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idempotent adjunction

Idempotent adjunctions

Idempotent adjunctions

Definition

Definition

Let F:CD:GF: C \rightleftarrows D : G be an adjunction with unit η\eta and counit ε\varepsilon. Then the following conditions are equivalent:

  1. FηF \eta is a natural isomorphism.

  2. εF\varepsilon F is a natural isomorphism.

  3. GεFG \varepsilon F is a natural isomorphism — i.e. the monad induced by the adjunction is idempotent.

  4. GFη=ηGFG F \eta = \eta G F.

  5. GFηG=ηGFGG F \eta G = \eta G F G.

  6. GεG\varepsilon is a natural isomorphism.

  7. ηG\eta G is a natural isomorphism.

  8. FηGF \eta G is a natural isomorphism — i.e. the comonad induced by the adjunction is idempotent.

  9. FGε=εFGF G \varepsilon = \varepsilon F G.

  10. FGεF=εFGFF G \varepsilon F = \varepsilon F G F.

  11. The adjunction can be factored as a composite

    CG 1F 1EG 2F 2D, C \stackrel{\overset{F_1}{\longrightarrow}}{\underset{G_1}{\hookleftarrow}} E \stackrel{\overset{F_2}{\hookrightarrow}}{\underset{G_2}{\longleftarrow}} D \,,

    where F 2F_2 and G 1G_1 are fully faithful, i.e. F 1G 1F_1\dashv G_1 is a reflection and F 2G 2F_2 \vdash G_2 is a coreflection.

When these conditions hold, the adjunction is said to be idempotent.

Remark

For an idempotent adjunction as in def. , the functors FF and GG restrict to an equivalence of categories between the full images of FF and of GG (which are, respectively, a coreflective subcategory of DD and a reflective subcategory of CC, both equivalent to the EE in the last item above). In other words, for an idempotent adjunction, the category of fixed points has a particularly simple form.

Remark

If an idempotent adjunction is monadic, then (up to equivalence) it consists of the inclusion and reflection of a reflective subcategory (i.e. the algebras for an idempotent monad). Dually, if it is comonadic, it consists of the inclusion and coreflection of a coreflective subcategory. Thus, the primary interest in isolating the notion of idempotent adjunction is when considering adjunctions which are neither monadic nor comonadic.

Examples

  • A categorification of the notion of idempotent adjunction is that of lax-idempotent 2-adjunction.

  • In an adjoint triple, one of the adjunctions is idempotent if and only if the other one is; see there for the proof.

References

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Last revised on December 19, 2017 at 07:34:06. See the history of this page for a list of all contributions to it.