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dominant geometric morphism

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Idea

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.

Definition

A geometric morphism f:f:\mathcal{F}\to\mathcal{E} is called dominant if the following equivalent conditions hold:

  • The direct image satisfies: f *( ) f_\ast(\emptyset_\mathcal{F})\cong\emptyset_\mathcal{E}.

  • The inverse image satisfies: from f *(Z) f^\ast(Z)\cong\emptyset_\mathcal{F} follows Z Z\cong\emptyset_\mathcal{E}.

  • The inverse image satisfies: from f *(Z) f^\ast(Z)\cong\emptyset_\mathcal{F} follows Z Z\cong\emptyset_\mathcal{E} for ZZ a subterminal object.

Examples

  • A geometric morphism f:Sh(X)Sh(Y)f:Sh(X)\to Sh(Y) between the toposes of sheaves on two topological spaces X,YX, Y is dominant iff the corresponding continuous map f:XYf:X\to Y has a dense image f(X)f(X) in YY.

  • A geometric embedding i:i:\mathcal{E}\hookrightarrow\mathcal{F} is dominant precisely when it exhibits \mathcal{E} as a dense subtopos.

Properties

Proposition

If f:f:\mathcal{F}\to\mathcal{E} is a surjection (i.e. f *f^\ast is faithful) then ff is dominant.

Proof: Suppose f *(Z) f^\ast(Z)\cong \emptyset_\mathcal{F} is initial, then Hom ( ,f *(Y))=Hom (f *(Z),f *(Y))Hom_\mathcal{F}(\emptyset_\mathcal{F},f^\ast(Y))=Hom_\mathcal{F}(f^\ast(Z),f^\ast(Y)) is a singleton for all YY\in\mathcal{E} , but by faithfulness of f *f^\ast this implies that Hom (Z,Y)Hom_\mathcal{E}(Z,Y) is a singleton for all YY\in\mathcal{E} , which says that ZZ is initial in \mathcal{E}. \qed

Proposition

Let f:f:\mathcal{F}\to\mathcal{E} be a geometric morphism and isi\circ s its surjection-inclusion factorization. f:f:\mathcal{F}\to\mathcal{E} is dominant iff i:Im(f)i:Im(f)\hookrightarrow\mathcal{E} is a dense inclusion.

Proof: Suppose ii is dense, hence dominant. ss as a surjection is dominant as well, and so is their composition ff.

Conversely, suppose f:f:\mathcal{F}\to\mathcal{E} is dominant and i *(Z) Im(f)i^\ast(Z)\cong \emptyset_{Im(f)}. Since s *s^\ast preserves colimits, s *( Im(f))s *i *(Z)=f *(Z)\emptyset_\mathcal{F}\cong s^\ast (\emptyset_{Im(f)})\cong s^\ast\circ i^\ast (Z)=f^\ast(Z) but ff is dominant by assumption, therefore Z Z\cong \emptyset_\mathcal{E}, hence ii is dense. \qed

The following is a slight generalization of the (dense,closed)-factorization employing dominant geometric morphisms:

Proposition

Let f:f:\mathcal{F}\to\mathcal{E} be a geometric morphism. Then ff factors as a dominant geometric morphism dd followed by a closed inclusion cc.

Proof: Let id 1i\circ d_1 be the surjection-inclusion factorization of ff. Since d 1d_1 is surjective, it is dominant (cf. above). Then we use the (dense,closed)-factorization to factor ii into cd 2c\circ d_2. Since both d id_i are dominant, so is d:=d 2d 1d:=d_2\circ d_1 and cdc\circ d yields the demanded factorization of ff. \qed

Remark

The concept can be generalized to morphisms in a topos \mathcal{E} by calling f:XYf:X\to Y dominant if the induced geometric morphism f * f:/X/Yf^\ast\dashv\prod_f:\mathcal{E}/X\to\mathcal{E}/Y is dominant, where f *f^\ast denotes the pullback functor.

References

Last revised on January 8, 2016 at 19:40:52. See the history of this page for a list of all contributions to it.