3
votes
1answer
276 views

Category of modules over commutative monoid in symmetric monoidal category

Let $\left(\cal{C},\otimes ,I\right)$ be a symmetric monoidal category (not necessarily closed) and $A$ a commutative monoid in $\cal{C}$. In his DAG III (page 95), Lurie writes: In many cases, ...
5
votes
2answers
233 views

What are TQFTs that are multiplicative under connected sums? Do bordisms with connected sum as monoidal product exist?

In general, one extracts a manifold invariant from a TQFT by interpreting the closed manifold as a bordism from the empty set to the empty set. The TQFT sends this bordism to a homomorphism of the ...
2
votes
0answers
89 views

Pseudomodules, “general coherence theorem”

A pseudomonoid is defined within a monoidal bicategory. It is like a monoid in a monoidal category except that the usual axioms hold up to coherent invertible 2-cells. Pseudomonoid is like a monoidal ...
5
votes
1answer
232 views

A general theory of quasi-functors, generalizing from dg-categories to $\mathcal V$-categories, with $\mathcal V$ monoidal model category

I employ the vast majority of the post to develop the notion of quasi-functor between dg-categories: I think it is important to get the idea. Let $k$ be a field, and let $\mathcal V =\mathbf C(k)$ ...
1
vote
0answers
64 views

symmetric monoidal dagger endofunctor categories

Take an arbitrary category $C$, and consider $End(C)$, any endofunctor category consisting of some or all endofunctors of $C$. I have several related questions: What restrictions must we impose on ...
6
votes
2answers
218 views

String diagrams for bimodules over noncommutative algebras?

I'm trying to do some calculations with bimodules (over Azumaya algebras, as it happens), and I need a string diagram notation that mixes the tensor product over the base ring (a symmetric monoidal ...
2
votes
1answer
164 views

Localization of symmetric monoidal category

Let $\mathcal M$ be a symmetric monoidal category, $S\subset \mathcal M$ a collection of objects and morphisms. I would like to construct the localization $\mathcal M \mathop{\longrightarrow}^T ...
2
votes
1answer
68 views

Does the category PCM (partial commutative monoids) have a closed symmetric monoidal product?

A partial commutative monoid (PCM) is, roughly speaking, a set with a partially defined binary operation that is as associative as it can be (given that not all products are defined) and commutative. ...
3
votes
1answer
131 views

How to construct a free 2-group on a groupoid?

Let G be a groupoid. I'm wondering how to construct the free 2-group on G. By the free 2-group I mean a 2-group $\mathcal{F}\left(G\right)$ equipped with a functor ...
3
votes
1answer
89 views

Strictifying strong monoidal functors

Let $C_1$ and $C_2$ be monoidal categories (not necessarily symmetric or strict) and let $\Psi : C_1 \rightarrow C_2$ be a strong monoidal functor. Is it possibly to construct a strict monoidal ...
10
votes
1answer
348 views

Categorical interpretation of disjoint cycle notation for tracing permutations

For a natural number $n\in\mathbb{N}$, let $\underline{n}$ denote the finite set $$\underline{n}:=\{1,2,\ldots,n\}.$$ A permutation $\sigma\in Aut(\underline{n})$ can be uniquely written (up to order) ...
1
vote
1answer
202 views

Inducing a Monoidal Structure using an Equivalence of Categories [closed]

Given an equivalence of categories $C \equiv D$, such that $C$ has a monoidal structure, is it clear that we can use the equivalence to induce a monoidal structure on $D$. Is there a standard ...
10
votes
1answer
130 views

Understanding the computation of the center of Tambara-Yamagami fusion categories when realized as C* categories

Recall that the Tambara-Yamagami categories are those with fusion ring $\mathbb{Z}[A \cup m]$ where $A$ is an abelian group and $m$ is a non-invertible (simple) object such that $ma = am = m$ for all ...
2
votes
1answer
99 views

When is an Eilenberg-Moore category or Kleisli category braided monoidal? When semisimple?

I have a braided monoidal, semisimple linear category $\mathcal{C}$. (Imagine representations of a semisimple quasitriangular Hopf algebra.) I also have a monad $(T,\mu,\eta)$ on it, however, $T$ is ...
4
votes
2answers
148 views

symmetric monoidal double categories?

Let me preface this by saying that I don't know much category theory. I am running into a situation where I have a double category and additionally there is a multiplication. Moreover, choosing ...
0
votes
0answers
85 views

Quantum Bayesian update and Bayesian update of a model category

We know that we can have internal categories in a model category. We also know that there is a notion of Quantum Bayesian update in a monoidal category. Does the Quantum Bayesian update (equation ...
3
votes
2answers
124 views

Why is a braided left autonomous category also right autonomous?

In Selinger, P. A survey of graphical languages for monoidal categories (New Structures for Physics, Springer, 2011, 813, 289-233), it is stated that: Lemma 4.17 ([23, Prop. 7.2]). A braided ...
5
votes
1answer
162 views

Making additive envelopes of monoidal categories monoidal

I have a monoidal category $(\mathcal{C},\otimes)$ enriched over abelian groups, for which I want to take the additive envelope $\mathcal{M}at\,\mathcal{C}$. (This is defined as the category with ...
2
votes
0answers
71 views

A question about braiding represented as pseudofunctors

I just asked a very similar question here (About symmetry, braids, and pseudo-functors.), still now I dont get a answere. I hope to express myself more clearly and correctly this time. Let ...
0
votes
3answers
271 views

Rank vanishing in tensor categories

Let $\mathcal{C}$ be a cocomplete $\mathbb{Q}$-linear symmetric monoidal category (if convenient, assume that it is locally finitely presentable and/or abelian). Recall that the rank (or dimension) of ...
10
votes
0answers
183 views

Does every commutative variety of algebras have a cogenerator?

By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes. Equivalently (i) every interpretation of ...
3
votes
3answers
327 views

When does a monoidal functor between ribbon categories preserve cups and caps, but not necessarily braidings?

Imagine a monoidal functor between ribbon categories (i.e. monoidal, with a braiding, a twist and compatible left and right duals). An important example would be the restriction functor from the ...
11
votes
1answer
378 views

Can one explain Tannaka-Krein duality for a finite-group to … a computer ? (How to make input for reconstruction to be finite datum?)

Consider a finite group. Tannaka-Krein duality allows to reconstruct the group from the category of its representations and additional structures on it (tensor structure + fiber functor). Somehow ...
4
votes
1answer
99 views

Does trace handle composition in a traced symmetric monoidal category?

Suppose that $(C,\otimes,I)$ is a traced symmetric monoidal category (TSMC) with symmetrizor $\sigma$ and trace $Tr$. Given two morphisms $f\colon A\to B$ and $g\colon B\to C$, I can tensor them to ...
12
votes
3answers
1k views

History and motivation for Tannaka, Krein, Grothendieck, Deligne et al. works on Tannaka-Krein theory?

I am trying to wrap my mind around Tannaka-Krein duality and it seems quite mysterious for me, as well, as its history. So let me ask: Question: What was the motivation and historical context for ...
2
votes
1answer
295 views

It looks so coKleisli, but it's not. What is it?

Fix a symmetric monoidal category $(M,\otimes,I)$ and a small discrete monoidal subcategory $M'\subseteq M$. Define a new symmetric monoidal category $C:=CoKl(M,M')$ as follows: $Ob(C):=Ob(M)$, and ...
7
votes
2answers
268 views

Is “being a modular category” a universal or categorical/algebraic property?

A semisimple braided category with duals is called modular when a certain matrix $S$ is invertible. The components $S_{AB}$ are indexed by (isomorphism classes of) simple objects of the category and ...
2
votes
1answer
203 views

Functors with an epi-mono factorization property

This is a simple question about terminology and a request for any related references. Specifically, what would you call a functor $F:\mathbf{D}\rightarrow\mathbf{C}$ with the following property? ...
4
votes
1answer
242 views

Two definitions of modules in monoidal category

The standard definition of a (left) simplicial module $V$ over some simplicial algebra $A$ is the map of simplicial vector spaces $A\otimes V\to V$ that gives the usual modules component-wise. Here ...
6
votes
1answer
243 views

About an embedding of abelian categories into categories of modules

Let $k$ be a field. Let $C$ be an abelian $k$-linear category with a symmetric tensor product $\otimes$ and internal homomorphisms, such that $\mathrm{End}(1)=k$. Let $M$ be another $k$-linear abelian ...
5
votes
1answer
163 views

Monoidal transformations are isomorphisms at dualizable objects

Here is a cute observation: Let $F,G : \mathcal{C} \to \mathcal{D}$ be a symmetric monoidal functors between symmetric monoidal categories, and let $\eta : F \to G$ be a monoidal transformation. Then ...
2
votes
2answers
178 views

Definitions and coherence in “rigid” monoidal categories

In "Cat├ęgories Tannakiennes" by Savedra Rivano (under A. Grothendieck supervision) at pag.78 he define a rigid category $\mathscr{C}$ as a monoidal symmetrical closed such that the natural morphisms ...
6
votes
1answer
242 views

Generalization of analytic functors

It's been a long time since I posted the following question on stackexchange. Now I think it's better to adress it to you, in the hope I will reach the right audience: Martin's comment, albeit ...
3
votes
2answers
186 views

Coherence in pseudo.monoids

In the article of A. Joyal and R. Street: Braided tensor categories, (Adv. Math. N.102 (1993), no. 1, 20-78) they define a tensor object (or pseudo.monoids) in a monoidal 2-category $\mathcal{C}$ ...
12
votes
2answers
399 views

Cartesian envelope of a symmetric monoidal category

Is there a left adjoint to the forgetful functor from cartesian monoidal categories to symmetric monoidal categories? And if so: how can it be described explicitely. In more detail: Given a symmetric ...
9
votes
1answer
361 views

Reference for “lax monoidal functors” = “monoids under Day convolution”

Suppose $A$ and $C$ two symmetric monoidal categories. Let's say that $A$ is small and $C$ is locally presentable, and let's assume also that the tensor product on $C$ preserves colimits separately in ...
0
votes
0answers
69 views

the category of right comodule of coalgebra is a monoidal category , why?

the category of right comodule of coalgebra is a monoidal category according the following the associativity constraint is defined as a_{U,V,W}((u\otimes v)\otimes w)=u_0\otimes (v_0\otimes ...
1
vote
2answers
269 views

“Wrong” strictification of symmetric monoidal categories

It is well-known that any symmetric monoidal category is equivalent to a strict symmetric monoidal category. The construction of this strict monoidal category is rather technical and it appears to me ...
11
votes
1answer
602 views

Are Lurie's operads special SMCs?

In Higher Operads, Higher Categories, Leinster nicely characterizes operads among monoidal categories (as PROs). Roughly speaking, a monoidal category comes from an operad if its set of objects is ...
5
votes
1answer
262 views

Unitalization internal to monoidal categories

Let $C$ be a monoidal category, not assumed to be symmetric. Assume that the underlying category of $C$ is nice enough, for example cocomplete, perhaps even presentable. A semigroup object in $C$ is a ...
8
votes
0answers
253 views

Twisted duality in a symmetric monoidal category

I would like to know if the following definition is already known or even well-known. Is there a reference in the literature? Do you know prominent examples? Definition. Let $\mathcal{C}$ be a ...
1
vote
0answers
128 views

Compatibility between strength and costrength of a monoidal monad

Let $C$ be a closed monoidal category, and let $T : C \to C$ be a monad on the underlying category. Let $\sigma$ be a tensorial strength of $T$ and let $\sigma'$ be a cotensorial strength of $T$. A ...
4
votes
1answer
225 views

Example of a non-closed cocomplete symmetric monoidal category

Background By a cocomplete symmetric monoidal category $C$ I mean a symmetric monoidal category whose underlying category is cocomplete and such that $- \otimes X : C \to C$ is cocontinuous for all ...
1
vote
2answers
317 views

Not-so-symmetric monoidal categories

Is the notion of a commutative monoidal category, where every product is isomorphic to its opposite, but not necessarily functorially, known not to be useful? I have not been able to find any ...
9
votes
3answers
572 views

Free symmetric monoidal category on a monoidal category

Consider the $2$-categories $\mathsf{MonCat}$ of monoidal categories, with strong monoidal functors and monoidal transformations, $\mathsf{SymMonCat}$ of symmetric monoidal categories, with strong ...
5
votes
2answers
464 views

What is the free monoidal category generated by a monoid?

In several places in a segment on cohomology (for example, here (PDF)) in John Baez's online lecture notes for a course in 2007 on quantum gravity, much is made of the fact that the simplex category ...
1
vote
1answer
184 views

About the closed structure on the modules of a monoidal closed symmetrical category

Let $(\mathscr{C}, \otimes , I)$ monoidal category, a monoid $(R, e_R, \mu )$ is a object $R \in \mathscr{C}$ with morphisms $e_R: I \to R$, $\mu: R \otimes R \to R$ with the well knowed unital ...
6
votes
1answer
371 views

Two questions about commutative theories

Let $\mathcal{T}$ be a commutative algebraic theory (for example sets, abelian groups, commutative monoids, but not groups etc.). References include the nlab and Borceux' Handbook of Categorical ...
3
votes
0answers
160 views

The multiplicative system in a symmetric monoidal category

Let $\mathcal{C}$ be a symmetric monoidal category. In the 1973 paper "Note on monoidal localisation" by Brian Day, the multiplicative system of morphism in $\mathcal{C}$ has been discussed. See also ...
7
votes
1answer
522 views

Saavedra's Definition of Tannakian Category

I have been reading Deligne-Milne's Tannakian Categories and got to the point at the end of part 3 where they discuss what went wrong with Saavedra's definition. Motivated by their counterexample, ...