generated $R$-bimodule $M$ satisfying
i) $M$ is projective as both a left and right $R$-module
ii) the right dual $\mathrm{Hom}_R(M,R)$ and the left dual ...
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Nonbraided rigid monoidal category where left and right duals coincide
In a braided rigid monoidal category $(\mathcal{M},\otimes)$ left and right duals coincide. What is an example of a rigid monoidal category where left and right duals coincide but there exist no ...