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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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Examples of closed categories which are not monoidal closed?

There is a solid definition of "closed category" axiomatizing the idea that we can assign something resembling a hom-object to each pair of objects of a category. However, I am struggling to ...
Morgan Rogers's user avatar
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Rigid symmetric monoidal category implies closed?

From def. 2.1 in Internal hom it is said that Let $(\mathcal{C},\otimes)$ be a tensor category. An internal hom in $\mathcal{C}$ is a functor $$\underline{\text{Hom}}(-,-):\mathcal{C}^{\text{opp}} \...
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Unique isomorphism between dual objects that preserve duality data.

In Tensor Categories, page 48, it is said that the isomorphism $\alpha:X_1^{*} \to X_2^{*}$ they construct, is the only isomorphism between (right) dual objects that preserve the evaluation and ...
Ben123's user avatar
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Zigzag/Snake-identities in monoidal categories

This is a rather elementary question; I have not put much time into understanding string diagrams, and still find them quite confusing to interpret. If $(Y,\text{ev},\text{coev})$ is the duality data ...
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Why the non-abelian 4-cocycle condition?

In a monoidal category it holds by definition (together with the identity coherence) an associativity coherence axiom, stating commutativity of the pentagon Now, if we categorify vertically, we can ...
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A diagrammatic proof of antipode being antihomomorphism in a Hopf algebra

Let $(H, \mu, \eta, \Delta, \epsilon, S)$ be a Hopf algebra with $S: H \to H$ denoting the antipode. By definition, $S$ is the convolution inverse of $1: H \to H$ in $\operatorname{End}(H)$, with the ...
Ray's user avatar
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Tensor functors on rigid categories.

Let $(\mathcal{C},\otimes)$ and $(\mathcal{C}',\otimes')$ be rigid tensor categories (in the sense of Deligne/Milne; see https://www.jmilne.org/math/xnotes/tc2022.pdf). My question is asked in the ...
Ben123's user avatar
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Enriched functor categories as $V$-objects

Under "Enriched functor categories" at https://ncatlab.org/nlab/show/end#enriched_functor_categories it is claimed that for $V$-enriched categories $C$ and $D$ the functor category $[C,D]$ (...
YordanToshev's user avatar
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Mac Lane Chapter 7 Section 2 Exercise 1

Let $\mathcal{C}$ be a monodical category, with the monodical product written $\otimes$, the associator denoted $\alpha$, and the left/right unitors denoted $\iota^\ell,\iota^r$ respectively. Mac ...
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A sort of Day convolution without enrichment

Some time ago I was trying to define a monoidal structure on a functor category $[\mathcal{C},\mathcal{D}]$ between two monoidal categories $\mathcal{C}$ and $\mathcal{D}$, such that the monoid ...
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Compatibility of adjunctions for closed monoidal category

If $\mathcal V$ is braided monoidal closed, meaning that for any object $A$ the functor $-\otimes A$ admits a right adjoint $[A,-]$, then $\mathcal V$ is enriched over itself by letting $\mathcal V(A,...
Nikio's user avatar
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Does free biproduct completion define a lax monoidal functor?

Does free biproduct completion (as described in Definition 2.3 of Coecke-Selby-Tull) define a lax monoidal functor from the category of semi-additive categories to itself ? I should clarify that here ...
Richard Southwell's user avatar
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Hopf algebra related to monoidal category

Recently, I heard that Braided rigid monoidal category corresponds to a quasi-triangular hopf algebra. I know in braided condition gives hexagonal equations and monoidal category gives pentagon/...
phy_math's user avatar
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Nerve theorem for small permutative categories

For small categories, there is a famous Nerve theorem: A simplicial set $X:\Delta^{op}\to Set$ is a nerve of a small category if and only if it satisfies the Segal conditions. For small permutative ...
xuexing lu's user avatar
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Kronecker product arising from a coproduct

I'm reading a paper that involves some background on linear algebra, and I came across a sentence that I'm trying to make sense of: "The tensor/Kronecker product $\otimes$ of representations ...
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Invertible objects in tensor categories.

In https://www.jmilne.org/math/xnotes/tc2018.pdf, page $7$ under the chapter on "Invertible objects" we call an object $L$ in a tensor category $(\mathcal{C},\otimes)$ (I will abbreviate ...
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Can one define an induced ordinary category from an enriched one?

I'm curious about Wikipedia's definition of an enriched category. An enriched category $\mathcal{C}$ over a monoidal category $\mathcal M$ is said to contain an object $\mathcal C(a, b)$ of $M$ for ...
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Associtivity of Tensor Product of Modules Over Algebras in a Tensor Category

I am attempting to prove that modules over a commutative algebra (monoid) $A$ in a fixed tensor category $\mathcal{T}$ form a tensor category $\mathcal{T}_A$. All of the references I have found say it ...
Dakota's Struggling's user avatar
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Monoidal category monoidally equivalent to skeletal monoidal category

The fact that any monoidal category is monoidally equivalent to a skeletal monoidal category is widely known(se e.g EGNO exercise 2.8.8) it seems that the argument most commonly used is: starting with ...
The exterminator's user avatar
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How to check Pentagon axiom with induced associator of skeletal category?

In the book Tensor Categorties by EGNO, there are Exercise 2.8 Show that any monoidal category $\mathcal{C}$ is monoidally equivalent to a skeletal monoidal category $\bar{\mathcal{C}}$. In the hint,...
MatrixBi's user avatar
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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/...
Ben123's user avatar
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Hilbert Spaces from Dagger Categories

Dagger compact closed categories are commonly said to be an abstraction of Hilbert spaces and is suppose to capture concepts such as unitary maps, scalars, basis, inner products. See for example the ...
Nanoputian's user avatar
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Bar construction for cocartesian monoidal structure is calculated by pushout

$\DeclareMathOperator\colim{colim}$ 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 $\text{CAlg}(\mathcal{C})$ is cocartesian. I ...
Xiong Jiangnan's user avatar
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Two algebra structures on endomorphisms

Let $(\mathcal{M}, \otimes, \mathbb{k})$ be a symmetric closed monoidal category, which in my application is the category of $dg$-modules over some commutative ring. Let $A$ be a bialgebra/bimonoid in ...
Brendan Murphy's user avatar
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1 answer
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The existence of "inverse" in monoidal category

In a monoidal category $\mathcal{C}$, Does any $f\in \operatorname{Hom}_{\mathcal{C}}(X\otimes \mathbf{1},Y\otimes \mathbf{1})$ can be expressed as $f=g\otimes \operatorname{Id}_{\mathbf{1}}$, where $...
liouville's user avatar
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1 answer
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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 $\mathcal{C}$ (is that enough? Or do we need something more?) are isomorphic, then $X$ and $Y$ have the same dimension, i.e., $$ \mathrm{...
NoetherNerd's user avatar
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What is a "functorial isomorphism"?

I am reading a lecture note about tensor category by P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik. The link is attached here https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-...
liouville's user avatar
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Is there a non-symmetric monoidal monad?

Recall that a monoidal monad on a monoidal category $(\mathcal{C}, \otimes, I)$ is a monad $(M, \eta, \mu)$ on $\mathcal{C}$ such that $M$ is also equipped with the structure of a lax monoidal functor ...
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Commutator of the unit object, in a symmetric monoidal category.

Let $(\mathcal{C},\otimes,\phi,\psi,U)$ be a symmetric monoidal category, where $\otimes:\mathcal{C} \times \mathcal{C} \to \mathcal{C}$ is a bifunctor. Let $X,Y,Z \in \mathcal{C}$, then $\phi:(X \...
Ben123's user avatar
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The collection of maps to a (commutative) monoid is a (commutative) monoid, via Eckmann-Hilton

A commutative monoid $M$ has the nice property that given a set $S$, the set of functions $S \to M$ forms a commutative monoid (under pointwise addition). The same statement without any mention of ...
Matthew Niemiro's user avatar
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1 answer
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Is the category of monoids $\textsf{Mon}(\mathcal{C})$ in a monoidal category $\mathcal{C}$ itself monoidal?

I have read about the forgetful functor $U$ from $\textsf{Mon}(\mathcal{C})$ to $\mathcal{C}$ (see e.g. this nLab page). I think I have read somewhere that this functor is monoidal, but I cannot find ...
user11718766's user avatar
11 votes
3 answers
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How much data does a category contain?

This might seem like a very vague question, but the details are really confusing me. So, for example, say we are studying the category of $A$-modules $\mathsf{Mod}_A$ where $A$ is a commutative unital ...
Anthony Lee's user avatar
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What does it mean that a morphism in a category is fully determined by another morphism

My question is taking place more specifically in a tensor category $(\mathcal{C},\otimes,\phi,\psi)$ (I will denote this category with $\mathcal{C}$ going forward) where $\otimes:\mathcal{C} \times \...
Ben123's user avatar
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If $A$ and $B$ are dualizable objects in a monoidal category, is the unit of the one duality the inverse of the counit of the other duality?

I'm currently trying to wrap my head around dualizable objects in monoidal categories and I was wondering whether the following claim holds: Let $A$ and $B$ be dualizable objects in a monoidal ...
user11718766's user avatar
1 vote
1 answer
59 views

Why does duality of objects $A$, $A^\ast$ in a symmetric monoidal category imply an adjunction $(-) \otimes A \dashv (-) \otimes A^\ast$?

Let $\mathcal{C}$ be a symmetric monoidal category and let $A$ and $A^*$ be dual in the sense of Definition 2.1 in nLab. Dold & Puppe (1984) show (Thm 1.3) that the map $$ \text{Hom}(X, Y \otimes ...
user11718766's user avatar
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1 answer
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Why is $\mathcal{C}$ equivalent to $\mathcal{C}^{\text{op}}$ when $\mathcal{C}$ is a compact category?

I came across the statement that a compact closed category $\mathcal{C}$ is equivalent to its dual category $\mathcal{C}^{\text{op}}$ (see e.g. this StackExchange post). This fact seems to be regarded ...
user11718766's user avatar
3 votes
1 answer
103 views

Forgetful functor $Z(C)\rightarrow C$ has Left Adjoint

The monoidal center $Z(C)$ of a monoidal category $C$ comes with a forgetful functor $F:Z(C)\rightarrow C$ defined $Z(X,\phi)=X.$ Does $F$ always admit a left adjoint? This is known (Section 3.2.) if $...
Peter's user avatar
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Singular Chain complex of Smash Product

\begin{align} \Delta(X\wedge Y,pt)=\quad? \end{align} Question: Is there a nice way (similar to the Eilenberg-Zilber map) to compute the singular chain complex of a smash product of pointed spaces up ...
Nico's user avatar
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Subtleties in commuting colimits

For context, I am reading Weibel's k-book and I am trying to express the homology of $BS^{-1}S$, the group completion of the classifying space of a symmetric monoidal category, as a colimit. In ...
DevVorb's user avatar
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How to compute braidings from modular data

I'm trying to compute the braiding morphisms between simple objects of the so-called metaplectic categories, i.e. the modular categories with fusion rules equivalent to those of $\operatorname{SO}(N)...
nanowillis's user avatar
2 votes
1 answer
48 views

How these two definitions of category of hypergraph are same?

In the paper "Homotopy, homology, and persistent homology using closure spaces Peter Bubenik, Nikola Milićević" HypGph denote the category of hypergraphs and hypergraph homomorphisms. In the ...
Rabia Sagheer's user avatar
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what does "relation" mean in the Category of hypergraphs?

in the ncatlab https://ncatlab.org/nlab/show/hypergraph they defined the category of hypergraphs: SimpHGrph has objects consisting of a pair of sets (V,H) equipped with a relation R⊆V×H, and morphisms ...
Rabia Sagheer's user avatar
6 votes
1 answer
101 views

Is there a notion of tensor-completion of a category?

Given an additive category $\mathcal{C}$, sometimes one wants to consider its Karoubi envelope or idempotent completion $\mathrm{Kar}(\mathcal{C})$, whose objects are summands of objects in $\mathcal{...
Alvaro Martinez's user avatar
1 vote
1 answer
51 views

Monoidal category's associator as a natural isomorphism

A monoidal category is equipped with the associator $ \alpha_{A,B,C}:A\otimes (B\otimes C)\cong (A\otimes B)\otimes C$ which is a natural isomorphism. What explicitly does it mean for this to be a ...
Mithrandir's user avatar
1 vote
1 answer
90 views

Understanding a proof of a lemma for rigid categories [closed]

I'm reading the proof of the lemma 3.4 in the Bruguieres' paper on Hopf monads which claims the following: Lemma Let $F,G: \mathcal{C} \rightarrow \mathcal{D}$ be two strong monoidal functors and $\...
Andres Felipe Vargas Mican's user avatar
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0 answers
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How to identify "irreducible factors" in a monoidal or cartesian category

Is there "a categorical way" of speaking of objects in a monoidal category which can't be written as a tensor product? (We have to be careful with multiplication by the unit.) The same can ...
Julián's user avatar
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2 votes
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Comonoidal categories?

Monoidal categories are everywhere. They can be defined as pseudomonoids in the monoidal bicategory $(\mathsf{Cat},\times)$ of categories, functors and natural transformations. By turning directions ...
Margaret's user avatar
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2 votes
1 answer
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Does Tensor Categories section 2.6 have a mistake?

I'm reading the book Tensor Categories by Etingof, Gelaki, Nikshych, and Ostrik. In Section 2.6 they discuss the monoidal functors between two categories $\mathcal{C}_{G_1}^{\omega_1}(A)$ and $\...
Leo's user avatar
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0 answers
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On categorical operads

A symmetric $1$-coloured operad can be defined in any symmetric monoidal category. In particular, it can be defined for the cartesian monoidal category $\mathsf{Cat}$ of small categories and functors. ...
Margaret's user avatar
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Canonical morphism $F(X,I)\otimes Y\to F(X,Y)$ in a closed symmetric monoidal category

Let $(\mathcal{C},\otimes,I,F)$ be a closed symmetric monoidal category, where $F$ is the 'internal Hom' functor, $I$ is the unit object and $\otimes$ is the monoidal product. I am reading Definition ...
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