The concept of dominance is a weakening of the concept of surjectivity for geometric morphisms. It generalizes the concept of a continuous function whose image is a dense subspace of its codomain in topology.
A geometric morphism is called dominant if the following equivalent conditions hold:
The direct image satisfies: .
The inverse image satisfies: from follows .
The inverse image satisfies: from follows for a subterminal object.
A geometric morphism between the toposes of sheaves on two topological spaces is dominant iff the corresponding continuous map has a dense image in .
A geometric embedding is dominant precisely when it exhibits as a dense subtopos.
If is a surjection (i.e. is faithful) then is dominant.
Proof: Suppose is initial, then is a singleton for all , but by faithfulness of this implies that is a singleton for all , which says that is initial in .
Let be a geometric morphism and its surjection-inclusion factorization. is dominant iff is a dense inclusion.
Proof: Suppose is dense, hence dominant. as a surjection is dominant as well, and so is their composition .
Conversely, suppose is dominant and . Since preserves colimits, but is dominant by assumption, therefore , hence is dense.
The following is a slight generalization of the (dense,closed)-factorization employing dominant geometric morphisms:
Let be a geometric morphism. Then factors as a dominant geometric morphism followed by a closed inclusion .
Proof: Let be the surjection-inclusion factorization of . Since is surjective, it is dominant (cf. above). Then we use the (dense,closed)-factorization to factor into . Since both are dominant, so is and yields the demanded factorization of .
The concept can be generalized to morphisms in a topos by calling dominant if the induced geometric morphism is dominant, where denotes the pullback functor.
Last revised on January 9, 2016 at 00:40:52. See the history of this page for a list of all contributions to it.