nLab monoid

Definitions

With a binary operation and an element

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

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

and the left and right unit laws:

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

In a monoidal category

See monoid in a monoidal category.

In terms of 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 $\mathbf{B}M$ for the corresponding category with single object $\bullet$ and with $M$ as its hom-set: the delooping of $M$, so that $M = Hom_{\mathbf{B}M}(\bullet, \bullet)$. 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. $\mathbf{Set}$-enriched) case.

For more on this see also group.

As a strict monoidal category

An alternate way to view a monoid as a category is as a discrete strict monoidal category $\mathbf{C}$ where the elements of the monoid are the objects of $\mathbf{C}$, the binary operation of the monoid provides the tensor product bifunctor, and the identity of the monoid is the unit object. Preordered monoids then yield (non-discrete) strict monoidal categories with the morphisms witnessing the preorder in the usual way.

$\mathcal{O}$-Monoids over an $(\infty,1)$-Operad

The notion of associative monoids discussed above are controled by the associative operad. More generally in higher algebra, for $\mathcal{O}$ any operad or (infinity,1)-operad, one can consider $\mathcal{O}$-monoids. (Lurie, def. 2.4.2.1)

These are closely related to (infinity,1)-algebras over an (infinity,1)-operad with respect to $\mathcal{O}$ (Lurie, prop. 2.4.2.5).

 Properties

Finite products (and sums)

Let $M$ be a monoid, and let $M^*$ be the free monoid on $M$ with canonical function $h:M \to M^*$ taking the elements of $M$ to the generators in $M^*$. The finite product operation on $M$ is a monoid homomorphism

$$\prod_{i = 0}^{\mathrm{len}(-) - 1}(-)(i):M^* \to M$$

from $M^*$ to $M$, where:

$$\prod_{i = 0}^{\mathrm{len}(\epsilon) - 1} \epsilon(i) = 1$$

$$\prod_{i = 0}^{\mathrm{len}(h(a)) - 1} (h(a))(i) = a$$

$$\left(\prod_{i = 0}^{\mathrm{len}(a) - 1} a(i)\right) \cdot \left(\prod_{i = 0}^{\mathrm{len}(b) - 1} b(i)\right) = \prod_{i = 0}^{\mathrm{len}(a b) - 1} (a b)(i)$$

If $M$ is written additively $(+, 0)$ instead of multiplicatively $(\cdot, 1)$, the operation is called finite sum, and is defined as

$$\sum_{i = 0}^{\mathrm{len}(-) - 1}(-)(i):M^* \to M$$

from $M^*$ to $M$, where:

$$\sum_{i = 0}^{\mathrm{len}(\epsilon) - 1} \epsilon(i) = 0$$

$$\sum_{i = 0}^{\mathrm{len}(h(a)) - 1} (h(a))(i) = a$$

$$\left(\sum_{i = 0}^{\mathrm{len}(a) - 1} a(i)\right) + \left(\sum_{i = 0}^{\mathrm{len}(b) - 1} b(i)\right) = \sum_{i = 0}^{\mathrm{len}(a b) - 1} (a b)(i)$$

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.)
• The set of endomorphisms of a given object in a category has a canonical monoid structure given by composition.

Remarks on notation

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

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

(where $\mathbf{E}G = G//G$ 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 $\mathbf{B}G \to Vect$, the latter functors $G \to Vect$. Both these are very important.

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

$$A \to 2Vect$$

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

and

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

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 $\mathbf{B}A$. We have notation to distinguish this, and we should use it.

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

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

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

$$\array{ V \\ \downarrow \\ V//G \\ \downarrow \\ \mathbf{B}G } \,,$$

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

$$\array{ V//G &\to& Vect_* \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\rho}{\to}& Vect } \,,$$

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 $\mathbf{B}G$ lives.

Related concepts

• monoid in a monoidal category, category of monoids

• monoidal symmetric proset

• monoid, internal monoid/monoid object,

  • commutative monoid, cancellative monoid

  • monoidal groupoid, braided monoidal groupoid, symmetric monoidal groupoid

  • k-tuply monoidal n-groupoid

  • monoid object in a (∞,1)-category

• topological monoid

• comonoid, cocommutative comonoid

• group, group object

• ring, ring object

• categorical algebra

• A3-type

algebraic structure	oidification
truth value	preorder
magma	magmoid
pointed magma with an endofunction	setoid/Bishop set
unital magma	unital magmoid
quasigroup	quasigroupoid
loop	loopoid
semigroup	semicategory
monoid	category
anti-involutive monoid	dagger category
associative quasigroup	associative quasigroupoid
group	groupoid
flexible magma	flexible magmoid
alternative magma	alternative magmoid
absorption monoid	absorption category
cancellative monoid	cancellative category
rig	CMon-enriched category
nonunital ring	Ab-enriched semicategory
nonassociative ring	Ab-enriched unital magmoid
ring	ringoid
nonassociative algebra	linear magmoid
nonassociative unital algebra	unital linear magmoid
nonunital algebra	linear semicategory
associative unital algebra	linear category
C-star algebra	C-star category
differential algebra	differential algebroid
flexible algebra	flexible linear magmoid
alternative algebra	alternative linear magmoid
Lie algebra	Lie algebroid
monoidal poset	2-poset
strict monoidal groupoid?	strict (2,1)-category
strict 2-group	strict 2-groupoid
strict monoidal category	strict 2-category
monoidal groupoid	(2,1)-category
2-group	2-groupoid/bigroupoid
monoidal category	2-category/bicategory

References

Exposition of basics of monoidal categories and categorical algebra:

• geometry of physics – categories and toposes, Section 2: Basic notions of categorical algebra

Properties of monoids expressed through properties of their toposes of presheaves:

• Jens Hemelaer, Morgan Rogers, Monoid Properties as Invariants of Toposes of Monoid Actions, arXiv:2004.10513.

Formalization of monoid objects as mathematical structures in proof assistants:

in a context of plain Agda:

• Martín Escardó, The Types of Magmas and Monoids, §4 in: Introduction to Univalent Foundations of Mathematics with Agda [arXiv:1911.00580, webpage]

in a context of cubical Agda:

• 1lab: Algebra.Monoid