# 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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### Monoidal structure preserving order

I am working with a category $\mathcal{C}$ in which the hom-sets are orders. Alternatively, we could look at it as a bicategory in which hom-sets for 2-cells are thin. Is there a notion of monoidal ...
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### What is a module over an algebra in the category theoretic sense?

Can someone please confirm this for me. In category theory one defines monoidal categories. One can then consider monoids in a monoidal category $C$. Given such a monoid $M$, one can then consider ...
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### Monoidal categories: strictness from coherence

A version of Mac Lane's Coherence Theorem states that every formal diagram (i.e., a diagram that involves only the associativity isomorphism, the unit isomorphisms, their inverses, identity morphisms, ...
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### What is the isomorphism from the identity functor to the double dual in a symmetric monoidal $(\infty,1)$-category?

I'm following example 2.4.12 in Lurie's note on the classification of TFTs. Let $\mathcal{C}$ be a symmetric monoidal $(\infty,1)$-category that has duals. The action of $O(1)$ on the largest $\infty$-...
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### The Eckmann-Hilton argument gives us an isomorphism between $Mon(Mon(C))$ and $CoMon(C)$ or just an equivalence?

I'm studying symmetric monoidal categories and I have seen some authors saying that, due to the Eckmann-Hilton argument, given some symmetric category $C$, the category $Mon(Mon(C))$ of monoidal ...
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### Completeness of the category of enriched categories

In the nLab entry on strict n-categories, one reads: For $V$ any complete and cocomplete closed monoidal category, also $VCat$ (the category of V-enriched categories) has these same properties. Is ...
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### What is the name of this identity relating the monoidal products of finite sets and finite-dimensional vector spaces?

Note: This question is almost certainly a duplicate. Since I don't know the terminology involved, I couldn't find the original question. If someone can find the original question and link to it, then ...
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### Hopf monoids in different categories

Hopf algebras are precisely Hopf monoids in the category of vector spaces. What are Hopf monoids in other common (to be interpreted by the reader) monoidal categories? In particular, are Hopf ...
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### Does this “leaky isomorphism” concept have a name?

I've been thinking about catalysis in chemical reaction networks while learning category theory at the same time, and it's given me a weird idea, which I'm asking about out of curiosity. Suppose I ...
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### Monoidal functor and the units II

In their book Tensor Categories Etingof, Gelaki, Nikshych and Ostrik give a different definition of a (strong) monoidal functor. The difference is that they do not set the isomorphism $F(1) \cong 1$ ...
In their book Tensor Categories Etingof, Gelaki, Nikshych and Ostrik define a monoidal functor between monoidal categories $(C,\otimes,1,\alpha,r,s)$ and $(C',\otimes', 1',\alpha',r',s')$ as the ...