nLab
monoid in a monoidal category

Contents

Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

Idea

Generalizing the classical notion of monoid, one can define a monoid (or monoid object) in any monoidal category (C,,I)(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 CC, CC itself must be a “categorified monoid” in some way. The natural requirement is that it be a monoidal category. In fact, it suffices if CC 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 CC is an object MM equipped with a multiplication μ:MMM\mu: M \otimes M \to M and a unit η:IM\eta: I \to M satisfying the associative law:

(MM)M α M(MM) 1μ MM μ1 μ MM μM \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:

IM η1 MM 1η MI λ μ ρ M \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 α\alpha is the associator in CC, while λ\lambda and ρ\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:MM\f: M \to M' between two monoid objects, satisfying the equations;

fμ=μ(ff)f \circ \mu = \mu' \circ (f \otimes f)

fη=ηf \circ \eta = \eta'

corresponding to the commutative diagrams;

MM ff MM μ μ M f M \array{ & M \otimes M & \stackrel{f \otimes f}{\longrightarrow} & M' \otimes M' \\ & {}_{\mu}\downarrow & & \downarrow_{\mu'} \\ & M & \stackrel{f}{\longrightarrow} & M' }
I η M η f 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 CC (with the obvious morphisms) is equivalent to the category of CC-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 there for more.

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 kk (with the usual tensor product of vector spaces) is an algebra over kk.
  • 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.
  • Given any monoidal category CC, a monoid in the monoidal category C opC^{op} is called a comonoid in CC.
  • In a cocartesian monoidal category, every object is a monoid object in a unique way.
  • For any category CC, the endofunctor category C CC^C has a monoidal structure induced by composition of endofunctors, and a monoid object in C CC^C is a monad on CC.

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

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

References

Categorical properties of monoid objects in monoidal categories are spelled out in sections 1.2 and 1.3 of

  • Florian Marty, Des Ouverts Zariski et des Morphismes Lisses en Géométrie Relative, Ph.D. Thesis, 2009, web

A summary is in section 4.1 of

See also MO/180673.

Last revised on September 5, 2019 at 05:21:01. See the history of this page for a list of all contributions to it.