# Tagged Questions

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/...

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### Monoidal structure on the opposite category of a symmetric monoidal category

I have two questions that confuse me a lot. First of all, can someone state the exact way in which, given a symmetric monoidal category $\mathcal{C}$, the opposite category $\mathcal{C}^{\text{op}}$ ...
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### Why not use the Cartesian product for the monoidal category of modules? Why use the tensor product?

At least for vector spaces, the Cartesian product is a direct sum (categorical product and coproduct) operation. Looking at the definition of monoidal category on Wikipedia, the Cartesian product/...
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### Is the tensor algebra functor a strong monoidal functor?

Let $K$ be a commutative ring. Is it true that $T(V \oplus W) \cong T(V) \otimes_K T(W)$ as $K$-algebras for any $K$-modules $V$ and $W$? The reason I ask is that I have heard it mentioned (I don't ...
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### Balanced Tensor Product of Module Categories

Let $C$ be a $k$-linear ($Vect_k$-enriched) monoidal category and consider the 2-category $Mod_{C}$ of $k$-linear $(C,C)$-bimodule categories in the sense of Ostrik (https://arxiv.org/abs/math/0111139....
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### Examples of asymmetrically braided monoid

From nCatlab https://ncatlab.org/nlab/show/braiding : Any braided monoidal category has a natural isomorphism $$B_{x,y} \;\colon\; x \otimes y \to y \otimes x$$ called the braiding. ...
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### How to call a “non-strict” monoidal category?

A monoidal category is a category $\mathsf{C}$ equipped with a bifunctor $\otimes : \mathsf{C} \times \mathsf{C} \to \mathsf{C}$, a unit object, an associator, and right and left unitors satisfying a ...
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### Enriching categories of simplicial objects

Let $C$ be a cocomplete category and $Simp(C)=C^{\triangle^{op}}$ the category of simplicial objects in $C$. I want to show that $Simp(C)$ is simplicially enriched but I don't understand how the ...
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### Epimorphisms and faithful functors in a rigid abelian tensor category

Let $\mathsf{C}$ be a rigid abelian tensor category in the sense of Deligne and Milne's notes (p.9). Let $\mathbf{1}$ denote the identity object in $\mathsf{C}$ with respect to $\otimes$. The abelian ...
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### Adjointness of internal contravariant Hom in symmetric monoidal categories.

Let consider a closed symmetric monoidal cateogry, $\mathscr C,\otimes$, with adjunction $(X\otimes-)\dashv\mathrm{Hom}(X,-)$ for all objects $X$. The following isomorphis, valid in categories such ...
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### Functor between Category and its Free Strict Monoidal Category

Let $C$ be a category and let $\sum(C)$ denote the free strict monoidal category over $C$. According to Wikipedia, the operation $\sum: C\rightarrow\sum(C)$ extends to a 2-monad on $Cat$. Can someone ...
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### Rings that cannot be representations rings

Given a monoidal category $\mathcal{C}$ one can define the Green ring (or representation ring) $r(\mathcal{C})$ as the abelian group generated by the isomorphism classes $[V]$ of $\mathcal{C}$ modulo ...
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### Extended Topological Quantum Field Theory, (ETQFT) basics ..

What is the functorial (categorical) definition of a 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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### When a monoidal category is equivalent to its center

The notion of the center of a monoidal category categorifies that of the center of a monoid. Similarly, the notion of a braided monoidal category is a categorification of that of a commutative monoid. ...
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### When is the tensor unit a generator?

Let $\mathcal{C}$ be a monoidal category with the tensor unit $I$. Then there is a "forgetful" functor from it to $Sets$: $$\mathrm{Hom}(I,-).$$ But in general, this ...
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### Free commutative ring functor

The free commutative ring on a set $X$ is the polynomial ring with variables the elements of $X$. This polynomial ring is the free (additive) abelian group on the free (multiplicative) abelian monoid ...
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### Understanding dual (noncommutative) modules / vector spaces functorially

For vector spaces over a fixed field $\Bbbk$, there's a natural transformation from the identity functor to the double dual functor. I think here's a way to construct it. Start from the identity arrow ...