Spahn étale topos (Rev #2, changes)

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~~$G:=\infty Grpd$~~ Let , $H$ ~~$E:=(H/X)_\Box$~~ be a topos, let ( $E:=(H/X)_\Box$ ~~$\Box$~~ (~~-closed/étale morphisms over~~ $\Box$ ~~$X$~~ -closed/étale morphisms over $X$ ).

~~$Str_G^{loc}(E)=Str_G(E)=Fun^{lex}(G,E)=Shv_{Ind(G^{op})}(E)$ (Remark 1.2.12 DAG V)~~

Lemma

In $E$ every monomorphism ist a strong monomorphism?.

~~$(S_L,R_L)$ factorization system on $Str_G(E)$ where $S_R=\{morhisms\, in\, Str_G^{loc}(E)\}=\{\Box-closed\, morphisms\}$ (Theorem 1.3.1 DAG V)~~

Proof

$E$ is a topos and hence any monomorphism in $H$ is strong. Let

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

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

Revision on December 9, 2012 at 20:14:09 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.