Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [monoidal-categories]

In mathematics, a monoidal category (or tensor category) is a category C equipped with a bifunctor ⊗ : C × C → C which is associative up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism. (Def: http://en.m.wikipedia.org/wiki/Monoidal_category)

0
votes
0answers
28 views

Role of the associativity isomorphism

Within monodial categories do the associativy isomorphisms get practically used in the following way: $(a \otimes b) \otimes c =$ associator$(x,y,z) \circ a \otimes (b \otimes c) \in \operatorname{...
1
vote
1answer
25 views

Monoidal categories and epimorphisms

The monoidal category $(\mathbf{Ab},\otimes,R)$ has monomorphisms $f:M\to N$ and $f':M'\to N'$ such that $f\otimes f':M\otimes M'\to N\otimes N'$ is not monic, for example $f:\mathbb{Z}\to \mathbb{Z},...
0
votes
0answers
42 views

decomposition of hom-functors in a self-enriched category

Let $\mathbb{C}$ be a self-enriched category (such as Set). The Functor $\mathbb{C}(X, \mathbb{C}(Y,\_))$ is the same than the composition of functors $\mathbb{C}(X,\_) \circ \mathbb{C}(Y,\_)$. In a ...
2
votes
0answers
40 views

If matching maps of a cosimplicial space are fibrations up to degree $n$, does the $n$-th partial totalization have the 'correct' homotopy type?

Let $X$ be a cosimplicial space and suppose that: the matching maps $$ s:X^{k}\rightarrow M^{k-1}X $$ are fibrations for all $k\leq n$. On one hand we can form the partial totalization $Tot_n(...
3
votes
0answers
54 views

Monoidal categories in which some tensor products of morphisms are equal

I have come across a certain monoidal category $(C,\otimes,I)$ (let us say it is strictly monoidal to simplify the notations, as usual it does not matter very much) which satisfies the following ...
1
vote
0answers
30 views

Does an adjoint of an internal Hom functor of a prounital closed category define a tensor product?

A closed category is a category equipped with internal Hom functors along with a unit object. Now this answer shows that if $C$ is a closed category whose internal Hom functor has a left adjoint, ...
1
vote
1answer
58 views

Is “monoidal category enriched over itself” the same as “closed monoidal category”?

If $M$ is a monoidal category, an enriched category over $M$ is a category $C$ whose hom-sets are viewed as objects in $M$. And a monoidal category $M$ is said to be closed if the tensor product ...
1
vote
0answers
12 views

Can I construct a monoidal category on locally convex modules over algebras with approximate identiy

I am faced with the following problem: Let $A$ be a complete locally convex algebra with a uniform approximation of identity, that is a net $e_\lambda$, such that $p(e_\lambda a-a)\rightarrow 0$ for ...
1
vote
1answer
36 views

Symmetric monoidal category which is not closed?

A monoidal category is symmetric if its tensor product is commutative up to natural isomorphism. And a symmetric monoidal category is closed if the tensor product functor has a right adjoint. We ...
3
votes
1answer
66 views

Does an adjoint of the Hom functor make a category monoidal?

In the category of modules, the tensor product functor is the left adjoint of the covariant Hom functor. Similarly in the category of sets, the Cartesian product functor is the left adjoint of the ...
0
votes
1answer
42 views

Enriched category over non-monoidal category

I'm finally learning homological algebra, and the notion of an enriched category seems to be the right setting in which to define the $\mathbf{Ext}$ functor. Yet, the definitions of an enriched ...
3
votes
0answers
32 views

For symmetric monoidal categories, the monoidal product is a monoidal functor

Has anyone got a reference for the following fact? If $\mathcal X$ is a symmetric monoidal category, then $\_\otimes\_\colon\mathcal X\times\mathcal X \to \mathcal X$ is a strong monoidal functor. ...
1
vote
1answer
35 views

Definition of unitor in monoidal category

From https://ncatlab.org/nlab/show/monoidal+category, a monoidal category requires a natural isomorphism a natural isomorphism $\lambda: (1 \otimes (-)) \rightarrow ^\cong (-)$ with components ...
3
votes
0answers
32 views

A monoidal category that preserves subobjects

Let $X$, $Y$ be objects in a monoidal category $\mathcal{C}$, s.t. the functors $X \otimes \_$ and $\_\otimes Y$ preserve monomorphisms. Moreover, let $A \hookrightarrow X$, $B \hookrightarrow Y$ be ...
2
votes
0answers
41 views

Examples of weak monoidal Quillen equivalences

Schwede-Shipley introduced the notion of weak monoidal Quillen equivalences between monoidal model categories in "Equivalences of monoidal model categories". Are there any examples of such Quillen ...
3
votes
0answers
48 views

Notation: why are categories enriched over $\mathcal V$?

In most references about enriched categories, $(\mathcal V, \otimes)$ is supposed to be a monoidal category and then $\mathcal V$-enriched categories are defined. Why is the letter $\mathcal V$ used ...
3
votes
0answers
45 views

Reference request for a “freeness” property of graded monoids

Let $I$ be a monoid and $G$ be an $I$-graded monoid, with multiplication $$ ( - \cdot - ) : G_i \times G_j \to G_{i+j}. $$ I'm interested in the following property of $G$: P: for any two indices $i,...
4
votes
1answer
63 views

A coend in the category of vector spaces

Let $Vect_k$ denote the category of (not necessarily finite-dimensional) $k$-vector spaces. Clearly, this category is closed symmetric monoidal with internal hom $[X,Y]=Hom_k(X,Y)$. Is it true that ...
0
votes
1answer
83 views

Stuck with Category Theory notation. What is the meaning of 'Corner brackets' 「 」?

While reading an article, I encountered this expression. Expression I was wondering if anyone knows what does the corner brackets 「(upper) and 」(down) in this expression do? Thank you.
3
votes
1answer
47 views

The coherence theorem for monoidal categories

I am reading the coherence theorem of monoidal categories. However I am confused by the following paragraph on page 165 of the book "Categories for the working mathmatician" $\bf{Here \ are \ my \ ...
1
vote
1answer
40 views

Existence and uniqueness of adjoints with respect to pairings

Let $V,W,L$ be $R$-modules over a commutative ring $R$. A pairing is an $R$-linear map $V\otimes W\to L$. An adjoint of an endomorphism $f:V\to V$ w.r.t a pairing $V\otimes W\overset{g}{\to}L$ is an ...
4
votes
1answer
47 views

The category $\bf{FinVect}$ of finite vector spaces is rigid.

I am following Pavel Etingof et al's book on tensor categories. They give FinVect as an example of a rigid monoidal category, with evaluation map given by $\text {ev}_V(\epsilon\otimes v)=\epsilon(v)$...
1
vote
0answers
20 views

A simple property of the $S$-matrix of a pre-modular category

I am following Pavel Etingof et al's book on tensor categories. In order to get used to the $S$-matrix of a pre-modular category and related concepts, I am trying to prove the following simple fact: ...
0
votes
0answers
84 views

Elements of the Monoid in the category of endofunctors

Quoting from Categories for the Working Mathematician by Saunders Mac Lane: All told, a monad in X is just a monoid in the category of endofunctors of X, with product × replaced by composition of ...
7
votes
1answer
134 views

Monoid in the category of endofunctors and Monoid as a category with one object

Quoting from Categories for the Working Mathematician by Saunders Mac Lane: All told, a monad in X is just a monoid in the category of endofunctors of X, with product × replaced by composition of ...
1
vote
1answer
30 views

Is $\text{Aut}_{\text{br}}(C)$ braided?

Let $C$ be a braided monoidal category. The category $\text{Aut}_{\text{br}}(C)$ of braided monoidal autoequivalences of $C$ is monoidal with tensor product functor given by the composition $\circ$. ...
2
votes
1answer
86 views

Lurie's reformulation of symmetric monoidal tensor categories in HA

In the introduction to Chapter 2 of Jacob Lurie's Higher Algebra (entitled "$\infty$-Operads"), a category $\mathcal{C}^\otimes$ is constructed from an arbitrary symmetric monoidal category $\mathcal{...
2
votes
1answer
42 views

Every under category with pushouts is monoidal

Maybe a stupid question, but I can't figure it out. Suppose we have some category $C$ where all pushouts exist, $c\in Ob C$ and $c\downarrow C$ an under category (i.e. a category with objects of the ...
3
votes
0answers
42 views

Dualizable presheaves with respect to Day convolution

Let $\mathcal{C}$ be a closed symmetric monoidal category and let $PSh(\mathcal{C}):=Fun(\mathcal{C}^{op}, Set)$ its category of presheaves regarded as a closed symmetric monoidal category via Day ...
5
votes
0answers
50 views

Intuition of factorization systems' role in the construction of free algebras

Kelly's article uses factorization systems for constructing free monoids. What's the intuitive reading of factorization systems in this case? Also, for example, Barr in this article cites a result by ...
2
votes
1answer
39 views

Precondition “small category” in functor category

I am currently working on a exercise sheet about categories. There are two exercises: In the first parts I have to show that the vertical composition and the horizontal composition of two natural ...
0
votes
1answer
17 views

Equality of maps in a monoidal category

Let $C_1,C_2$ be monoidal categories (aka tensor categories) with tensor bifunctor $$\otimes_i: C_i\times C_i\to C_i$$ and tensor units $1_i$. Assume I have a monoidal functor $F:C_1\to C_2$, it ...
4
votes
1answer
160 views

Commuting diagram for (monoidal) functors

Assume we have two categories $\cal A,B$ and four functors $F,G:\cal A\to B$, $D:\cal A\to A$, $E:\cal B\to B$. In order to show an equality like $$E\circ F=G\circ D\quad (1)$$ I have to prove that ...
3
votes
1answer
99 views

Definition of monoidal categories in Etingof’s “Tensor Categories”

I am quite confused on page 22,23 of the unit axiom of a monoidal category. Unit axiom: $L_1:X \mapsto 1 \otimes X$ and $R_1:X \mapsto X \otimes 1$ are autoequivalences of $C$. (end of pg 22)...
2
votes
1answer
51 views

$(\mathrm{Vect}_k,\otimes_k)$ as a non-strict monoidal category

What is the easiest way to see (or understand) that the category of vector spaces over a field $k$, endowed with its usual modoidal structure $\otimes_k$, is not a strict monoidal category.
0
votes
1answer
28 views

Unit isomorphism in SVECT

What is the unit isomorphism $$X\otimes\mathbb{C}^{1|0}\cong X$$ in the monoidal caregory of super-vector spaces? Is it $$x\otimes\lambda\mapsto \lambda x$$ like in the monoidal category of vector ...
2
votes
0answers
60 views

Adjunction of functors between monoidal categories induces adjunction between their categories of internal monoids

Here's a similar question, but I'm not sure if what I'll ask is already answered there. Let $(C,\odot), (D,\square)$ be monoidal categories. If $F \colon D \to C$ is left adjoint to $U \colon C \to D$...
2
votes
1answer
82 views

Affine space as a ringed space; is this the correct definition?

Let $k$ denote a field and let $\mathbb{A}^n$ denote affine $n$-space. Then if I understand correctly, it's best to view $\mathbb{A}^n$ as a ringed space. I'm a bit unsure as to what the structure ...
1
vote
0answers
85 views

Group homomorphism yields to monoidal functor

Consider a group $G$. We can get to a monoidal category $\cal G$ with objects being the group elements, tensor product given by group multiplication and just the identity morphisms as morphisms. ...
1
vote
0answers
58 views

Finding the Drinfeld centre of a category

I have the following unitary monoidal spherical category C: Simple objects: $\{1,x,y\}$. Non-trivial Fusion Rules: $$x\otimes y=x=y\otimes x$$ $$x\otimes x=1 \oplus 2x \oplus y$$ I would like to ...
2
votes
1answer
94 views

Is the class of dualizable objects in an abelian monoidal category closed under sums, kernels and cokernels?

Goodmorning to everybody. I am in the following situation. I have been told that in an abelian monoidal category (I assume this means an abelian category $\mathscr{A}$ with a monoidal structure $(\...
0
votes
1answer
75 views

In a tensor category, does $X\otimes Y\cong 0$ imply $Y\cong 0$ for non-zero $X$?

By a tensor category I mean a locally finite rigid $k$-linear abelian category with bilinear tensor product, and such that $\operatorname{Hom}(1,1)\cong k$.$^1$ Suppose we fix some non-zero object $...
3
votes
0answers
49 views

When is a monoidal structure on $\mathrm{Mod}_A$ induced by a bialgebra structure on $A$?

Fix a field $k$, and let $A$ be a (commutative, coassociative, counital) $k$-bialgebra. Write $\otimes = \otimes_k$. The category $\mathrm{Mod}_A$ of $A$-modules admits the structure of a monoidal ...
1
vote
1answer
132 views

Monoids as Categories With One Object of Specific Type

I was reading David Spivak's excellent book on category theory, where he uses a category to define a monoid. Another text by Tom Leinster defines monoids in the same way (I've only read through the "...
0
votes
1answer
30 views

Smallest abelian braided monoidal subcategory containing an object $V$

Let $\mathcal{C}$ be an abelian braided monoidal category with countable direct sums compatible with the tensor product (i.e. $X\otimes \bigoplus_{i \in \mathbb{N}} V_i \cong \bigoplus_{i \in \mathbb{...
4
votes
0answers
69 views

Constructing free monoid from free semigroup

Given a monoidal category $(\mathcal{C}, \otimes, I)$ with coproducts, the free monoid on an object $A \in \mathcal{C}$ is usually constructed by first constructing the free pointed object on $A$, i.e....
1
vote
0answers
34 views

the Verlinde formula

The Verlinde formula writes the fusion coefficient in terms of S matrix. My question is that for fusion category without braiding, is there a similar formula which gives the fusion coefficient in ...
1
vote
1answer
38 views

Equations on the free monoid??

While studying a bit of Joachim Lambek's calculus and some other applications of formal languages to the study of the structure of human language, I have come accross a reference to what authors like ...
3
votes
0answers
43 views

Monoidal Equivalence: Two Definitions

One has the definition of a monoidal equivalence as in definition 12 of Baez's Some Definitions Everyone Should Know. I have also seen monoidal equivalence defined as a monoidal functor between ...
3
votes
1answer
100 views

Why are the monoid objects in Mon(C) the commutative monoids?

Let $(C, \otimes, 1, \alpha, l, r)$ be a symmetric monoidal category. Can someone explain me why the monoids in the category of monoids Mon(C) are the abelian monoids ? Thanks for your help.