Questions tagged [monoidal-categories]
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570 questions
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Bicategories in which the composition functors $\circ_{A, B, C}$ admit right adjoints
In Bénabou's 1967 Introduction to Bicategories, he mentions that, in a forthcoming part II, he would study bicategories $\mathcal K$ in which each composition functor $$\circ_{A, B, C} \colon \mathcal ...
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Are cofibrant objects flat with respect to Day convolution?
Question
Let $\mathcal{C}$ be a small symmetric monoidal category. The category $\mathsf{sSet}^{\mathcal{C}}$ of simplicial precosheaves on $\mathcal{C}$ is a symmetric monoidal model category with ...
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Duals and direct summands in an abelian monoidal category
This question may be seen as a continuation of Duals and sub-objects in a monoidal category.
In an abelian monoidal category, i.e. an abelian category with biadditive monoidal product, if $X \oplus Y$ ...
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Connected Frobenius algebras non-semisimple as an object
A Frobenius algebra object $A$ in a tensor category $\mathcal C$ is said to be connected if $\text{Hom}_{\mathcal C}(\mathbb{1}, A)$ is a one dimensional vector space, where $\mathbb {1} $ denotes the ...
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Fourier-like transforms for a Day convolution?
The presheaf category on a monoidal category inherits the monoidal structure via the Day convolution. Moreover you can inherit (bi)closed monoidal structure.
In the study of Fourier analysis we can ...
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Monoidal categories with canonical left-strengths of monads
It is well-known that every monad on Set is left-strong and that left-strength on a Set-monad is unique.
Is there an abstract characterization of monoidal categories $C$ for which every monad on $C$ ...
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Day Convolution of Sheaves
In Bodil Biering's master's thesis, Conjecture 4.3.3 conjectures that Day convolution of presheaves does not generally preserve sheaf conditions, with an incomplete attempt at a counter-example given. ...
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Categorical duals for Yetter-Drinfeld modules [duplicate]
Yetter-Drinfeld (YD) modules appear naturally in the theory of Hopf algebras. They are both modules and comodules at the same time, satisfying a certain compatibility condition, as presented here. The ...
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Duals and sub-objects in a monoidal category
Consider $\mathcal{M}$ a monoidal category. Let $V$ be an object that admits a left/right dual. If $U$ is a subject of $V$ then does it also admit a left right dual?
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Tensor product of intersections in an abelian rigid monoidal category
To what extent can the following results in ${\rm Vec}$ stated in [Ro08, Chapter 14, Exercises 10-12] be extended to abelian rigid monoidal categories?
(10) Let $S_1, S_2$ be subspaces of $U$. Then $$...
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Describing monoidal categories of positive-weight representations geometrically
Let $G=\mathbb{G}_m.$ The monoidal category $\mathcal{C}=\text{Rep}(G)$ of $G$-representations (also known as the category $\text{Gr}$ of graded vector spaces) can be written geometrically as $\...
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Intersection of subalgebras in an abelian monoidal category
$\require{AMScd}$Let $\mathcal{C}$ be an abelian monoidal category, and let $(M,m,e)$ be an algebra in $\mathcal{C}$, where
$M$ is an object in $\mathcal{C}$,
$m: M \otimes M \to M$ is the ...
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Bimodule endomorphisms of a bimodule over a noncommutative ring
Let $R$ be a noncommutative ring and $M$ an $R$-$R$-bimodule that is projective as left $R$-module. We know that the bimodule of left $R$-module endomorphisms ${}_REnd(M)$ is isomorphic to the tensor ...
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Intersection of Frobenius subalgebra objects of a connected Frobenius algebra object
This post is the connected version of this one.
$\require{AMScd}$Let $\mathcal{C}$ be a tensor category over $\mathbb{C}$ and let $M$ be a Frobenius algebra object in $\mathcal{C}$. Recall that a ...
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Intersection of Frobenius subalgebra objects
$\require{AMScd}$Let $\mathcal{C}$ be a tensor category and let $M$ be a Frobenius algebra object in $\mathcal{C}$. A Frobenius subalgebra object of $M$ is a Frobenius algebra object $X$ equipped with ...
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Pullback of monomorphisms between selfdual objects in a tensor category
$\require{AMScd}$ Let $\mathcal{C}$ be a tensor category. An object $X$ in $\mathcal{C}$ will be called selfdual if $X^* = X$. Let $A,B,M$ be selfdual objects in $\mathcal{C}$. Let $i_X: X \to M$, ...
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Reference for the monoidal category structure $X \otimes Y = X + Y + X \times Y$ on a distributive category
Given a distributive category $\mathscr C$ (more generally a rig category), we can define a (semicocartesian) monoidal category structure on $\mathscr C$ with tensor product given by $X \otimes Y := X ...
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Why does the definition of a braided monoidal category not mention the braid equation?
Let $\mathcal{M}$ be a braided monoidal category (BMC) with braiding $\gamma$. In the definition of a BMC $\gamma$ is required to satisfy the two hexagon identities. However since "braided" ...
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$G$-crossed (braided) fusion categories and Tannaka duality
Many important concepts in tensor category theory have their counterpart in Hopf algebra theory under Tannaka duality. They have the general form: let $A$ be an XX-algebra, and let Rep$(A)$ denote the ...
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Are there noncartesian monoidal categories with $A \otimes B = A \times B$?
Can we find a non-cartesian monoidal category where for each pair of objects the tensor product $A \otimes B$ is a cartesian product of $A$ and $B$?
Let me explain.
A monoidal category $(\mathsf{C}, \...
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Tamari lattice and bicategory coherence
Reference links:
Tamari lattice (Wikipedia): https://en.wikipedia.org/wiki/Tamari_lattice
Associahedra: https://en.wikipedia.org/wiki/Tamari_lattice#/media/File:Tamari_lattice.svg
The Tamari lattice ...
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Is there a synthetic approach to (symmetric) monoidal infinity-categories?
Recent work of Riehl and Verity (e.g. the book "Elements of $\infty$-category theory") has established a "synthetic" / model-independent approach to the study of $\infty$-...
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Equivalence between bialgebras and finite ring categories with fibre functor
$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Rec{Rec}\DeclareMathOperator\End{End}\DeclareMathOperator\Mod{Mod}\newcommand\fdMod[1]{#1\text-{\Mod}^\text{fd}}$In the celebrated [EGNO, Thm 5.2.3], ...
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Enriched tensor product of chain complexes
Question (idea): Is there a notion of tensor product of chain complexes in a $\mathcal{V}$-enriched monoidal category $\mathcal{C}$, for $\mathcal{V}$ a linear symmetric monoidal category?
Let me ...
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Coherence between associator and unitor in a monoidal category
I have a hard time trying to prove a coherence relation in a monoidal category. I'll start by giving the definition I'm working with:
A monoidal category is a sextuple $(C, \otimes, \mathbf{1}, \alpha,...
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Are dualizable topological vector spaces finite-dimensional?
Consider the symmetric monoidal category TVS of complete Hausdorff topological vector spaces equipped with the completed projective, injective, or inductive tensor product.
Every finite-dimensional ...
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Is the symmetry compatibility condition in Fox's theorem necessary?
Let $(\mathscr V, \otimes, 1, \sigma)$ be a symmetric strict monoidal category whose unit is terminal. Suppose that every object $A$ is equipped with the structure of a cocommutative comonoid $1 \...
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If a strong monoidal functor $F$ has an ambidextrous adjoint, then how close is the adjoint to being strong monoidal?
Let $F : C \to D$ be a strong (symmetric, say) monoidal functor. Suppose that $G : D \to C$ is both left and right adjoint to $F$ (an ambidextrous adjunction). Then by doctrinal adjunction $G$ is both ...
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Permutative Gray monoid that can not be strictified
Strict monoidal 2-categories do not suffice to model all connected 3-types: the non-trivial coherence cells $\Sigma: (f \otimes 1)\circ (1 \otimes g) \cong (1 \otimes g)\circ (f \otimes 1) $ of a Gray ...
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Equivalences induced from invertible objects in transported bifunctors along an adjoint pair
I'm interested in the following problem, similar in vein to this other question. To put it simply, I have an adjoint pair $F\dashv G$ between categories $\mathrm{C}$ and $\mathrm{D}$ and I suppose ...
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How general is $TX \otimes X \simeq X \otimes TX$?
Let $C$ be a monoidal category, $X \in C$ an object, and $TX$ the free monoid object on $X$, assuming it exists. How often do we have an isomorphism $TX \otimes X = X \otimes TX$? is there a canonical ...
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Group completion of a monoid (braid groups)
Let $B_n$ be the braid group on $n$ strands, $B_{\infty}$ the direct limit of braid groups.
For a discrete group $G$, we let $BG$ to be the classifying space of $G$.
After reading this question, I was ...
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Are the minimal nondegenerate extensions universal?
We reference [EGNO] for the concept of a braided fusion category. Following the conventions in [JFR], let $\mathcal{C}$ denote a braided fusion category equipped with a braiding $\beta$, and let $\...
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Endomorphisms of simple dualizable objects in a linear abelian monoidal categories
In When is an object in a linear or abelian category simple? Or: How should I define fusion categories? endomorphisms of simple objects in a k-linear abelian category are discussed. In the answer it ...
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In a monoidal category with duals is the coevaluation map determined by the evaluation?
For a monoidal category $(\mathcal{M},\otimes,1)$ and an object $X$ with a left dual $X^*$. Let $(e,c)$ be a pair of (co)evaluation maps for the pair $(X,X^*)$. Is it possible to have another map $c': ...
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The evaluation and coevaluation maps for an object isomorphic to a dualisable object
Let $X$ be an object in a monoidal category with dual $X^*$ and evaluation and coevaluation maps $e$ and $c$. Now if we have an isomorphism $\sigma:X \to Y$, for some other object $Y$, then $Y$ must ...
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Monoidal structure on presheaves
I am confused about the following monoidal structure, which gives a symmetric monoidal structure on R-modules (that I think is not Cartesian), even if R is not commutative.
Let $C$ be a small category....
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Coherence of the graphical language for pivotal categories
Throughout I follow A survey of graphical languages for monoidal categories, Peter Selinger, arXiv.
A pivotal category is a monoidal category where each object $A$ has a dual $A^*$, together with a ...
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Bar construction in commutative algebras is calculated by pushout
$\DeclareMathOperator\colim{colim}$
Also asked in MathStackexchange here
This is a statement in Lurie's Higher Algebra 5.2.2.4.
Proposition 3.2.4.7 in HA said that the monoidal structure on $\text{...
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Isomorphic objects have the same dimension (pivotal categories)
I want to prove that if two objects $X,Y$ in a pivotal category $\mathcal{C}$ are isomorphic, then $X$ and $Y$ have the same dimension, i.e.,
$$
\mathrm{dim}(X) = \mathrm{Tr}^{L}(\mathrm{id}_{X}) = \...
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Are the Drinfeld doubles of twist equivalent Hopf algebras twist equivalent?
Let $H_1$ and $H_2$ be finite dimensional Hopf algebras that are twist equivalent, i.e. $H_2$ is obtained from $H_1$ using a Drinfeld twist. My question is: are the Drinfeld doubles $D(H_1)$ and $D(...
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1-categorical universal properties for the smash product of pointed sets
Question I. Is the following statement, inspired by this one, true?
Universal Property I. The symmetric monoidal structure on the category $\mathsf{Sets}_*$ of pointed sets and pointed maps between ...
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Different notions of equivalences of $\mathcal{O}$-monoidal $\infty$-categories
I am currently reading Higher Algebra by Jacob Lurie and I have a question regarding equivalences of $\mathcal{O}$-monoidal categories.
Let $\mathcal{O}$ be an $\infty$-operad. Suppose that I have two ...
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Is the category of simplicial $R$-modules closed monoidal?
I am trying to understand if the simplicial mapping space for simplicial $R$-Modules (or at least simplicial vector spaces) is adjoint to the (level-wise) tensor product. For simplicity, let me state ...
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Self-enrichment for a closed monoidal bicategory
First, there are two possible generalization of the notion of closed category, vertical and horizontal.
I'm interested in the vertical one, something saying, I guess, that a monoidal bicategory $\...
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Does the Gray tensor product exhibit Gray as a monoidal Gray-category?
Crans constructs a lax Gray tensor product for strict omega-categories that defines a monoidal structure on the category of strict omega-categories $\omega Cat$. We write Gray for this monoidal ...
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Yetter-Drinfeld modules for Hopf monads
1. Context.
1.1. Classical Yetter-Drinfeld modules.
Let $H$ a bialgebra in a braided monoidal category $\mathcal{C}$. A left-right Yetter-Drinfeld module over $H$ is a triple $(V,\rho,\Delta)$ ...
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One-object lax natural transformation
A lax natural transformation $\alpha$ between two pseudofunctors $F,G: \mathcal{K} \to \mathcal{L}$ between bicategories $\mathcal{K}, \mathcal{L}$ consists of the following data:
For every object $A ...
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Dual objects in an abelian monoidal category
Let $(\mathcal{M},\otimes)$ be a monoidal category that is also abelian, and assume that $\otimes$ is bilinear with respect to the biproduct. Let $X$ and $Y$ be two objects that admit (left) duals ${}^...
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Day convolution and sheafification
$\DeclareMathOperator\Psh{Psh}\DeclareMathOperator\Sh{Sh}\newcommand\copower{\mathrm{copower}}$I was looking through Bodil Biering's thesis On the Logic of Bunched Implications - and its relation to ...
