Spahn étale topos (Rev #3, changes)

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

Lemma

In Like every reflective subcategory of a toposEE every is monomorphism closed ist under a limits and colimits.strong monomorphism?.

Proof Lemma

EEIn is a topos and hence any monomorphism in EEHH every monomorphism ist a is strong. Letstrong monomorphism?.

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

be a solved lifting problem with Y→WY\to W an etale monomorphism, Z→WZ\to W an etale morphism, and Q→ZQ\to Z an epimorphism. Then by the left cancellation property also Z→YZ\to Y is etale. This remains true if we consider the lifting problem in H/XH/X.
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 Y→WY\to W an etale monomorphism, Z→WZ\to W an etale morphism, and Q→ZQ\to Z an epimorphism. Then by the left cancellation property also Z→YZ\to Y is etale. This remains true if we consider the lifting problem in H/XH/X.

Revision on December 9, 2012 at 22:40:43 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

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