# Monoidal Categories and Monoids

A monoid is a concept in mathematics similar to that of a group (see Groups), except that every element need not have an inverse. Therefore, a monoid is a set, equipped with a law of composition which is associative, and an identity element. An example of a monoid is the natural numbers (including zero) with the law of composition given by addition.

In this post, we will introduce certain concepts in category theory (see Category Theory) that are abstractions of the classical idea of a monoid.

A monoidal category is given by a category $\mathbf{C}$, a bifunctor $\Box: \mathbf{C}\times\mathbf{C}\rightarrow\mathbf{C}$, an object $I$ of $\mathbf{C}$, and three natural isomorphisms $\alpha$ (also known as the associator), $\lambda$ (also known as the left unitor), and $\rho$ (also known as the right unitor), with components

$\displaystyle \alpha_{A,B,C}:A\Box (B\Box C)\cong (A\Box B)\Box C$

$\displaystyle \lambda_{A}:I\Box A\cong A$

$\displaystyle \rho_{A}:A\Box I\cong A$

satisfying the conditions

$\displaystyle 1_{A}\Box\alpha_{A,B,C}\circ\alpha_{A,B\Box C,D}\circ\alpha_{A,B,C}\Box 1_{D}=\alpha_{A,B,C\Box D}\circ\alpha_{A\Box B,C,D}$

for any four objects $A$, $B$, $C$, and $D$ of $\mathbf{C}$, and

$\displaystyle \alpha_{A,I,B}\circ 1_{A}\Box \lambda_{B}=\rho_{A}\Box 1_{B}$

for any two objects $A$ and $B$ in $\mathbf{C}$.

The following “commutative diagrams” courtesy of user IkamusumeFan of Wikipedia may help express these conditions better (the symbol $\otimes$ is used here instead of $\Box$ to denote the bifunctor; this is very common notation, but we use $\Box$ following the book Categories for the Working Mathematician by Saunders Mac Lane in order to differentiate it from the tensor product, which is just one specific example of the bifunctor in question; I hope this will not cause any confusion):

If the natural isomorphisms $\alpha$, $\lambda$, and $\rho$ are identities, then we have a strict monoidal category.

A monoid object, or monoid in a monoidal category $(\mathbf{C},\Box,I)$ is an object $M$ of $\mathbf{C}$ together with two morphisms $\mu:M\Box M\rightarrow M$ and $\eta:I\rightarrow M$ satisfying the conditions

$\displaystyle \mu\circ 1\Box\mu\circ\alpha=\mu\circ\mu\Box 1$

$\displaystyle \mu\circ \eta\Box 1=\lambda$

$\displaystyle \mu\circ 1\Box\eta=\rho$

Again we can use the following commutative diagrams made by User IkamusumeFan of Wikipedia to help express these conditions:

As examples of monoidal categories, we have the following:

$\displaystyle (\mathbf{Set},\times,1)$

$\displaystyle (\mathbf{Ab},\otimes,\mathbb{Z})$

$\displaystyle (K\mathbf{-Mod},\otimes_{K},K)$

$(\mathbf{Cat},\times,\mathbf{1})$

$(\mathbf{C}^{\mathbf{C}},\circ,\text{Id})$    ($\mathbf{C}^{\mathbf{C}}$ denotes the category of functors from $\mathbf{C}$ to itself)

The monoids in these monoidal categories are given respectively by the following:

Ordinary monoids

Rings

$K$-algebras

Strict monoidal categories

Among the important kinds of monoidal categories with extra structure are braided monoidal categories and symmetric monoidal categories. A braided monoidal category $\mathbf{C}$ is a monoidal category equipped with a natural isomorphism $\gamma$ (also known as a commutativity constraint) with components $\gamma_{A,B}:A\Box B\cong B\Box A$ satisfying the following coherence conditions

$\displaystyle \alpha_{B,C,A}\circ\gamma_{A,B\Box C}\circ\alpha_{A,B,C}=1_{B}\Box\gamma_{A,C}\circ\alpha_{B,A,C}\circ \gamma_{A,B}\Box 1_{C}$

$\displaystyle \alpha_{C,A,B}^{-1}\circ\gamma_{A\Box B,C}\circ\alpha_{A,B,C}^{-1}=\gamma_{A,C}\Box 1_{B}\circ\alpha_{A,C,B}^{-1}\circ 1\Box\gamma_{A}\gamma_{A,B}$

which can be expressed in the following commutative diagrams (once again credit goes to User IkamusumeFan of Wikipedia):

The category $\mathbf{C}$ is a symmetric monoidal category if the isomorphisms $\gamma_{A,B}$ satisfy the condition $\gamma_{B,A}\circ\gamma_{A,B}=1_{A\Box B}$. We have already encountered an example of this category in The Theory of Motives in the form of tensor categories, defined as a symmetric monoidal categories whose Hom-sets (the sets of morphisms from a fixed object $A$ to another object $B$) form a vector space (the term “tensor category” is sometimes used to refer to other concepts in mathematics though, including symmetric monoidal categories themselves).

Another important kind of monoidal category is a closed monoidal category. A closed monoidal category is a monoidal category where the functor $-\Box B$ has a right adjoint (see Adjoint Functors and Monads) also known as the “internal Hom functor”, which is like a Hom functor that takes values in the category itself instead of in sets, and is denoted by $(\ )^{B}$. We have already seen an example of a closed monoidal category in Adjoint Functors and Monads, given by the category of $R$-modules for a fixed commutative ring $R$. There $A^{B}$ was given by $\text{Hom}(A,B)$ (this is the set of $R$-linear transformations from $A$ to $B$, which itself is an $R$-module).

We see therefore that the concepts of monoidal categories and monoids can be found everywhere in mathematics. Studying these structures are not only interesting for their own sake, but can also help us find or construct other useful new concepts in mathematics.

References:

Monoidal Category on Wikipedia

Monoid on Wikipedia

Braided Monoidal Category on Wikipedia

Symmetric Monoidal Category on Wikipedia

Closed Monoidal Category on Wikipedia

Image by User IkamusumeFan of Wikipedia

Image by User IkamusumeFan of Wikipedia

Image by User IkamusumeFan of Wikipedia

Image by User IkamusumeFan of Wikipedia

Image by User IkamusumeFan of Wikipedia

Image by User IkamusumeFan of Wikipedia

Categories for the Working Mathematician by Saunders Mac Lane