Topos

Definition

In mathematics, a topos is a category that behaves like the category of sheaves of sets on a topological space. Topoi behave much like the category of sets and possess a notion of localization. The Grothendieck topoi find applications in algebraic geometry, and more general elementary topoi are used in logic.

Related concepts

2-functor 2-group 2-ring 3-category Abelian category Additive category Adjoint functor Adjoint functors Alexander Grothendieck Algebraic category Algebraic geometry Algebraic stack Algebraic structure American English Anabelian geometry Axiom of choice Axiom of reducibility Axiomatic system Barry Mazur Bicategory British English Cartesian closed category Categorification Category (mathematics)Category of abelian groups Category of groups Category of magmas Category of modules Category of relations Category of rings Category of sets Category theory Charles Wells (mathematician)Classifying topos Coequalizer Coherent sheaf Cokernel Colimit Colin McLarty Comma category Commutative diagram Complete category Concrete category Cone (category theory)Conservative functor Constructive set theory Contravariant functor Coproduct Crystal (mathematics)Dependent type Derived functor Descriptive set theory Determinacy Diagonal functor Diagram (category theory)Direct limit En-ring End (category theory)Enriched category Epimorphism Equaliser (mathematics)Equivalence of categories Equivalence relation Equivalent categories Eric Friedlander Essentially surjective functor Exact functor Exponential object F. William Lawvere Fiber product First-order logic Flat topology Forgetful functor Formal system Foundation of mathematics Foundations of mathematics Frames and locales Free category Full and faithful functors Function (mathematics)Functor Functor category Generalized space Generator (category theory)Geometric logic Geometrically unibranch Girard's paradox Giraud's theorem Glossary of category theory Grothendieck Grothendieck site Group (mathematics)Group action Group action (mathematics)Groupoid Higher-dimensional algebra Higher category theory Hilbert system History of topos theory History of type theory Homomorphism Homotopy Homotopy category Homotopy hypothesis Homotopy type theory Ieke Moerdijk Image (mathematics)Initial and terminal objects Intuitionistic type theory Inverse limit Inverse system Isomorphism Isomorphism of categories Jean-Louis Verdier Jean Giraud (mathematician)John Baez John Lane Bell Kan extension Kernel (category theory)Kleisli category Law of excluded middle Left adjoint Limit (category theory)List of category theory topics List of mathematical logic topics List of set theory topics Localization of a category Luc Illusie M. Cristina Pedicchio Mathematical constructivism Mathematical induction Mathematical logic Mathematical proof Mathematics Michael Artin Michael Barr (mathematician)Modal logic Model category Model theory Module (mathematics)Monoidal category Monoidal functor Monomorphism Morphism Multidigraph N-group (category theory)N-monoid Naive set theory Natural deduction Natural transformation Nisnevich topology Noetherian scheme Olivia Caramello Opposite category Outline of category theory Overcategory Pathological (mathematics)Peano axioms Peter Johnstone (mathematician)Pierre Deligne Pointed set Powerset Pre-abelian category Preadditive category Preorder Presheaf (category theory)Princeton University Press Pro-finite Pro-simplicial set Product (category theory)Product category Pseudo-functor Pullback (category theory)Pushout (category theory)Quasi-category Quasitopos Quotient category Regular category Representable functor Ring object Ringed space Ringed topos Robert Goldblatt Russell's paradox Saunders Mac Lane Scheme (mathematics)Second-order logic Separable field extension Set (mathematics)Set theory Sheaf (mathematics)Simplex category Site (mathematics)Smooth functor Springer-Verlag Stack (mathematics)Stephen Schanuel Steve Vickers (computer scientist)Strict 2-category Strict n-category String diagram Structuralism (philosophy of mathematics)Subalgebra Subcategory Subgroup Subobject classifier Subset Symmetric monoidal category Tetracategory Topological space Topos (disambiguation)Traced monoidal category Tricategory Type theory Univalent foundations Universal algebra Universal construction Universal property Weak n-category Yoneda Lemma Yoneda lemma Zermelo set theory Zermelo–Fraenkel set theory Zero morphism Étale fundamental group Étale homotopy theory Étale morphism Étale topology Étale topos ∞-groupoid ∞-topos

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