idempotent monoid in a monoidal category



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In higher category theory

Monoid theory



An idempotent monoid is a monoid, which is idempotent in that “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.

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 September 25, 2021 at 21:07:24. See the history of this page for a list of all contributions to it.