# Questions tagged [symmetric-monoidal-categories]

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### Automated rewriting of string diagrams in symmetric monoidal categories

Many algebraic structures, such as Frobenius algebras, or quasi-triangular Hopf algebras, can be formulated in an arbitrary symmetric monoidal category. They are given by a collection of morphisms, ...
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### Oplax monoidal functors of $\infty$-categories

In Higher Algebra, a notion of lax symmetric monoidal functors (in what follows, I'll remove the adjective "symmetric", but I'm mainly interested in the symmetric situation) is defined : if you have ...
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### Does Cantor Bernstein hold in a Closed Symmetric Monoidal Category?

In a closed symmetric monoidal category with $[I,X] \cong X$ for all $X$ is it true that having monomorphisms $m :A \rightarrow B$ and $m: B \rightarrow A$ is enough to imply $A \cong B$ ? I tried to ...
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### Very basic question about category theory

I have a symmetric monoidal category, and go to another one by replacing objects with isomorphism classes of objects. What's that called in category theory language? To give an example, consider the ...
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### Can the effective topos be seen as symmetric monoidal?

In Example(s) of monoidal symmetric closed category with NNO without infinite coproducts? user Zhen Lin states the effective topos is locally cartesian closed. On nLab we have that locally ...
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### 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 $\mathbb{Z}$-...
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### 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 $\overline U \oplus V$...
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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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### 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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### 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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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 $\... 1answer 395 views ### Functors with an epi-mono factorization property This is a simple question about terminology and a request for any related references. Specifically, what would you call a functor$F:\mathbf{D}\rightarrow\mathbf{C}$with the following property?$(*)...
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 ...
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 \$[...