# Tagged Questions

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### Coherence theorem for symmetric lax monoidal functors

Let $V$ and $W$ be symmetric monoidal categories. Let $F:V\to W$ be a lax symmetric monoidal functor with multiplication $\nabla:FA\otimes FB \to F(A\otimes B)$. Consider the following statements: 1) ...
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### t-structures on the tensor product of stable $\infty$-categories

It is a matter of checking definitions that the tensor product of presentable $\infty$-categories restrict to a tensor product between stable presentable $\infty$-categories; I was wondering if there ...
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### Seeking more information regarding the “rigoidal category” of $\mathbb{N}$-graded sets

Definitions. By an $\mathbb{N}$-graded set, I mean a set $X$ together with a function $|\Box|_X:\mathbb{N} \leftarrow X,$ called the grading. These will simply be called graded sets hereafter. If $Y$...
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### Non-trivial but simple concrete examples for some categories related to Tensor/Fusion categories

I'm writing a note on Tensor and Fusion categories, the readers of which are physicists rather than mathematicians. So instead of giving abstract definitions I have to give examples to inspire each ...
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### Mac Lane strictness theorem and categorifiability of fusion rings

The Mac Lane strictness theorem states that any monoidal category is monoidally equivalent to a strict monoidal category (see here section 2.8). Q1: Is it true that any fusion category is monoidally ...
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### The definition of unitary fusion category

I just come across a definition of the unitary fusion category: A fusion category $\mathcal{C}$ over the complex number is said to be unitary if we have: We have a Hilbert space structure on each ...
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### Semisimple monoidal category with duals

We say that an abelian category is semisimple if every object is a semisimple object, which is to say, a direct sum of finitely many simple objects. Let $({\cal C},\otimes,*)$ be a semisimple ...
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### About a closed strucure on profunctors

Let $Prof$ the bicategory with profunctors (on small categories), arrows are like $D: \mathscr{A} \dashrightarrow \mathscr{B}$ and this means that $D: \mathscr{A}^{op} \times \mathscr{B}\to Set$. ...
Given two abelian monoidal categories ${\cal C,D}$ (where the monoidal operation is bilinear) and an additive monoidal functor $F:{\cal C} \to {\cal D}$. Will $F$ always admit an adjoint?