研究生一年级Category Theory课程已上线
Elementary topos theory
Elementary toposes as categories that behave like categories of sets (finite limits, exponentials and a subobject classifier); presheaf and sheaf toposes; internal intuitionistic logic; and Lawvere's ETCS as a categorical axiomatisation of sets.
math.cat.elementary-topoi
前置知识
33 项必需的前置知识（一直追溯到底）
你将学会
- 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.
定义
- 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.
定理
- 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.
常见误解
- 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.
可选发现
- ∞-categories ∞-toposes extend topos theory to homotopy theory, and they model homotopy type theory.
- ← ZFC: axiomatic set theory
- ← Representable functors
- ← Forcing and independence
- ← Sheaves
延伸阅读
- 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)