# Questions tagged [symmetric-monoidal-categories]

The symmetric-monoidal-categories tag has no usage guidance.

**11**

votes

**3**answers

253 views

### Inequivalent compact closed symmetric monoidal structures on the same category

I am looking for interesting examples of categories admitting multiple monoidally inequivalent closed (or compact closed) symmetric monoidal structures.
We know how to construct disconnected toy ...

**10**

votes

**1**answer

231 views

### Is every symmetric monoidal combinatorial model category symmetric monoidally Quillen equivalent to a simplicial such?

Is every symmetric monoidal combinatorial model category symmetric monoidally Quillen equivalent (by a zig-zag of symmetric monoidal Quillen equivalences)
to a symmetric monoidal combinatorial ...

**12**

votes

**2**answers

479 views

### Status of Thomason's idea for a symmetric monoidal model of stable homotopy - from his last paper

In 1995, Robert Thomason published “Symmetric monoidal categories model all connective spectra” in TAC. On page 2, he argues that symmetric monoidal categories are more convenient than “May’s ...

**5**

votes

**0**answers

186 views

### Derived tensor products and Tor of commutative monoids

Two commutative monoids $M,N$ have a tensor product $M\otimes N$ satisfying the universal property that there is a tensor-Hom adjunction for any other commutative monoid $L$: $$\text{Hom}(M\otimes N,L)...

**18**

votes

**5**answers

666 views

### If a $\otimes$-idempotent object has a dual, must it be self-dual?

Let $C$ be a symmetric monoidal category.
Recall that a dual for $X \in C$ is an object $X^\vee$ and maps $\eta: I \to X \otimes X^\vee$ and $\varepsilon: X^\vee \otimes X \to I$ (where $I$ is the ...

**8**

votes

**1**answer

319 views

### Monoidal tensor product which preserves directed limits

Given a symmetric monoidal category $Q$, is there a construction of a (preferably full and faithful strong) monoidal embedding of $Q$ into some symmetric monoidal closed category $M$ which has all ...

**15**

votes

**1**answer

420 views

### What are the advantages of various “models” for the motivic stable homotopy category

People use several distinct models for the motivic stable homotopy category (so, there are some choices for the underlying category and a collection of available model structures). I would like to ask ...

**13**

votes

**0**answers

240 views

### Representation categories and homology

Let $G$ be a finite group.
Let $\mathcal{C}=Rep-G$ be the rigid $\mathbb{C}$-linear symmetric monoidal category of finite dimensional complex representations of $G$.
Can we recover some homological ...

**2**

votes

**0**answers

164 views

### Free commutative monoid monad

Has the monad induced by the free commutative monoid functor already been studied anywhere? Does it have any particular properties (other than not being cartesian)?
I would prefer a reference on ...

**4**

votes

**1**answer

176 views

### Left adjoint for categories of commutative monoids?

The $n$Lab writes (prop. 2.2 in https://ncatlab.org/nlab/show/category+of+monoids) :
Let $C$ be a monoidal category with countable coproducts that are preserved by the tensor product. Then the ...

**2**

votes

**1**answer

109 views

### How nontrivial can “central extensions of ribbon fusion categories” be?

In a sense, this is a follow up on this question, but one PhD programme later.
Let $\mathcal{C}$ be ribbon fusion. By $\mathcal{C}'$, we denote the symmetric centre, i.e. the full subcategory of ...

**5**

votes

**0**answers

76 views

### Temporal semantics for string diagrams

Suppose I have a string diagram $D$ which involves a set of strings $S$ and atomic processes $A$. Formally, we should think of this as a canonically chosen map in the free symmetric monoidal category (...

**3**

votes

**0**answers

60 views

### Branching behavior in string diagrams/monoidal categories?

I am currently working through Peter Selinger's paper "Towards a Quantum Programming Language", and trying to connect it with what I already know about monoidal categories and string diagrams.
...

**2**

votes

**0**answers

96 views

### Chevalley property for Tannakian categories

Are all Tannakian categories have the Chevalley property? I know that the categories $\mathrm{Rep}_k(G)$ have the Chevalley property for affine algebraic groups $G$ over a field $k$, but I don't know ...

**3**

votes

**1**answer

186 views

### Deligne tensor product of semisimple tensor categories

Let $T_1, T_2$ be two semisimple tensor categories over a field $k$ (i.e. symmetric rigid monoidal abelian $k$-linear). Then is the Deligne product, $T_1\otimes T_2$ also a $k$-tensor category?
...

**17**

votes

**1**answer

703 views

### Exponentiation of vector spaces?

This question occurred to me while thinking on another one here, Name for an operation on matrices?
Can one define in an invariant way a binary operation on finite-dimensional vector spaces - let us ...

**9**

votes

**2**answers

584 views

### The category of elements, enrichment, and weighted limits

This is a crosspost of this MSE question.
Every so often, when reading notes online or skimming through books, the category of elements and the Grothendieck construction pop up. I don't know anything ...

**12**

votes

**2**answers

568 views

### Adding inverses to a symmetric monoidal category (Reference?)

As we all know, the forgetful functor $\mathsf{Ab} \to \mathsf{CMon}$ from abelian groups to commutative monoids has a left adjoint, the Grothendieck group. I would like to categorify this ...

**6**

votes

**1**answer

275 views

### Categorical definition of infinite symmetric product

Let $(C,\otimes,I)$ be a symmetric monoidal category with coequalizers and directed colimits.
Fix some object $X$ and morphism $\tau\colon I\to X.$
Using $\tau$ one can construct a sequence of ...

**14**

votes

**2**answers

689 views

### Reference for an unbiased definition of a symmetric monoidal category

In the references I know, a symmetric monoidal category is defined as a monoidal category equipped with some additional data, so in particular the monoidal product is a functor
$$
\mathcal{C}\times \...

**13**

votes

**1**answer

405 views

### Is there something like “Noncommutative geometry internal to a category”?

I have heard that one can do algebraic geometry internal to symmetric monoidal categories. Topological quantum field theories also exist internal to symmetric monoidal categories, and the usual ...

**7**

votes

**0**answers

255 views

### Is there a classification of 2d extended TQFTs with defects?

Chris Schommer-Pries has classified 2d extended TQFTs (topological quantum field theories) in his PhD thesis. The result is a (not necessarily abelian) separable symmetric Frobenius algebra (possibly ...

**3**

votes

**0**answers

126 views

### Proofs in monoidal categories [closed]

I have to do some pretty ugly proofs in monoidal categories. Basically, I have some long identities that I would like to prove. A random example:
$$(a\otimes b)\circ (c\otimes d) \circ q = q $$
Are ...

**3**

votes

**2**answers

646 views

### When is the endofunctor category of a monoidal category braided? When is it ribbon? Fusion? Modular?

Given a category $\mathcal{C}$, we can define the category of endofunctors $\operatorname{Cat}(\mathcal{C})$, with objects functors $F: \mathcal{C} \to \mathcal{C}$ and morphisms natural ...

**5**

votes

**1**answer

234 views

### Is there a quotient or exact sequence of symmetric, premodular (ribbon fusion) and modular categories?

In Walker and Wang's article about (3+1)-TQFTs from premodular categories, they say on page 14 that you can take a quotient of a premodular category $\mathcal{C}$ by its symmetric fusion subcategory $...

**0**

votes

**0**answers

176 views

### Shift functor and the origin of the linear decalage isomorphism

Let $(\mathbf{Vec}_\mathbb{Z}(\mathbb{K}),\otimes,\tau)$ be the symmetric monoidal category of $\mathbb{Z}$-graded $\mathbb{K}$-vector spaces, where $\otimes$ is the tensor product of $\mathbb{Z}$-...

**0**

votes

**1**answer

172 views

### If a (linear) relation maps Lagrangian subspaces to Lagrangian subspaces, is it a Lagrangian relation?

Let $(U,\omega),(V,\rho)$ be symplectic vector spaces. Call a relation $U \to V$ a (linear) Lagrangian relation (also Lagrangian correspondence) if it is a Lagrangian subspace of $\overline U \oplus V$...

**9**

votes

**2**answers

411 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 ...

**6**

votes

**2**answers

317 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 ...

**4**

votes

**1**answer

255 views

### How do we handle the symmetry condition in nCob and TQFTs?

A $(n+1)$-topological quantum field theory $\mathcal{T}$ is a rigid symmetric monoidal functor from the category $(n+1)$-Cob of $n$-manifolds and $(n+1)$-cobordisms to FdVect.
My question is about the ...

**2**

votes

**2**answers

287 views

### Example(s) of monoidal symmetric closed category with NNO without infinite coproducts?

The question is in the title, here is my motivation:
$\require{AMScd}$Let $(\mathcal C,\otimes,I)$ be a monoidal symmetric closed category. Then, the tensor product commutes with colimits, and if $\...

**2**

votes

**1**answer

357 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?
$(*)...

**1**

vote

**1**answer

127 views

### lfp property for dagger symmetric monoidal categories and their internal categories

We can define internal categories in a monoidal category like this. Let $C$ be a dagger symmetric monoidal category. Will $C$ be locally finitely presentable? Let $C_{int}$ be the category of ...

**2**

votes

**2**answers

304 views

### Definitions and coherence in “rigid” monoidal categories

In "Catégories 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 $[...

**24**

votes

**1**answer

2k views

### Analogy between the exterior power and the power set

The symmetric algebra of an object exists in every cocomplete $\otimes$-category. For the category of sets $\mathrm{Sym}(X)$ is the set of multi-subsets of $X$.
The usual definition of the exterior ...

**12**

votes

**1**answer

747 views

### Classification of symtrivial modules over a PID

Let us call a module $M$ over a commutative ring $R$ symtrivial if the symmetry $M \otimes M \to M \otimes M, a \otimes b \mapsto b \otimes a$ equals the identity (the same notion applies to arbitrary ...

**15**

votes

**3**answers

1k 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 ...

**4**

votes

**2**answers

661 views

### Gamma spaces and monoidal categories II

This question is kind of a follow-up of this one.
Suppose I have a topological category $\mathcal{C}$ (objects and morphisms topological spaces, source and target map continuous, etc.) together with ...

**5**

votes

**2**answers

561 views

### Module categories over symmetric/braided monoidal categories

Given an algebraically closed field $k$ and a finitely generated commutative $k$-algebra $A$, all simple modules over $A$ are 1-dimensional
What is the analogous statement for symmetric monoidal $k$-...

**5**

votes

**3**answers

445 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 ...