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) μ MM μ μ MM μM \array{ & (M \otimes M) \otimes M & \stackrel{\alpha}{\longrightarrow} & M \otimes (M \otimes M) & \stackrel{\mu}{\longrightarrow} & M \otimes M \\ & {}_{\mu}\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.

Examples

A monoid in a category of modules is an associative unital algebra. A monoid in a category of endofunctors where tensor product is defined by composition, is a monad.

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 March 11, 2019 at 02:39:02. See the history of this page for a list of all contributions to it.