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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 12th 2011
   • (edited Apr 12th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have expanded the entry [[formally smooth morphism]]: I have first of all added the general-abstract formalization by Kontsevich-Rosenberg, taking the liberty of polishing it a bit from Q-category language to genuine (cohesive) topos-theoretic language and making contact with the notion of infinitesimal cohesion . Then I added their theorems about how the general abstract topos-theoretic definitions do reproduce the traditional explicit notions. Except for one clause : in prop. 5.8.1 of _Noncommutative spaces_ they show that the correct notion of formal smoothness for morphisms is reproduced in the non-commutative case (via the relative Cuntz-Quillen condition). But for the commutative case I see the corresponding statement only for objects (in section 4.1) not for morphisms. Zoran, do you know if they also discuss the relative version in the commutative case? Maybe it's trivial, I haven't thought it through yet.

   I have expanded the entry formally smooth morphism:

   I have first of all added the general-abstract formalization by Kontsevich-Rosenberg, taking the liberty of polishing it a bit from Q-category language to genuine (cohesive) topos-theoretic language and making contact with the notion of infinitesimal cohesion .

   Then I added their theorems about how the general abstract topos-theoretic definitions do reproduce the traditional explicit notions.

   Except for one clause : in prop. 5.8.1 of Noncommutative spaces they show that the correct notion of formal smoothness for morphisms is reproduced in the non-commutative case (via the relative Cuntz-Quillen condition). But for the commutative case I see the corresponding statement only for objects (in section 4.1) not for morphisms.

   Zoran, do you know if they also discuss the relative version in the commutative case? Maybe it’s trivial, I haven’t thought it through yet.