nLab dominant functor

Dominant functors

Dominant functors


A functor F:CDF\colon C \to D is dominant if it is “surjective on objects up to retracts”.


F:CDF\colon C \to D is dominant if for every object yy of DD, there exists an object xx of CC for which yy is a retract of FxF x, i.e. there is a pair of morphisms yFxyy \to F x \to y composing to the identity.



  • Let LRIL \dashv R I be an adjunction. If II is dominant and full, then ILRI L \dashv R. In this case, the two adjunctions induce the same monad. This is Proposition 1.5 of DFH75. (For a converse, see fully faithful functor.)

  • Let LRL \dashv R. If LL is dominant, then RR is faithful. RR is full if it is full on the image of LL. This is Proposition 1.7 of DFH75.

  • Let G:ABG : A \to B be a functor between small categories. The induced functor [G op,Set]:[B op,Set][A op,Set][G^{op}, Set] : [B^{op}, Set] \to [A^{op}, Set] between presheaf categories is a surjective geometric morphism if and only if GG is dominant, if and only if [G op,Set][G^{op}, Set] is monadic. This is Example A4.2.7(b) of Sketches of an Elephant.

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Last revised on May 31, 2023 at 12:29:46. See the history of this page for a list of all contributions to it.