A functor is dominant if it is “surjective on objects up to retracts”.
is dominant if for every object of , there exists an object of for which is a retract of , i.e. there is a pair of morphisms composing to the identity.
Let be an adjunction. If is dominant and full, then . 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 . If is dominant, then is faithful. is full if it is full on the image of . This is Proposition 1.7 of DFH75.
Let be a functor between small categories. The induced functor between presheaf categories is a surjective geometric morphism if and only if is dominant, if and only if is monadic. This is Example A4.2.7(b) of Sketches of an Elephant.
basic properties of…
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