nLab
monoid

Definition

Classical

Classically, a monoid is a set $M$ equipped with a binary operation $\mu :M\times M\to M$ and special element $1\in M$ such that $1$ and $x\cdot y=\mu (x,y)$ satisfy the usual axioms of an associative product with unit, namely the associative law:

$$(x\cdot y)\cdot z=x\cdot (y\cdot z)$$(x \cdot y) \cdot z = x \cdot (y \cdot z)

and the left and right unit laws:

$$1\cdot x=x=x\cdot 1.$$1 \cdot x = x = x \cdot 1 .

In a monoidal category

More generally, we can define a monoid (sometimes called monoid object) in any monoidal category $(C,\otimes ,I)$. Namely, a monoid in $C$ is an object $M$ equipped with a multiplication $\mu :M\otimes M\to M$ and a unit $\eta :I\to M$ satisfying the associative law:

and the left and right unit laws:

Here $\alpha$ is the associator? in $C$, while $\lambda$ and $\rho$ are the left and right unitors?.

Classical monoids are of course just monoids in $(\mathrm{Set},\times \mathrm{,1})$.

In string diagrams

The data of a monoid may be written in string diagrams as:

Thanks to the distinctive shapes, one can usually omit the labels:

The axioms $\mu \cdot (\eta \otimes M)=1_M=\mu \cdot (M\otimes \eta )$ and $\mu \cdot (M\otimes \mu )=\mu \cdot (\mu \otimes M)$ then appear as:

As a one-object category

Equivalently, and more efficiently, we may say that a (classical) monoid is the hom-set of a category with a single object, equipped with the structure of its unit element and composition.

More tersely, one may say that a monoid is a category with a single object, or more precisely (to get the proper morphisms and $2$-morphisms) a pointed category with a single object. But taking this too literally may create conflicts in notation. To avoid this, for a given monoid $M$, we write $BM$ for the corresponding category with single object $•$ and with $M$ as its hom-set: the delooping of $M$, so that $M={\mathrm{Hom}}_{BM}(•,•)$. This realizes every monoid as a monoid of endomorphisms.

Similarly, a monoid in $(C,\otimes ,I)$ may be defined as the hom-object of a $C$-enriched category with a single object, equipped with its composition and identity-assigning morphisms; and so on, as in the classical (i.e. $\mathrm{Set}$-enriched) case.

For more on this see also group.

Examples

• A monoid in which every element has an inverse is a group. For that reason monoids are often known (especially outside category theory) as semi-groups. (But this term is often extended to monoids without identities, that is to sets equipped with any associative operation.)
• A monoid object in Ab (with the usual tensor product of $\mathbb{Z}$-modules as the tensor product) is a ring. A monoid object in the category of vector spaces over a field $k$ (with the usual tensor product of vector spaces) is an algebra over $k$.
• For a commutative ring $R$, a monoid object in the category of $R$-modules (with its usual tensor product) is an $R$-algebra.
• A monoid object in Top (with cartesian product as the tensor product) is a topological monoid.
• A monoid object in the category of monoids (with cartesian product as the tensor product) is a commutative monoid. This is a version of the Eckmann-Hilton argument.
• A monoid object in the category of complete join-semilattices (with its tensor product that represents maps preserving joins in each variable separately) is a unital quantale.
• Given any monoidal category $C$, a monoid in the monoidal category $C^{\mathrm{op}}$ is called a comonoid in $C$.
• For any category $C$, the endofunctor category $C^C$ has a monoidal structure induced by composition of endofunctors, and a monoid object in $C^C$ is a monad on $C$.

These are examples of monoids internal to monoidal categories. More generally, given any bicategory $B$ and a chosen object $a$, the hom-category $B(a,a)$ has the structure of a monoidal category. So, the concept of monoid makes sense in any bicategory $B$: we define a monoid in $B$ to be a monoid in $B(a,a)$ for some object $a\in B$. This often called a monad in $B$. The reason is that a monad in Cat is the same as monad on a category.

A monoid in a bicategory $B$ may also be described as the hom-object of a $B$-enriched category with a single object.

Remarks on notation

(originally from category algebra)

It can be important to distinguish between a $k$-tuply monoidal structure and the corresponding $k$-tuply degenerate category, even though there is a map identifying them. The issue appears here for instance when discussing the universal $G$-bundle in its groupoid incarnation. This is

$$G\to EG\to BG$$G \to \mathbf{E}G \to \mathbf{B}G

(where $EG=G//G$ is the action groupoid of $G$ acting on itself). On the left we crucially have $G$ as a monoidal 0-category, on the right as a once-degenerate 1-category. Without this notation we cannot even write down the universal $G$-bundle!

Or take the important difference between group representations and group 2-algebras, the former being functors $BG\to \mathrm{Vect}$, the latter functors $G\to \mathrm{Vect}$. Both these are very important.

Or take an abelian group $A$ and a codomain like $2\mathrm{Vect}$. Then there are 3 different things we can sensibly consider, namely 2-functors

$$A\to 2\mathrm{Vect}$$A \to 2Vect

$$BA\to 2\mathrm{Vect}$$\mathbf{B}A \to 2Vect

and

$$B^{2}A\to 2\mathrm{Vect}.$$\mathbf{B}^2A \to 2Vect \,.

All of these concepts are different, and useful. The first one is an object in the group 3-algebra of $A$. The second is a pseudo-representation of the group $A$. The third is a representations of the 2-group $BA$. We have notation to distinguish this, and we should use it.

Finally, writing $BG$ for the 1-object $n$-groupoid version of an $n$-monoid $G$ makes notation behave nicely with respect to nerves, because then realization bars $\mid \cdot \mid$ simply commute with the $B$s in the game: $\mid BG\mid =B\mid G\mid$.

This behavior under nerves shows also that, generally, writing $BG$ gives the right intuition for what an expression means. For instance, what’s the “geometric” reason that a group representation is an arrow $\rho :BG\to \mathrm{Vect}$? It’s because this is, literally, equivalently thought of as the corresponding classifying map of the vector bundle on $BG$ which is $\rho$-associated to the universal $G$-bundle:

the $\rho$-associated vector bundle to the universal $G$-bundle is, in its groupoid incarnations,

$$\begin{array}{c}V\\ \downarrow \\ V//G\\ \downarrow \\ BG\end{array},$$\array{ V \\ \downarrow \\ V//G \\ \downarrow \\ \mathbf{B}G } \,,

where $V$ is the vector space that $\rho$ is representing on, and this is classified by the representation $\rho :BG\to \mathrm{Vect}$ in that this is the pullback of the universal $\mathrm{Vect}$-bundle

$$\begin{array}{ccc}V//G& \to & {\mathrm{Vect}}_{*}\\ \downarrow & & \downarrow \\ BG& \stackrel{\rho }{\to }& \mathrm{Vect}\end{array},$$\array{ V//G &\to& Vect_* \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\rho}{\to}& Vect } \,,

In summary, it is important to make people understand that groups can be identified with one-object groupoids. But next it is important to make clear that not everything that can be identified should be, for instance concerning the crucial difference between the category in which $G$ lives and the 2-category in which $BG$ lives.