Spahn étale topos (Rev #4, changes)

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Let HH be a topos, let E:=(H/X) E:=(H/X)_\Box (\Box-closed/étale morphisms over XX).

Lemma

Like every reflective subcategory of a toposEEreflective subcategory of a topos EE is closed under limits and colimits.

Lemma

In EE every monomorphism ist a strong monomorphism?.

Proof

EE is a topos and hence any monomorphism in HH is strong. Let

Q Y Z W\array{ Q&\to&Y \\ \downarrow&\nearrow&\downarrow \\ Z&\to&W }

be a solved lifting problem with YWY\to W an etale monomorphism, ZWZ\to W an etale morphism, and QZQ\to Z an epimorphism. Then by the left cancellation property also ZYZ\to Y is etale. This remains true if we consider the lifting problem in H/XH/X.

Revision on December 9, 2012 at 23:24:00 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.