nLab dominant functor

Idea

A functor $F\colon C \to D$ is dominant if it is “surjective on objects up to retracts”. Such functors are also called co-conservative or liberal (see codiscrete cofibration).

Definition

$F\colon C \to D$ 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.

Examples

• Every essentially surjective on objects functor is dominant.

Properties

• 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 follows from Example A4.2.7(b) of Sketches of an Elephant, which states that under these assumptions, $[G^{op}, Set]$ is conservative, from which monadicity follows because $[G^{op}, Set]$ creates coequalisers, so that the crude monadicity theorem applies.

Related concepts

basic properties of…

• functors

  • faithful functor

  • full functor

  • full and faithful functor

  • injective-on-objects functor

  • essentially injective functor

  • embedding of categories

  • essentially surjective functor

    • surjective-on-objects functor

    • bijective-on-objects functor

    • dominant functor

  • split essentially surjective functor

  • essentially surjective and full functor

  • equivalence of categories

  • conservative functor

  • localization functor

  • pseudomonic functor

  • (co)continuous functor

  • final functor, dense functor

• 2-functors

  • full and faithful 2-functor

  • locally fully faithful 2-functor

  • equivalence of 2-categories

• (∞,1)-functors:

  • fully faithful (∞,1)-functor

  • essentially surjective (∞,1)-functor

  • equivalence of (∞,1)-categories

  • homotopy final functor, final (∞,1)-functor

References

• Aristide Deleanu?, Armin Frei, Peter Hilton, Idempotent triples and completion, Mathematische Zeitschrift 143 (1975) 91-104 [doi:10.1007/BF01173053]

Called liberal in (see Proposition 3.1):

• Aurelio Carboni and Scott Johnson? and Ross Street and Dominic Verity, “Modulated bicategories” (MB 1994) MR.

Called quasi-surjective on objects functors in:

• Gabriella Böhm, Steve Lack, Ross Street, Idempotent splittings, colimit completion, and weak aspects of the theory of monads, Journal of Pure and Applied Algebra 216:2 (2012) 385–403 doi