# Questions tagged [monoidal-categories]

A monoidal category, also called a tensor category, is a category $\mathcal{C}$ equipped with a bifunctor $\otimes\colon \mathcal{C}\times\mathcal{C}\to \mathcal{C}$ which is associative up to a natural isomorphism, and an object $\mathbb{1}$ which is both a left and right identity for $\otimes$ up to a natural isomorphism.

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### Examples of closed categories which are not monoidal closed?

There is a solid definition of "closed category" axiomatizing the idea that we can assign something resembling a hom-object to each pair of objects of a category. However, I am struggling to ...
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### What is a "functorial isomorphism"?

I am reading a lecture note about tensor category by P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik. The link is attached here https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-...
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### Is there a non-symmetric monoidal monad?

Recall that a monoidal monad on a monoidal category $(\mathcal{C}, \otimes, I)$ is a monad $(M, \eta, \mu)$ on $\mathcal{C}$ such that $M$ is also equipped with the structure of a lax monoidal functor ...
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### If $A$ and $B$ are dualizable objects in a monoidal category, is the unit of the one duality the inverse of the counit of the other duality?

I'm currently trying to wrap my head around dualizable objects in monoidal categories and I was wondering whether the following claim holds: Let $A$ and $B$ be dualizable objects in a monoidal ...
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### Why does duality of objects $A$, $A^\ast$ in a symmetric monoidal category imply an adjunction $(-) \otimes A \dashv (-) \otimes A^\ast$?

Let $\mathcal{C}$ be a symmetric monoidal category and let $A$ and $A^*$ be dual in the sense of Definition 2.1 in nLab. Dold & Puppe (1984) show (Thm 1.3) that the map  \text{Hom}(X, Y \otimes ...
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### Why is $\mathcal{C}$ equivalent to $\mathcal{C}^{\text{op}}$ when $\mathcal{C}$ is a compact category?

I came across the statement that a compact closed category $\mathcal{C}$ is equivalent to its dual category $\mathcal{C}^{\text{op}}$ (see e.g. this StackExchange post). This fact seems to be regarded ...
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### How these two definitions of category of hypergraph are same?

In the paper "Homotopy, homology, and persistent homology using closure spaces Peter Bubenik, Nikola Milićević" HypGph denote the category of hypergraphs and hypergraph homomorphisms. In the ...
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### what does "relation" mean in the Category of hypergraphs?

in the ncatlab https://ncatlab.org/nlab/show/hypergraph they defined the category of hypergraphs: SimpHGrph has objects consisting of a pair of sets (V,H) equipped with a relation R⊆V×H, and morphisms ...
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### How to identify "irreducible factors" in a monoidal or cartesian category

Is there "a categorical way" of speaking of objects in a monoidal category which can't be written as a tensor product? (We have to be careful with multiplication by the unit.) The same can ...
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### Comonoidal categories?

Monoidal categories are everywhere. They can be defined as pseudomonoids in the monoidal bicategory $(\mathsf{Cat},\times)$ of categories, functors and natural transformations. By turning directions ...
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