nLab monoid in a monoidal category

Idea

Generalizing the classical notion of monoid, one can define a monoid (or monoid object) in any monoidal category $(C,\otimes,I)$. Classical monoids are of course just monoids in Set with the cartesian product.

By the microcosm principle, in order to define monoid objects in $C$, $C$ itself must be a “categorified monoid” in some way. The natural requirement is that it be a monoidal category. In fact, it suffices if $C$ is a multicategory. Contrast this with a group object, which can only be defined in a cartesian monoidal category (or a cartesian multicategory).

Definition

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:

$$\array{ & (M \otimes M) \otimes M & \stackrel{\alpha}{\longrightarrow} & M \otimes (M \otimes M) & \stackrel{1 \otimes \mu}{\longrightarrow} & M \otimes M \\ & {}_{\mu \otimes 1}\searrow && && \swarrow_{\mu} & \\ && M \otimes M & \stackrel{\mu}{\longrightarrow} M && }$$

and the left and right unit laws:

$$\array{ & I \otimes M & \stackrel{\eta \otimes 1}{\longrightarrow} & M \otimes M & \stackrel{1 \otimes \eta}{\longleftarrow} & M \otimes I \\ & & {}_{\lambda}\searrow & {}_{\mu}\downarrow & \swarrow_{\rho} & \\ & & & M & & }$$

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

Morphism of monoids

The analogue of a monoid homomorphism, called a morphism of monoids, is a morphism, $\f: M \to M'$ between two monoid objects, satisfying the equations;

$f \circ \mu = \mu' \circ (f \otimes f)$

$f \circ \eta = \eta'$

corresponding to the commutative diagrams;

$$\array{ & M \otimes M & \stackrel{f \otimes f}{\longrightarrow} & M' \otimes M' \\ & {}_{\mu}\downarrow & & \downarrow_{\mu'} \\ & M & \stackrel{f}{\longrightarrow} & M' }$$

$$\array{ & I & \stackrel{\eta}{\longrightarrow} & M \\ & & {}_{\eta'}\searrow & \downarrow_{f} \\ & & & M' }$$

As categories with one object

Just as the category of regular monoids is equivalent to the category of locally small (i.e. Set-enriched) categories with one object, the category of monoids in $C$ (with the obvious morphisms) is equivalent to the category of $C$-enriched categories with one object.

Properties

Preservation by lax monoidal functors

Monoid structure is preserved by lax monoidal functors. Comonoid structure by oplax monoidal functors. See lax monoidal functor for more details.

Category of monoids

For special properties of categories of monoids, see category of monoids.

Examples

• 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 $k$.
• A monoid in a category of modules is an associative unital algebra.
• A monoid object in Top (with cartesian product as the tensor product) is a topological monoid.
• A monoid object in Ho(Top) is an H-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.
• The category of pointed sets has a monoidal structure given by the smash product. A monoid object in this monoidal category is an absorption monoid.
• Given any monoidal category $C$, a monoid in the monoidal category $C^{op}$ is called a comonoid in $C$.
• In a cocartesian monoidal category, every object is a monoid object in a unique way.
• 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.

Related concepts

• monoid
• commutative monoid in a symmetric monoidal category
• idempotent monoid in a monoidal category
• module over a monoid
• category of monoids
• monoid in a monoidal (infinity,1)-category

References

Original reference (including the case of a commutative monoid in a symmetric monoidal category):

• Jean Bénabou, Algèbre élémentaire dans les catégories (1964), C. R. Acad. Sci. Paris 258 (1964) pp.771-774, gallica

• Saunders MacLane, Section III.6 in: Categories for the Working Mathematician, Springer (1971)

• Francis Borceux, George Janelidze, Gregory Maxwell Kelly, p. 7 in: Internal object actions, Commentationes Mathematicae Universitatis Carolinae (2005) Volume: 46, Issue: 2, page 235-255 (dml:249553)

• Florian Marty, Dections 1.2, 1.3 in: Des Ouverts Zariski et des Morphismes Lisses en Géométrie Relative, Ph.D. Thesis, 2009 (web)

• Martin Brandenburg, Section 4.1 of: Tensor categorical foundations of algebraic geometry (arXiv:1410.1716)

See also MO/180673.

In cartesian monoidal categories:

• John Michael Boardman, Algebraic objects in categories, Chapter 7 of: Stable Operations in Generalized Cohomology (pdf) in: Ioan Mackenzie James (ed.) Handbook of Algebraic Topology Oxford 1995 (doi:10.1016/B978-0-444-81779-2.X5000-7)

and here formalized 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