# Questions tagged [monoidal-categories]

In mathematics, a monoidal category (or tensor category) is a category C equipped with a bifunctor ⊗ : C × C → C which is associative up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism. (Def: http://en.m.wikipedia.org/wiki/Monoidal_category)

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### decomposition of hom-functors in a self-enriched category

Let $\mathbb{C}$ be a self-enriched category (such as Set). The Functor $\mathbb{C}(X, \mathbb{C}(Y,\_))$ is the same than the composition of functors $\mathbb{C}(X,\_) \circ \mathbb{C}(Y,\_)$. In a ...
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### Monoidal categories in which some tensor products of morphisms are equal

I have come across a certain monoidal category $(C,\otimes,I)$ (let us say it is strictly monoidal to simplify the notations, as usual it does not matter very much) which satisfies the following ...
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### Extended Topological Quantum Field Theory (ETQFT) by Jacob Lurie

What is the functorial (categorical) definition of TQFT (Topological Quantum Field Theory), which Jacob Lurie "had extended", for his ETQFT ? Actually I just need to know what are basic tools, to ...
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### Does an adjoint of an internal Hom functor of a prounital closed category define a tensor product?

A closed category is a category equipped with internal Hom functors along with a unit object. Now this answer shows that if $C$ is a closed category whose internal Hom functor has a left adjoint, ...
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### Is “monoidal category enriched over itself” the same as “closed monoidal category”?

If $M$ is a monoidal category, an enriched category over $M$ is a category $C$ whose hom-sets are viewed as objects in $M$. And a monoidal category $M$ is said to be closed if the tensor product ...
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### Does an adjoint of the Hom functor make a category monoidal?

In the category of modules, the tensor product functor is the left adjoint of the covariant Hom functor. Similarly in the category of sets, the Cartesian product functor is the left adjoint of the ...
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### Can I construct a monoidal category on locally convex modules over algebras with approximate identiy

I am faced with the following problem: Let $A$ be a complete locally convex algebra with a uniform approximation of identity, that is a net $e_\lambda$, such that $p(e_\lambda a-a)\rightarrow 0$ for ...
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### Symmetric monoidal category which is not closed?

A monoidal category is symmetric if its tensor product is commutative up to natural isomorphism. And a symmetric monoidal category is closed if the tensor product functor has a right adjoint. We ...
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### question about monoidal structure of a 2-category

Consider an extension of the 1-category of vector spaces and linear maps down to a 2-category $\mathcal{C}$ whose objects are $k$-linear categories. What is the symmetric monoidal structure on the ...
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### Enriched category over non-monoidal category

I'm finally learning homological algebra, and the notion of an enriched category seems to be the right setting in which to define the $\mathbf{Ext}$ functor. Yet, the definitions of an enriched ...
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### Definition of unitor in monoidal category

From https://ncatlab.org/nlab/show/monoidal+category, a monoidal category requires a natural isomorphism a natural isomorphism $\lambda: (1 \otimes (-)) \rightarrow ^\cong (-)$ with components ...
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### For symmetric monoidal categories, the monoidal product is a monoidal functor

Has anyone got a reference for the following fact? If $\mathcal X$ is a symmetric monoidal category, then $\_\otimes\_\colon\mathcal X\times\mathcal X \to \mathcal X$ is a strong monoidal functor. ...
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### A monoidal category that preserves subobjects

Let $X$, $Y$ be objects in a monoidal category $\mathcal{C}$, s.t. the functors $X \otimes \_$ and $\_\otimes Y$ preserve monomorphisms. Moreover, let $A \hookrightarrow X$, $B \hookrightarrow Y$ be ...
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### Examples of weak monoidal Quillen equivalences

Schwede-Shipley introduced the notion of weak monoidal Quillen equivalences between monoidal model categories in "Equivalences of monoidal model categories". Are there any examples of such Quillen ...
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### Notation: why are categories enriched over $\mathcal V$?

In most references about enriched categories, $(\mathcal V, \otimes)$ is supposed to be a monoidal category and then $\mathcal V$-enriched categories are defined. Why is the letter $\mathcal V$ used ...
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### Every under category with pushouts is monoidal

Maybe a stupid question, but I can't figure it out. Suppose we have some category $C$ where all pushouts exist, $c\in Ob C$ and $c\downarrow C$ an under category (i.e. a category with objects of the ...
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### Equality of maps in a monoidal category

Let $C_1,C_2$ be monoidal categories (aka tensor categories) with tensor bifunctor $$\otimes_i: C_i\times C_i\to C_i$$ and tensor units $1_i$. Assume I have a monoidal functor $F:C_1\to C_2$, it ...
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### Commuting diagram for (monoidal) functors

Assume we have two categories $\cal A,B$ and four functors $F,G:\cal A\to B$, $D:\cal A\to A$, $E:\cal B\to B$. In order to show an equality like $$E\circ F=G\circ D\quad (1)$$ I have to prove that ...
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### $(\mathrm{Vect}_k,\otimes_k)$ as a non-strict monoidal category

What is the easiest way to see (or understand) that the category of vector spaces over a field $k$, endowed with its usual modoidal structure $\otimes_k$, is not a strict monoidal category.
I am quite confused on page 22,23 of the unit axiom of a monoidal category. Unit axiom: $L_1:X \mapsto 1 \otimes X$ and $R_1:X \mapsto X \otimes 1$ are autoequivalences of $C$. (end of pg 22)...