Elementary topos theory

Se basa en

Requisitos previos obligatorios, hasta el fondo: 33

Aprenderás a

• Verify the topos axioms for Set and for presheaf categories [Cᵒᵖ, Set], computing their subobject classifiers.
• Show that the internal logic of a topos is intuitionistic, and give a presheaf topos whose logic is not Boolean.
• State the axioms of ETCS and compare them with ZFC.

Definición

• An elementary topos is a category with finite limits, exponentials and a subobject classifier.
• A subobject classifier is a morphism true : 1 → Ω such that every monomorphism is the pullback of true along a unique morphism to Ω.
• A Grothendieck topos is a category equivalent to the category of sheaves on a site.

Teorema

• Every presheaf category [Cᵒᵖ, Set] on a small category C is a topos.
• The subobjects of any object of a topos form a Heyting algebra.
• Set is a Boolean topos, while the topos of presheaves on the poset {0 < 1} is not Boolean.
• ETCS without a replacement axiom is strictly weaker than ZFC in consistency strength.

Errores frecuentes

• Thinking a topos is a kind of topological space; every space gives a topos of sheaves, but many toposes do not arise in this way.
• Believing the internal logic of every topos is classical because that of Set is.

Descubrimientos opcionales

• ∞-categories ∞-toposes extend topos theory to homotopy theory, and they model homotopy type theory.
• ← ZFC: axiomatic set theory
• ← Representable functors
• ← Forcing and independence
• ← Sheaves

Lecturas complementarias

• Saunders Mac Lane and Ieke Moerdijk, Sheaves in Geometry and Logic (1992)
• Robert Goldblatt, Topoi: The Categorial Analysis of Logic (1979)
• Peter T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium (2002)