nLab monoid in a monoidal category

Contents

Context

Categorical algebra

Monoidal categories

Monoid theory

Idea
Definition
Morphism of monoids
As categories with one object
Properties

Preservation by lax monoidal functors
Category of monoids

Examples
Related concepts
References

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

Example

A monoid in a monoidal category of modules $R Mod$ (over any ground ring $R$ and equipped with the tensor product of modules) is an associative unital algebra over $R$ .

Example

As the special case of Exp. for $R = \mathbb{Z}$ the integers:

A monoid object in the monoidal category Ab of abelian groups with the tensor product of abelian groups, is a ring.

Example

As the special case of Exp. for $R = k$ a field:

A monoid object in the category Vect of vector spaces (over any ground field $k$ ) with the tensor product of vector spaces is an associative unital algebra over $k$ .

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

(co)monad name	underlying endofunctor	(co)monad structure induced by
reader monad	$W \to (\text{-})$ on cartesian types	unique comonoid structure on $W$
coreader comonad	$W \times (\text{-})$ on cartesian types	unique comonoid structure on $W$
writer monad	$A \otimes (\text{-})$ on monoidal types	chosen monoid structure on $A$
cowriter comonad	$\array{A \to (\text{-}) \\ \\ A \otimes (\text{-})}$ on monoidal types	chosen monoid structure on $A$ chosen comonoid structure on $A$
Frobenius (co)writer	$\array{A \to (\text{-}) \\ A \otimes (\text{-})}$ on monoidal types	chosen Frobenius monoid structure

References

Original references (including the case of a commutative monoids in a symmetric monoidal category, but see there for more):

Jean Bénabou, Algèbre élémentaire dans les catégories (1964), C. R. Acad. Sci. Paris 258 (1964) pp.771-774, gallica
Saunders MacLane, Sections III.6 & VII.3 in: Categories for the Working Mathematician, Springer (1979, 2nd ed.) [doi:10.1007/978-1-4757-4721-8]
Hans-Joachim Baues, Mamuka Jibladze, Andy Tonks, Cohomology of monoids in monoidal categories, in: Operads: Proceedings of Renaissance Conferences, Contemporary Mathematics 202, AMS (1997) 137-166 [doi:10.1090/conm/202, preprint:pdf]
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)

Discussion for commutative monoids in a symmetric monoidal category including proof that/when the category of module objects is itself closed symmetric monoidal:

Mark Hovey, Brooke Shipley, Jeff Smith, pp. 15 in: Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149-208 [arXiv:math/9801077, doi:10.1090/S0894-0347-99-00320-3]