A functor $F\colon C \to D$ is dominant if it is “surjective on objects up to retracts”.
$F\colon C \to D$ is dominant if for every object $y$ of $D$, there exists an object $x$ of $C$ for which $y$ is a retract of $F x$, i.e. there is a pair of morphisms $y \to F x \to y$ composing to the identity.
Let $L \dashv R I$ be an adjunction. If $I$ is dominant and full, then $I L \dashv R$. 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 $L \dashv R$. If $L$ is dominant, then $R$ is faithful. $R$ is full if it is full on the image of $L$. This is Proposition 1.7 of DFH75.
Let $G : A \to B$ be a functor between small categories. The induced functor $[G^{op}, Set] : [B^{op}, Set] \to [A^{op}, Set]$ between presheaf categories is a surjective geometric morphism if and only if $G$ is dominant, if and only if $[G^{op}, Set]$ is monadic. This is Example A4.2.7(b) of Sketches of an Elephant.
basic properties of…
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