Monoidal Categories

Monoidal Categories

Monoidal Category

A monoidal category is a tuple $(\mathsf{C},\otimes ,1,\alpha ,\lambda ,\rho )$ where

• $\mathsf{C}$ is a category;
• functor $\otimes :\mathsf{C}\times \mathsf{C}\to \mathsf{C}$ is called the monoidal product;
• $1\in \text{obj}\mathsf{C}$ is called the monoidal unit;
• $\alpha$ is a natural isomorphism called an associator: ${\alpha }_{X,Y,Z}:(X\otimes Y)\otimes Z\cong X\otimes (Y\otimes Z)$for all $\mathsf{C}$-objects $X,Y,Z$, called the associativity isomorphism;
• $\lambda$ and $\rho$ are also natural isomorphisms that $\lambda :1\otimes X\cong X,\rho :X\otimes 1\cong X$for all $\mathsf{C}$-objects $X$, called the left unit isomorphism and the right unit isomorphism respectively.

And the following axioms hold:

• Middle Unity Axiom:
  
• Pentagon Axiom:
  

Moreover, a strict monoidal category is a monoidal category in which the components of $\alpha$, $\lambda$ and $\rho$ are all identity morphisms.

Monoid Object

A monoid object in a monoidal category $(\mathsf{C},\otimes ,1,\alpha ,\lambda ,\rho )$ is an object $M$ with two morphisms

• multiplication $\mu :M\otimes M\to M$;
• unit $\eta :\text{one}\to M$,

such that the unit diagram and the pentagon diagram hold:

Dually, a comonoid object is a monoid object in the dual category ${\mathsf{C}}^{\text{op}}$.

e.g.

• It is called monoidal because the structure is “monoid-like”. Any monoid $(M,\cdot ,1)$ forms a small monoidal category with object set $M$, $\cdot$ as monoidal product and the identity of $M$ as its identity object.
• The category of abelian groups $(\mathsf{Ab},\otimes ,\mathbb{Z})$ is a monoidal category with the usual tensor product of abelian groups and the group of integers $\mathbb{Z}$ as the monoidal unit. Rings are monoid objects.
• The category of sets $(\mathsf{Set},\bigsqcup ,\varnothing )$ is a monoidal category. The monoidal product is the disjoint union of sets, and the monoidal unit is the empty set.
• The category sets can carry another monoidal category $(\mathsf{Set},\times ,\{\ast \})$, where the monoidal product is the Cartesian product of sets, and the monoidal unit is the singleton set $\{\ast \}$. Monoids are monoid objects; Every set $X$ has a unique comonoid structure given by the diagonal map $X\to X\times X$ and the unique map $X\to \{\ast \}$.
• The category of vector spaces over a field $\mathbb{K}$ $({\mathsf{Vect}}_{\text{K}},{\otimes }_{\text{K}},\text{K})$ is a monoidal category with the usual tensor product of vector spaces and the field $\text{K}$ as the monoidal unit. Algebras are monoid objects; coalgebras are comonoid objects.
  

Monoidal Functor

A monoidal functor between monoidal categories $(\mathsf{C},\otimes ,1,\alpha ,\lambda ,\rho )$ and $(\mathsf{D},{\otimes }^{\prime },1^{\prime },{\alpha }^{\prime },{\lambda }^{\prime },{\rho }^{\prime })$ is a functor $F:\mathsf{C}\to \mathsf{D}$ together with a morphism $F_0:1^{\prime }\to F(1)$ in $\mathsf{D}$ and a natural isomorphism $F_2:F(-){\otimes }^{\prime }F(-)\Rightarrow F(-\otimes -)$ such that the following diagrams commute for all objects $X,Y,Z$ in $\mathsf{C}$:

Braided Monoidal Category

Braided Monoidal Category

A braided monoidal category is a monoidal category with an additional structure, a braiding, which is a natural isomorphism ${\tau }_{A,B}:A\otimes B\to B\otimes A$ that is compatible with the associativity and unit constraints. That is, the following hexagon diagrams commute for all objects $X,Y,Z$:

A symmetric monoidal category is a braided monoidal category with the braiding satisfying the symmetry axiom: ${\tau }_{A,B}={\tau }_{B,A}^{-1}$.

Braided Commutative Monoid Object

A monoid object $(M,\mu ,\eta )$ in a braided monoidal category is braided commutative if the following diagram commutes:

e.g.

Category	Universal Property	Categorical Equivalence
$(\mathsf{\Delta },+,0)$	Free monoidal category on a single monoid object $1$	$\mathsf{MonFunc}(\mathsf{\Delta },\mathsf{C})\cong \mathsf{Mon}(\mathsf{C})$
$(\mathsf{FinCard},+,0)$	Free symmetric monoidal category on on a single commutative monoid $1$	$\mathsf{SymMonFunc}(\mathsf{FinCard},\mathsf{C})\cong \mathsf{cMon}(\mathsf{C})$
$(\mathsf{PlanGraph},\bigsqcup ,0)$	Free monoidal category on a single Frobenius object $1$	$\mathsf{MonFunc}(\mathsf{PlanGraph},\mathsf{C})\cong \mathsf{Frob}(\mathsf{C})$
$(\mathsf{Braid},+,0)$	Free braided monoidal category on a single braided monoid	$\mathsf{BrMonFunc}(\mathsf{Braid},\mathsf{C})\cong \mathsf{C}$
Skeleton of $(2Cob,\bigsqcup ,\varnothing )$	Free symmetric monoidal category on a single commutative Frobenius object $S^1$	$\mathsf{SymMonFunc}(2Cob,\mathsf{C})\cong \mathsf{cFrob}(\mathsf{C})$

The Category of Monoid Objects

Monoid Homomorphism

Proposition

If

Enriched Categories

Enriched Category

We call a category $\mathsf{C}$ a $\mathsf{V}$-category, or a category enriched in $\mathsf{V}$, where $\mathsf{V}=(\mathbf{V},\otimes ,1,\alpha ,\lambda ,\rho )$ is a monoidal category, if

• For each pair of objects $X,Y$ in $\mathsf{C}$, there is an object $\mathsf{C}(X,Y)$ in $\mathsf{V}$, called the hom object with domain $X$ and codomain $Y$. Every morphism $f:X\to Y$ in $\mathsf{C}$ can be uniquely described by $\tilde{f}:1\to \mathsf{C}(X,Y)$ in $\mathsf{V}$.
• For each triple of objects $X,Y,Z$ in $\mathsf{C}$, there is a morphism $m_{X,Y,Z}:\mathsf{C}(Y,Z)\otimes \mathsf{C}(X,Y)\to \mathsf{C}(X,Z)$ in $\mathsf{V}$, called the composition;
• For each object $X$ in $\mathsf{C}$, there is a morphism $i_X:1\to \mathsf{C}(X,X)$ in $\mathsf{V}$, called the identity of $X$.

And they make the associativity diagram

and the unity diagram

commutes for all objects $W,X,Y,Z$ in $\mathsf{C}$.

Preadditive Category

A preadditive category is a category enriched over the monoidal category of abelian groups.