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.

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 $1\cdot y=y$.

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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

   $$((W\otimes X)\otimes Y)\otimes Z(W\otimes (X\otimes Y))\otimes Z(W\otimes X)\otimes (Y\otimes Z)W\otimes ((X\otimes Y)\otimes Z)W\otimes (X\otimes (Y\otimes Z))$$

   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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