Monoidal_Categories.md

Monoidal Categories

Now things are getting interesting! We are ready to start exploring things in more depth. In case a definition is left outside of this it can be found on Abelian Categories or on Categories. These notes follow EGNO - Tensor Categories very closely.

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Definitions

As every proper set of notes in category theory we start with a set of definitions. The intuition behind monoidal categories is some kind of categorification of monoids. These objects are defined as.

Definition: A monoid is a set $C$ with an associative multiplication map $(x,y) \mapsto x\cdot y$ with an element $1 \in C$ such that $1^2 = 1$ and $x\mapsto 1\cdot x$ and $x\mapsto x\cdot 1$ are bijections $C\to C$.

Corollary: A monoid is a semi-group, and $1\cdot x = x\cdot 1 = x$.

Proof: Since $1\cdot 1 = 1$ we have that $$ 1 \cdot ( 1 \cdot x) = (1\cdot 1) \cdot x = 1\cdot x $$ now since $1\cdot x$ is a bijection let $y = 1\cdot x$ for some $x$. This implies $1 \cdot y = y$. $$ \begin{equation}\tag*{$\Box$}\end{equation} $$ This bijectivity property might seem an unnecessary complication in the definition of a monoid, but in reality it is a nice path to lead us to the correct abstract concept. Also recall, that equivalently a monoid is a category with a single object.

Definition: A category $\mathcal{C}$ is monoidal if there exists a functor $\otimes : \mathcal{C}\times \mathcal{C} \to \mathcal{C}$ called the tensor product with a natural isomorphism $\alpha:(\cdot \otimes \cdot)\otimes \cdot \to^\ast \cdot \otimes (\cdot \otimes \cdot)$ implementing associativity, and a unit object $1 \in \mathcal{C}$ be an object with an isomorphism $1\otimes 1 \to 1$ such that

1. (Pentagon Axiom) the following diagram commutes $$ \xymatrix{ & ((W\otimes X)\otimes Y) \otimes Z\ar[dl]{\alpha{W,X,Y}\otimes \text{Id}Z}\ar[dr]^{\alpha{W\otimes X,Y,Z}} &\ (W\otimes (X\otimes Y)) \otimes Z\ar[d]^{\alpha_{W,X\otimes Y, Z}} & & (W\otimes X)\otimes (Y \otimes Z)\ar[d]{\alpha{W,X,Y\otimes Z}}\ W\otimes ((X\otimes Y) \otimes Z)\ar[rr]^{\text{Id}W \otimes \alpha{X,Y,Z}} & & W\otimes (X\otimes (Y \otimes Z)) } $$ for all objects $W,X,Y,Z \in \mathcal{C}$ and with suitably chosen natural isomorphisms $\alpha$.

2. (Unit Axiom) The functors $X \mapsto 1\otimes X$ and $X \mapsto X\otimes 1$ are equivalences in $\mathcal{C}$.

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