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

**9**

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**1**answer

250 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

217 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 ...

**8**

votes

**2**answers

312 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 ...

**12**

votes

**1**answer

261 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 ...

**6**

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**0**answers

143 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**

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**0**answers

108 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 ...

**1**

vote

**0**answers

94 views

### When is the category of endofunctors 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 ...

**4**

votes

**1**answer

130 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

63 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 ...

**-1**

votes

**1**answer

352 views

### $E_n$ structures on Symmetric Monoidal Stable infinity-categories [closed]

Let $C$ be a stable $∞$-category with a monoidal structure on it, hence a monoidal stable $∞$-category; if this monoidal structure is "maximally symmetric" (in Wikipedia's sense) $C$ is a symmetric ...

**0**

votes

**1**answer

97 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 ...

**5**

votes

**2**answers

286 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

240 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

231 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

227 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

234 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

105 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

207 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 ...

**20**

votes

**1**answer

1k 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 ...

**11**

votes

**1**answer

527 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 ...

**11**

votes

**3**answers

681 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

561 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

453 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 ...

**5**

votes

**3**answers

376 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 ...