# nLab idempotent monoid in a monoidal category

Contents

### Context

#### Categorical algebra

internalization and categorical algebra

universal algebra

categorical semantics

#### Monoidal categories

monoidal categories

With braiding

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

#### Monoid theory

monoid theory in algebra:

# Contents

## Idea

An idempotent monoid is a monoid which is idempotent in that it “squares to itself” in the evident category-theoretic sense.

## Definition

An idempotent monoid $(A,\mu,\eta)$ in a monoidal category $\mathcal{C}$ is a monoid in $\mathcal{C}$ whose multiplication morphism $\mu\colon A\otimes_{\mathcal{C}}A\to A$ is an isomorphism.

Similarly, an idempotent semigroup in $\mathcal{C}$ (also called a non-unital idempotent monoid in $\mathcal{C}$) is a semigroup $(A,\mu)$ in $\mathcal{C}$ with $\mu$ an isomorphism.

Note. This is different from the usual algebraic notion of an idempotent monoid (namely, one in which $a \cdot a = a$). One can make sense of that notion in any monoidal category with diagonals $\Delta_A : A \to A \otimes_{\mathcal{C}} A$ by requiring that $\mu \circ \Delta_A = \id_A$.

We write $\mathsf{IdemMon}(\mathcal{C})$ for the full subcategory of $\mathsf{Mon}(\mathcal{C})$ spanned by the idempotent monoids in $\mathcal{C}$.

### Strict idempotency

An idempotent monoid $(A,\mu,\eta)$ or semigroup $(A,\mu)$ is strict if $\mu\colon A\otimes_{\mathcal{C}}A\to A$ is not only an isomorphism, but in fact the identity morphism of $A$.

## Properties

### Preservation by strong monoidal functors

A strong monoidal functor (resp. strict monoidal functor) $F\colon\mathcal{C}\to\mathcal{D}$ induces a functor

$\mathsf{IdemMon}(F)\colon\mathsf{IdemMon}(\mathcal{C})\to\mathsf{IdemMon}(\mathcal{D}),$

and hence “preserves” idempotent monoids (resp. strict idempotent monoids).

Similarly, strong (strict) semigroupal functors (“non-unital strong monoidal functors”) “preserve” (strict) idempotent semigroups.

### As strong monoidal functors from the punctual category

(Strict) idempotent monoids in $\mathcal{C}$ are the same as (strict) strong monoidal functors from the punctual monoidal category $\mathsf{pt}$.

Similarly, (strict) idempotent semigroups in $\mathcal{C}$ may be identified with (strict) strong semigroupal functors functors $(F,F^\otimes)\colon\pt\to\mathcal{C}$.

### As strictly unitary strong monoidal functors from the Boolean monoid

(Strict) idempotent semigroups in $\mathcal{C}$ are also the same as strictly unitary strong monoidal functors (resp. strict monoidal functors) from $\mathbb{B}_\mathsf{disc}$ to $\mathcal{C}$, where $\mathbb{B}=(\{0,1\},\vee,1)$ is the “Boolean monoid”, the initial monoid with an idempotent element.

## Examples

• An idempotent semigroup in $\left(A_{\mathsf{disc}},\cdot_A,1_A\right)$ for $A$ an ordinary monoid (in $\mathsf{Set}$) is an idempotent element of $A$, i.e. an element $a\in A$ such that $a^2=a$.

• An idempotent semigroup in $\left(\mathrm{End}_{\mathcal{C}}(X)_{\mathsf{disc}},\circ^{\mathcal{C}}_{X,X,X},\mathrm{id}_X\right)$ with $\mathrm{End}_{\mathcal{C}}(X)$ the monoid of endomorphisms of $\mathcal{C}$ at $X$ is an idempotent morphism $f\colon X\to X$ of $\mathcal{C}$, satisfying $f\circ f=f$.

• An idempotent monoid in abelian groups $\left(\mathsf{Ab},\otimes_{\mathbb{Z}},\mathbb{Z}\right)$ is a solid ring.

• An idempotent monoid in an endomorphism functor category $\left(\mathsf{Fun}(\mathcal{C},\mathcal{C}),\circ,\mathrm{id}_\mathcal{C}\right)$ is an idempotent monad.

• An idempotent monoid in the monoidal category $\mathsf{End}_{\mathcal{C}}(X)$ of endomorphisms of a bicategory $\mathcal{C}$ at $X\in\mathrm{Obj}(\mathcal{C})$ is an idempotent $1$-morphism $f\colon X\to X$ of $\mathcal{C}$, satisfying $f\circ f\simeq f$ up to a coherent $2$-isomorphism.

• An idempotent monoid in Spectra $\left(\mathsf{Sp},\otimes_{\mathbb{S}},\mathbb{S}\right)$ is a “solid ring spectrum” as in Gutierrez 2013, Section 4. See also MO #298435.

• An idempotent monoid in the category $\mathsf{Fun}(\mathcal{C},\mathsf{Sets})$ equipped with the Day convolution monoidal structure is a strong monoidal functor. Similarly, strict idempotent monoids in $\mathsf{Fun}(\mathcal{C},\mathsf{Sets})$ recover strict monoidal functors.

## References

Last revised on February 18, 2024 at 11:39:56. See the history of this page for a list of all contributions to it.