nLab
idempotent adjunction

Idempotent adjunctions

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 2DC \underoverset{G_1}{F_1}{\rightleftarrows} E \underoverset{G_2}{F_2}{\rightleftarrows} 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. It then follows that FF and GG restrict to an equivalence of categories between the full images of FF and of GG (which are, respectively, a reflective subcategory of DD and a coreflective subcategory of CC).

Note that 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

References

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Revised on November 22, 2013 09:54:09 by Mike Shulman (192.195.154.58)