Questions tagged [symmetric-monoidal-categories]
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101 questions
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Examples of (co)commutativity of Frobenius algebras via ambijunctions
This question is related to the paper "Frobenius algebras and ambidextrous adjunctions" by Aaron Lauda (https://arxiv.org/abs/math/0502550). Below $\Sigma\mathrm{Vect}$ is the one-object ...
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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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Base change for module categories? ($E_\infty$-modules in $\mathrm{Cat}$)
I'm working on a project where I would like to consider the category of symmetric monoidal categories. Though I suspect it will be easier easier to consider the $\infty$-category of symmetric monoidal ...
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The importance of the Balmer spectrum
Why are Balmer spectra important? Can someone give examples of reconstruction a category by its spectrum (in some sense)?
It would also be interesting to see applications of Balmer spectra to the ...
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Box tensor product in the correspondence category
I am currently reading Peter Scholze's note on six-functors formalism, where for an infinity category $C$ and a nice class of morphism $E$ in $C$, we can define the correspondence category $Corr(C,E)$ ...
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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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Deligne/Milne Tannakian Categories, prop. 1.17
In Tannakian Categories (page 14, prop. 1.17) we are given the following:
Let $(\mathcal{C},\otimes)$ be a rigid abelian tensor category. If $U$ is a subobject of $\mathbf{1}$, then $\mathbf{1} = U \...
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Subobjects in rigid abelian tensor categories
$\require{AMScd}$
In Tannakian Categories (prop. 1.17) we are given the following:
Let $(\mathcal{C},\otimes)$ be a rigid abelian tensor category. If $U$ is a subobject of $\mathbf{1}$, then $\mathbf{...
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Adjoint to "strict twocategory of strict twofunctors"
Let C be the category of strict twofunctors, featuring the addition of a Grothendieck universe. Strict twocategories are categories enriched over the category of categories.
C has an internal hom ...
user531303
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Morphism of tensor functors in rigid tensor categories
This is a cross-post from MSE.
$\require{AMScd}$
The following proposition (1.13) is given in Tannakian Categories (loosely paraphrased with some change in notation).
Assume that $(F,c),(G,d):\...
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Trace morphism in Deligne/Milne's "Tannakian categories"
I originally posted this on MSE, but only got a comment linking an article (Bontea and Nikshych's "Pointed braided tensor categories"). So I'll repost the question in full here:
Is there a ...
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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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Invertible objects in tensor categories (with P. Deligne/J.S. Milne's definition)
$\newcommand\id{\mathrm{id}}$I've asked this question on MSE, but the only response I've gotten so far is a comment, which I failed to understand. I'll add another question I have, related to this.
...
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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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Is a symmetric monoidal category ("tensor-category" in P. Deligne & J.S. Milne's vocabulary) neccessarily locally small?
Let $(\mathcal{C},\otimes,\mathbf{1},\phi,\psi)$ (I will denote this by just $(\mathcal{C},\otimes)$) be a tensor-category (in P. Deligne & J.S. Milne's vocabulary, see https://www.jmilne.org/math/...
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$R$-Module objects in cartesian closed categories
I am looking for a reference for the following statement.
Theorem. Let
$C$ be a regular, well-powered, countably complete cartesian closed category,
$R$ be a (commutative) ring object in $C$,
$R\...
user103549
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Tensor product of modules in model vs. infinity categories
Let $C$ be a combinatorial symmetric monoidal model category and let $A$ be a associative algebra object in $C$, that is cofibrant as an object in $C$. In Higher Algebra 4.3.3.17, Lurie proves an ...
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Symmetric-monoidal-associative smash product up to homotopy
I am thinking about sequential spectra. I am trying to figure out if the smash product here is symmetric monoidal associative up to homotopy. See definition 3.16 in the above.
Recall that a sequential ...
user30211
4
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A result on symmetric closed monoidal categories
In this discussion from the categories mailing there is mention of the following result by Robin Houston, supposedly proved in 2006:
Theorem. Let $\mathcal{C}$ be a symmetric closed monoidal category,...
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Is the free algebra functor over an $\infty$-operad symmetric monoidal?
Suppose $F: \mathcal{O}^\otimes \to \mathcal{P}^\otimes$ is a map of $\infty$-operads, and $\mathcal{C}$ is a symmetric monoidal $\infty$-category that admits small colimits, such that the tensor ...
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The change-of-monoid adjunction between categories of modules induced by a morphism of monoids
Let $\mathcal{M}$ be a cocomplete closed symmetric monoidal category. Let $A, B$ be monoids in $\mathcal{M}$ and $f: A \rightarrow B$ be a morphism of monoids. The morphism $f$ induces the extension ...
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Is the symmetric monoidal product on the $\infty$-category of $R$-modules unique?
In Higher Algebra 4.2.8.19, Lurie shows that the symmetric monoidal structure on spectra is uniquely defined (on the $\infty$-category level) by the following properties:
The sphere spectrum is the ...
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Why are enriched (co)ends defined like that?
I'm mainly following references such as Kelly, Loregian and the nLab, and it seems customary there to generalize (co)ends to the enriched context (over a symmetric monoidal category $\mathcal{V}$) by ...
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Nerve functor for symmetric monoidal category
The nerve $N(\mathsf{C})$ of a category $\mathsf{C}$ can be seen as a geometric realization of it (via n-simplices). This defines a functor $N: \mathsf{Cat} \rightarrow \mathsf{SSet}$ called nerve ...
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Properties of semisimple monoidal category
In my work, I have constructed a semisimple category which has two monoidal structures: the usual direct sum; and a new "tensor product". This "tensor product" have several nice ...
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Symmetric monoidal structures on the functor taking presheaves
Let $\mathrm{Cat}_\infty$ be the $\infty$-category of small $\infty$-categories, $\mathrm{Pr}^\mathrm{L}$ be the $\infty$-category of presentable $\infty$-categories, and $\mathcal P(\mathcal C)$ be ...
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"Closed $\mathscr{V}$-modules are uniquely (co)tensored $\mathscr{V}$-categories": shouldn't we assume they are also "mixed monoidal"?
$\newcommand{\M}{\mathcal{M}}\newcommand{\V}{\mathscr{V}}\newcommand{\hom}{\mathsf{hom}}$Throughout this post, $\V$ refers to some "cosmos", where I borrow the word "cosmos" from ...
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When a monoidal closed category is cartesian closed
Let $C$ be a monoidal closed category with tensor $\otimes$ and internal hom $[-, -]$.
Suppose that
$C$ acts by adjoint monads, i.e. $- \otimes X$ is a comonad and $[X, -]$ is a monad, and each $F : ...
user30211
4
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$\ast$-autonomous categories with non-invertible dualizing object?
1. Definition
Firstly, recall the following nLab-definition of a $\ast$-autonomous category:
A $\ast$-autonomous category is a symmetric closed monoidal category $(C,\otimes,I,\multimap)$ with a ...
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Reference for free symmetric monoidal categories with duals on symmetric monoidal categories
The inclusion $U$ of the (2,1)-category of symmetric monoidal categories with duals into the (2,1)-category of symmetric monoidal categories admits a left 2-adjoint functor $L$ for formal reasons.
In ...
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Tannaka without Yoneda?
I am studying enriched categories, and as I wrote in my previous question How is the morphism of composition in the enriched category of modules constructed?, this is very difficult because there are ...
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How is the morphism of composition in the enriched category of modules constructed?
I asked this a week ago at MSE, but without success.
I am studying enriched categories and I have a feeling that I am doing something wrong because all the way each step, each elementary proposition, ...
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Taking the homology of a chain complex, seen as a symmetric monoidal functor
I've found, in quite some places online (e.g. it's the last example in the wikipedia page about monoidal functors), a statement similar to this:
Homology can be seen as a symmetric, monoidal functor $...
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Day convolution for bimonoidal categories
Semiring categories, also called rig categories or bimonoidal categories, are pseudomonoids in the symmetric monoidal bicategory $(\mathsf{SymMonCats},\otimes_{\mathbb{F}},\mathbb{F})$¹. These are a ...
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Symmetric monoidal structure on categorical nerves
There are several notions of nerves, including nerves of categories, $2$-categories, and simplicial categories. These define functors
\begin{align*}
\mathrm{N} &\colon \mathrm{Cats}_{(2,1)} ...
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What is an invertible operad?
Let $\mathcal V$ be a nice symmetric monoidal ($\infty$-)category, and consider the ($\infty$-)category $Op(\mathcal V)$ of $\mathcal V$-enriched (symmetric) operads, symmetric monoidal under the ...
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Functors that preserve monoids
In the comments section of this question there was a question that I don't know if it has been asked on the site. It is well-known and easily proved that lax monoidal functors preserve monoids. So the ...
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Functors between module categories that comes from restriction
Suppose you have two $k$ algebras $A, B$ (say also finitely generated if this helps) and a functor $F: A-mod \to B-mod $ such that $| F(M) |= |M|$. Here $|U|$ denotes the underlying $k$ vector space.
...
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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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Is every $\otimes$-invertible object "coherently sym-central"?
Let $\mathcal C$ be a symmetric monoidal $\infty$-category, and let $L \in \mathcal C$ be a $\otimes$-invertible object. Then the braiding $L \otimes L \to L \otimes L$ is simply multiplication by $\...
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Symmetric monoidal structure(s) on the $\infty$-category of dg-categories
Let $k$ be a commutative ring with $1$, and let $\mathsf{dgCat}_k$ be the category of $k$-linear dg-categories, as defined in [1, Section 2]. We may equip $\mathsf{dgCat}_k$ with the Morita model ...
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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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Formal completion of a quotient stack
$\newcommand{\Rep}{\operatorname{Rep}}$
$\newcommand{\mo}{\operatorname{-mod}}$
$\renewcommand{\hat}{\widehat}$
I apologize in advance if this is a naive question but my background in algebraic ...
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422
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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^\...
- 15.7k
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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 ...
- 3,001
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R-matrices and symmetric fusion categories
Given a $\mathbb{C}$-linear braided fusion category $\mathcal{C}$ containing a fusion rule of the form e.g.
\begin{equation}X\otimes Y\cong A\oplus B \oplus C\end{equation}
(where $A,B, C, X$ and $Y$ ...
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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, ...
- 3,001
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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 ...
- 15.7k
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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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Very basic question about category theory
I have a symmetric monoidal category, and go to another one by replacing objects with isomorphism classes of objects. What's that called in category theory language?
To give an example, consider the ...
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