Some logical aspects of topos theory and examples from algebraic - PowerPoint PPT Presentation

Some logical aspects of topos theory and examples from algebraic geometry (Part II) Matthias Hutzler Universit¨ at Augsburg June 29, 2020

Recap of Part I

Interpreting IZF in Sh ( X )

Classifying toposes

De fi nition A geometric formula is a fi rst order formula φ that only uses the logical connectives � = , � , ∧ , ⊥ , ∨ , ∃ , . A geometric theory is a fi rst-order theory with axioms φ � � A ψ x : � where φ and ψ are geometric formulas.

Why geometric logic? Every continuous map f : X → Y induces f ∗ Sh ( X ) − Sh ( Y ) . ← This f ∗ preserves fi nite limits and all colimits. ⇒ f ∗ preserves models of geometric theories. In fact, every geometric morphism f : E � → E has such a “pull-back part” f ∗ E � − E . ←

De fi nition A geometric theory T is classi fi ed by a topos E T if there is a universal model U T ∈ Mod E T ( T ), that is: ∼ Geom( E � , E T ) Mod E � ( T ) = f ∗ U T f �→ is an equivalence of categories for every topos E � . Aside: A point of E is a geometric morphism Set → E .

Example What is the classifying topos of sub-singletons? � one sort A � no function/relation symbols � one axiom: � � x , y : A x = y

Theorem Every geometric theory has a unique (up to equivalence) classifying topos. Every topos classi fi es some geometric theory. Theorem A geometric sequent φ � � A ψ is ful fi lled for U T i ff it is x : � provable modulo T .

Examples from algebraic geometry

De fi nition A site is a category together with a “notion of covering” (a Grothendieck topology ). site for Sh ( X ) site for Zar objects U ⊆ X open fi nitely presented ring A morphisms V ⊆ U ring homomorphism B ← A ( A [ a − 1 covers U = � i U i i ] ← A ) i =1 ... n whenever a 1 + . . . + a n = 1

Theorem A 1 is the universal local ring. (So Zar classi fi es local rings.) Proof sketch.

In fi nitesimals? No problem!

Theorem The big in fi nitesimal topos classi fi es the theory of in fi nitesimally thickened local rings. Proof sketch.