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4
votes
0answers
254 views

Reshetikhin-Turaev and links with a distinguished component

Hi, This question came up to me when reading the paper of Cartier on Vassiliev invariants, but it can probably be turned into a more general question. Let $T$ be the category whose objects are ...
4
votes
1answer
246 views

Definition of enriched caterories or internal homs without using monoidal categories.

I know this question may seem nonsensical at first but let me exlain what i have in mind: In enriched category theory we define categories enriched over a monoidal category $(\mathcal{V},\otimes, ...
6
votes
2answers
405 views

Does the dual of an object with trivial symmetry also have trivial symmetry?

Let $C$ be a symmetric monoidal category. I am interested in objects $X \in C$ such that the symmetry $S_{X,X} : X \otimes X \cong X \otimes X$ is equal to the identity. There are many examples of ...
14
votes
0answers
656 views

Is the category of smooth manifolds equivalent to the opposite category of the category of commutative monoids of some additive symmetric monoidal category?

This is a followup to my previous question, which asked whether the category of commutative or noncommutative C*-algebras or von Neumann algebras is equivalent to the category of commutative or ...
9
votes
1answer
707 views

Gamma spaces and monoidal categories

In his paper "Categories and cohomology theories" Graeme Segal gives examples how to construct a Gamma category and therefore also a Gamma space from a strict monoidal category like finite chain ...
29
votes
0answers
660 views

Enriched Categories: Ideals/Submodules and algebraic geometry

While working through Atiyah/MacDonald for my final exams I realized the following: The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed ...
4
votes
1answer
859 views

Understanding Penrose diagrammatical notation

I arrived to Penrose's paper Applications of negative dimensional Tensors after reading some bits of Baez's Prehistory (link) and the first two chapters of Turaev's Quantum invariants of knots and ...
3
votes
0answers
128 views

Show that duality functor is anti-monoidal

Let $\mathcal{C}$ be a right rigid (not strict) monoidal category with associativity constraint $\Phi$. Let $J_{UV}: U^*\otimes V^*\to (V\otimes U)^*$ the canonical isomorphism for every objects ...
4
votes
0answers
255 views

When is the cofibrant replacement of a product the product of the cofibrant replacements?

I'm in a situation where I'd like to prove $Q(E\otimes E) \simeq QE \otimes QE$ for a monoid $E$ in a symmetric monoidal model category. I know it's not true in general that $Q(E\otimes F)\simeq QE ...
5
votes
1answer
247 views

Ends as a “cotrace” operation on profunctors

As mentioned here, there is a trace operation on the monoidal category of profunctors given by taking coends: for any profunctor $F : A\times X \nrightarrow B \times X$, there is a profunctor $Tr^X(F) ...
7
votes
1answer
865 views

Exterior powers in tensor categories

Let $\mathcal{C}$ be a cocomplete $R$-linear tensor category. Many notions of commutative algebra can be internalized to $\mathcal{C}$. For example a commutative algebra is an object $A$ in ...
2
votes
3answers
439 views

Action on tensor power and “element notation” in monoidal categories

Let $C$ be a symmetric monoidal category. Fix an object $X$, let $S$ denote the symmetry $X \otimes X \to X \otimes X$. Also define $X^{\otimes n}$ by induction on $n$: $X^{\otimes 0} = 1$, ...
1
vote
0answers
125 views

A simple example of a tame tensor functor

Let $C$ be a $R$-linear cocomplete tame abelian tensor category (see here for definitions) with unit $1$, then there is a unique $R$-linear cocontinuous tensor functor $F : \text{Mod}(R) \to C$, ...
5
votes
1answer
291 views

representing tensor-C*-categories in BIM

Given a factor M (=von Neumann alg. with center ℂ), let us write BIM for the ⊗-C*-category of M-M-bimodules. Which ⊗-C*-categories can one faithfully embed into BIM? ⓵ Are ...
4
votes
1answer
360 views

Is the functor of divided powers a weakly monoidal functor?

Let $R$ be a commutative ring with identity $e$. For every $R$-module $M$ the algebra of divided powers $D(M)$ is defined as follows. The generators of $D(M)$ are the symbolls $m^{(k)}$ for every ...
3
votes
1answer
484 views

Tame abelian tensor categories

In the article "Tannaka duality for geometric stacks" (arxiv, see nlab for a summary) Jacob Lurie introduced the notion of a tame abelian tensor category. An abelian tensor category is called tame if ...
5
votes
1answer
232 views

Do plain functors out of monoidal categories factor into a nontrivial monoidal part and a plain part?

Given symmetric monoidal closed categories $C, D$, a symmetric monoidal (not closed!) functor $F:C\to D$ factors into two parts: the first is a symmetric monoidal closed functor from $C$ to a ...
0
votes
1answer
529 views

universal property of module categories internal to a tensor category

By a tensor category, I mean here a cocomplete $k$-linear symmetric tensor category, where $k$ is a fixed ground ring. Tensor functors are assumed to be $k$-linear and cocontinuous. I will denote the ...
3
votes
1answer
135 views

Epimorphisms between line objects

Let $\mathcal{A}$ be a $k$-linear abelian symmetric tensor category with unit $\mathcal{O}_A$; here $k$ is a comm. ring. By that I assume implicitly that $\otimes$ is finitely cocontinuous in each ...
5
votes
3answers
375 views

Does one of the hexagon identities imply the other one?

Suppose we have a monoidal category equipped with additional data that almost makes it a braided monoidal category except that only one of the hexagon identities is satisfied. Can we then prove the ...
8
votes
2answers
475 views

Gerbes and Z-graded symmetric monoidal categories

Let me first recall a known fact. Suppose $X$ is a complex algebraic variety, $\mathscr{L}$ is a line bundle on $X$ and $L^\times$ is the total space of $\mathscr{L}$ with the zero section removed. ...
5
votes
1answer
381 views

graded ring associated to a line bundle in a tensor category

Let $\mathcal{A}$ be an abelian tensor category with unit $\mathcal{O}$. An object $\mathcal{L}$ is called invertible or a line bundle if there is some $\mathcal{L}^{-1}$ such that $\mathcal{L} ...
10
votes
1answer
724 views

Alternative definition of monoidal categories

My question is about monoidal categories. To motivate it, let me first recall something about group objects. Assume you define a group object in a category $C$ with products by an object $G$ together ...
5
votes
2answers
291 views

Is the box product of morphisms associative?

Suppose $(C,\otimes)$ is a symmetric monoidal finitely-cocomplete category such that $\otimes$ preserves colimits. Given two morphisms $a:A_1\to A_2$ and $b:B_1\to B_2$, define $a\Box b$ to be the ...
3
votes
0answers
192 views

Symmetric monoidal structure on cosimplicial spaces

Is there a monoidal structure on the category $Spc^{\Delta}$ of cosimplicial spaces such that in the adjunction $$ \Delta^{\bullet}\otimes-\colon Spc\leftrightarrows Spc^{\Delta}\colon Tot(-) $$ the ...
2
votes
2answers
328 views

The condition End(1) = k in Tannakian Categories

A neutral Tannakian category over a field $k$ is a rigid $k$-linear abelian tensor category $\mathcal{C}$ whose unit $1$ satisfies $\mathrm{End}(1) \simeq k$, and is moreover equipped with an exact ...
3
votes
3answers
393 views

What is the proper name for “compact closed” multiplicative intuitionistic linear logic?

Multiplicative intuitionistic linear logic (MILL) has only multiplicative conjunction $\otimes$ and linear implication $\multimap$ as connectives. It has models in symmetric monoidal closed ...
7
votes
2answers
492 views

From tensor algebras in monoidal categories to (commutative?) monoids

Let $D$ be a monoidal category (without the structure of a symmetric monoidal category), with unit object $Id$, and let $L$ be an invertible object in $D$, so that $L$ is dualizable and the pairing ...
2
votes
2answers
205 views

A better way to compute the mapping spaces of the category of spans in an enriched tensored category?

Let X be a tensored and cotensored V-category, where V is a fixed complete, cocomplete, closed symmetric monoidal category. Define $C:=Span(X)$ to be the category of spans in X (this is the functor ...
5
votes
1answer
240 views

What is the right way to define the nerve of an unbiased monoidal category?

I've been toying around with unbiased composition in higher categorical structures on and off for a while now. In particular, I've been playing around with unbiased monoidal 2-categories. One ...
8
votes
3answers
609 views

Skeleton of a braided monoidal category

Does every (lax) braided monoidal category have a braided monoidal skeleton? That is, I want to define a (lax) braided monoidal structure on a skeleton so that it is braided monoidal equivalent to ...
1
vote
2answers
300 views

In which categories is every coalgebra a sum of its finite-dimensional subcoalgebras?

Let $\mathcal V$ be a reasonably nice category — I'm interested in the case when $\mathcal V$ is $\mathbb K$-linear for some field $\mathbb K$, abelian, and has all products and coproducts ...
6
votes
4answers
714 views

180˚ vs 360˚ Twists in String Diagrams for Ribbon Categories

Ribbon categories are braided monoidal categories with a twist or balance, $\theta_B:B\to B$, which is a natural transformation from the identity functor to itself. In the string diagram calculus for ...
4
votes
1answer
307 views

De-equivariantization by Rep(G)

I'm trying to understand Proposition 2.9 of this paper on weakly group theoretical fusion categories. First of all I have a problem with understanding the settings for de-equivariantization process. ...
20
votes
8answers
4k views

Resources for graphical languages / Penrose notation / Feynman diagrams / birdtracks?

There is an idea I've recently gotten interested in that doesn't seem to have a good agreed-upon name ("diagrammatic algebra?"). It centers around the use of two-dimensional diagrams of dots, ...
8
votes
4answers
2k views

What is the universal enveloping algebra?

Let ${\mathfrak g}$ be a Lie algebra in a symmetric monoidal category enriched over $K$-vector spaces, i.e., in particular, hom-s are $K$-vector spaces (where $K$ is a field of characteristic zero). ...
7
votes
3answers
689 views

Tannakian description of a semi-direct product

Let $H$ and $K$ be affine group schemes over a field $k$ of caracteristic zero. Let $\varphi:H\to Aut(K)$ be a group action. Then we can form the semi-direct product $G = K\ltimes H$. Problem: ...
4
votes
1answer
430 views

Is assigning the endomorphism object in some sense functorial?

Let $\mathcal V$ be a monoidal category and let $\mathcal C$ be a $\mathcal V$-category. Let's denote the $\mathcal V$-valued hom-functor $[-,-]$. Now for every object $X\in\mathcal C$ we have it's ...
3
votes
0answers
164 views

What is the underlying graphical calculus of the Interactions-Round-a-Face lattice model?

Background Let $\mathcal{L}$ be an $m \times n$ square lattice on a torus, and let $\Sigma$ be a finite set. We think of $\Sigma$ as the possible spin values that can be assigned to the points of the ...
7
votes
1answer
352 views

Category of copresheaves over commutative monoids

Let C be a symmetric monoidal category. Let Comm(C) be the category of commutative monoids in C. Consider the topos X = CoPSh(Comm(C)) of covariant functors from Comm(C) to the category Set of sets. ...
4
votes
2answers
461 views

Closed monoidal structure on the derived category of sheaves

Given a topological space X, i'd like to find Der X - the derived category of sheaves of abelian groups on X - to be a closed monoidal category. Hom should be cohomological and the internal-hom should ...
5
votes
5answers
437 views

What are the correct axioms for a “weakly associative monoidal functor”?

Definitions and the main question Recall that a category $\mathcal C$ is monoidal if it is equipped with the following data (two functors, three natural transformations, and some properties): a ...
5
votes
1answer
474 views

Is there a relative version of Tannakian reconstruction?

According to some form of Tannakian reconstruction, given a finite tensor category with a fiber functor to the category of vector spaces, one determines a Hopf algebra by considering tensor ...
16
votes
1answer
2k views

Calculating 6j-symbols (aka Racah-Wigner coefficients) for quantum groups

Which $6j$-symbols for quantised enveloping algebras are known explicitly? The quantum $6j$-symbols for $sl(2)$ are well-known. The references are Masbaum and Vogel and Frenkel and Khovanov. What ...
7
votes
2answers
375 views

Effects of “weak” vs. “strict” categories in Eckmann-Hilton arguments

A standard example for demonstrating the need for genuinely weak n-categories is that a weak 3-category with unique 0- and 1-cells amounts to the same thing as a braided monoidal category (by an ...
10
votes
4answers
846 views

Why is a monoid with closed symmetric monoidal module category commutative?

Given a symmetric monoidal category and a monoid object A in it, one can form the category of modules over this monoid object, i.e. objects are $A \otimes M \rightarrow M$ satisfying the natural ...
24
votes
4answers
2k views

Understanding the definition of the Lefschetz (pure effective) motive

For all those who are unlikely to have answers to my questions, I provide some Background: In some sense, pure motives are generalisations of smooth projective varieties. Every Weil cohomology ...
16
votes
1answer
805 views

Semiadditivity and dualizability of 2

Short version: Let (C, ⊗, 1) be a locally presentable closed symmetric monoidal category with a zero object, and write 2 = 1 ∐ 1. Suppose the object 2 has a dual. Does it follow that C is a ...
2
votes
1answer
247 views

When does a certain natural construction on monoidal categories yield a Hopf algebra?

Let $\mathcal C = (\mathcal C_0,\mathcal C_1)$ be a (small) strict monoidal category. Pick a field $\mathbb K$, and let $\mathbb K[\mathcal C_1]$ be the vector space with basis the morphism of ...
12
votes
3answers
631 views

Model Structure/Homotopy Pushouts in topological monoids?

Let C be the category of topological monoids, that is, the category of monoids in (Top, $\times$). Can the model category structure on Top (Serre fibrations, cofibrations, weak homotopy ...