nLab idempotent monoid in a monoidal category

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.

Related concepts

• monoid

• commutative monoid in a symmetric monoidal category

• idempotent monad

References

• Peter Hines, The categorical theory of self-similarity, 1999. Theory and Applications of Categories. (abstract, pdf, dvi, ps.)

• Javier J. Gutiérrez, On solid and rigid monoids in monoidal categories, Applied Categorical Structures 23, no. 2 (2015), 575-589 (arXiv:1303.5265)