The symmetric-monoidal-catego tag has no wiki summary.

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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107 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 ...

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### 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 ...

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126 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 ...

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59 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 ...

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348 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 ...

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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 ...

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### 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 ...

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237 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 ...

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**1**answer

230 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 ...

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225 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 ...

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**1**answer

227 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?
...

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### 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 ...

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203 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 ...

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### 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 ...

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520 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 ...

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### 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 ...

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559 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 ...

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### 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 ...

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### 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 ...