nLab idempotent monoid in a monoidal category



Categorical algebra

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory

Monoid theory



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


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

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

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

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

Strict idempotency

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


Preservation by strong monoidal functors

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


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

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

As strictly unitary strong monoidal functors from the Boolean monoid

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


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

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

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

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

  • An idempotent monoid in the monoidal category End 𝒞(X)\mathsf{End}_{\mathcal{C}}(X) of endomorphisms of a bicategory 𝒞\mathcal{C} at XObj(𝒞)X\in\mathrm{Obj}(\mathcal{C}) is an idempotent 11-morphism f:XXf\colon X\to X of 𝒞\mathcal{C}, satisfying ffff\circ f\simeq f up to a coherent 22-isomorphism.

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


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