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.
637 questions
1
vote
0
answers
20
views
Müger: semisimplicity implies duals are two-sided
In Müger's Modular Categories , end of page 3, he claims that if a $k$-linear monoidal category which is object semisimple, with absolutely simple monoidal unit satisfies that all objects have left ...
- 1,664
0
votes
0
answers
28
views
Categorification of "sub" in monoidal category
As we know, we have monoid objects (or ring objects) in monoidal categories. For example, the monoid objects in $\mathrm{Vect}\; k$ are $k-$algebras.
I want to question that can we define a "...
- 43
0
votes
0
answers
36
views
Monoidality of Strict Free Cocompletion
For a small category $\mathcal{C}$, the free cocompletion is given by the category $\mathrm{PSh}(\mathcal{C})$ of presheaves $\mathcal{C}^{op} \to \mathrm{Set}$. Alternatively, it can be described as ...
- 1
0
votes
1
answer
37
views
Internal $\operatorname{Hom}(1, X)=X$ in a closed monoidal category? [duplicate]
If $R$ is a graded-commutative ring, we know that $\operatorname{Hom}^*(R,M)\simeq M$ for each graded $R$-module $M$. Let $\mathcal{C}$ be a closed monoidal category (could also assume that it is ...
- 3,538
4
votes
2
answers
257
views
modules over monoids：trouble in a specific example
I‘m recently learning about monoidal categories and the monoid & module object in monoidal category. After reading the definitions, I hope to give a specific example about $kG$-mod category.($kG$ ...
- 43
0
votes
1
answer
43
views
Modifying morphisms to track inputs and outputs
Suppose I have a functor $F$ from the category ${\bf 3}=(1 \overset{j_1}\to 2 \overset{j_2}\to 3)$ to the Kleisli category $\text{Kl}(D)$ of the distribution monad $D$ (the target of $F$ can be ...
- 1,823
3
votes
1
answer
60
views
Order of the images of Dehn twists in the quantum representation of Reshetikhin-Turaev TQFT and relation to Vafa's theorem
Short version
Let $\mathcal T$ be the modular functor of a Reshetikhin-Turaev TQFT defined over a modular (semisimple) category $\mathcal C$ and a ring $K$, and $\Sigma$ be some compact (closed) ...
- 33
1
vote
1
answer
66
views
Explicit definition of cartesian multicategories
In the definition of cartesian multicategories of the nlab it says "symmetric multicategory [equiped with contraction and deletion operations ...] which satisfy certain evident axioms".
In ...
- 1,556
0
votes
1
answer
22
views
Understanding symmetry of the definition of exponential graph $H^G$
I am reading Hom complexes and homotopy theory in the category of graphs and am trying to understand the definition of an exponential graph for simple, undirected graphs (with loops allowed).
...
- 839
0
votes
0
answers
53
views
Category Glued to its Dual Category
Suppose I have a category $C$. Then I can construct its dual $C^\bot$, by way of a contravariant equvalence $C \to C^\bot$. Suppose that $C$ has an initial object $0 : C$ so that $0^\bot$ is ...
- 11
1
vote
1
answer
38
views
Is a direct summand of a dualisable object itself dualisable?
If an object $X$ in an abelian monoidal category is the direct summand of a dualisable object, is $X$ itself dualisable? This is true in the category of modules over a commutative ring, since then a ...
- 2,133
2
votes
1
answer
95
views
Exercise 4.4. from "Categories for Quantum Theory": Existence of left duals implies left-closedness
I am stuck on exercise 4.4 from Heunen and Vicary: "Categories for Quantum Theory":
Let $A$ and $B$ be objects in a monoidal category. Their exponential
is an object $B^A$ together with a ...
- 41
0
votes
1
answer
37
views
Why is a functor $\mathcal{V}_{0}(I,-): \mathcal{V}_{0} \to \textbf{Set}$ is the "underlying-set" functor?
I'm reading Kelly's 'Basic Concepts Of Enriched Category Theory' and there is a bit in the introductory section about monoidal categories that confuses me
A monoidal category $\mathcal{V} = (\mathcal{...
- 33
5
votes
0
answers
60
views
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 ...
- 1,299
1
vote
1
answer
48
views
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}} \...
- 1,687
1
vote
1
answer
57
views
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 ...
- 1,687
1
vote
0
answers
31
views
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 ...
- 1,687
5
votes
1
answer
98
views
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 ...
- 1,070
4
votes
0
answers
87
views
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 ...
- 1,370
1
vote
1
answer
137
views
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 ...
- 1,687
2
votes
1
answer
48
views
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]$ (...
- 99
0
votes
1
answer
90
views
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 ...
- 1,825
3
votes
1
answer
63
views
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 ...
- 27.1k
2
votes
1
answer
60
views
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,...
- 1,070
1
vote
0
answers
52
views
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 ...
- 1,823
2
votes
1
answer
69
views
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/...
- 6,550
1
vote
1
answer
91
views
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 ...
- 11
0
votes
1
answer
34
views
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 ...
- 1,523
3
votes
0
answers
103
views
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 ...
- 1,687
2
votes
1
answer
54
views
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 ...
- 6,899
1
vote
0
answers
52
views
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 ...
2
votes
1
answer
54
views
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 ...
6
votes
1
answer
115
views
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,...
- 147
1
vote
1
answer
59
views
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/...
- 1,687
2
votes
0
answers
55
views
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 ...
- 491
0
votes
0
answers
47
views
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 ...
- 510
1
vote
0
answers
40
views
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 ...
- 991
1
vote
1
answer
79
views
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 $...
- 103
3
votes
1
answer
114
views
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{...
- 669
0
votes
0
answers
69
views
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-...
- 103
3
votes
1
answer
133
views
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 ...
- 3,550
2
votes
1
answer
74
views
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 \...
- 1,687
5
votes
0
answers
64
views
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 ...
- 577
2
votes
1
answer
69
views
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 ...
- 411
11
votes
3
answers
1k
views
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 ...
- 407
0
votes
0
answers
55
views
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 \...
- 1,687
1
vote
1
answer
35
views
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 ...
- 411
2
votes
1
answer
72
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 ...
- 411
4
votes
1
answer
81
views
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 ...
- 411
3
votes
1
answer
108
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 $...
- 881