nLab Sketches of an Elephant



Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory


This page collects material related to the book

on topos theory.

Like Gravitation, the title can be taken to refer not only to the subject matter but also to the immense size and scope of the book itself. Like The Lord of the Rings, it consists of 6 parts arranged evenly into 3 volumes (but without appendices). Actually, Volume 3 has not yet been published (so who knows? it may have appendices after all!).

The Elephant is a good reference for anything related to topos theory, and we may often cite it here. However, it introduced many terminological changes, some of which may not be widely accepted or even known. (Fortunately, it will tell you about these in the text.)


A Toposes as Categories

A1 Regular and cartesian closed categories

A1.1 Preliminary assumptions

A1.2 Cartesian categories

A1.3 Regular categories

A1.4 Coherent categories

A1.5 Cartesian closed categories

A1.6 Subobject classifiers

A2 Toposes - basic theory

A2.1 Definition and examples

A2.2 The monadicity theorem

A2.3 The Fundamental Theorem

A2.4 Effectiveness, positivity and partial maps

A2.5 Natural number objects

A2.6 Quasitoposes

A3 Allegories

A3.1 Relations in regular categories

A3.2 Allegories and tabulations

A3.3 Splitting symmetric idempotents

A3.4 Division allegories and power allegories

A4 Geometric morphisms - basic theory

A4.1 Definition and examples

B 2-Categorical Aspects of Topos Theory

B1 Indexed categories and fibrations

B1.1 Review of 2-categories

B1.2 Indexed categories

B1.3 Fibrations

B1.4 Limits and colimits

B1.5 Descent conditions and stacks

B2 Internal and locally internal categories

B2.1 Review of enriched categories

B2.2 Locally internal categories

B2.3 Internal categories and diagram categories

B2.4 The indexed adjoint functor theorem

B2.5 Discrete opfibrations

B2.6 Filtered colimits

B2.7 Internal profunctors

B3 Toposes over a base

B3.1 𝒮\mathcal{S}-Toposes as 𝒮\mathcal{S}-indexed categories

B3.2 Diaconescu’s theorem

B3.3 Giraud’s theorem

B3.4 Colimits in Top

  • Topos

  • The paragraph before B3.4.8 refers to A4.1.13, but should probably refer instead to A4.1.15.

C Toposes as Spaces

C1 Sheaves on a locale

C1.1 Frames and nuclei

C1.2 Locales and spaces

C1.3 Sheaves, local homeomorphisms and frame-valued sets

C1.4 Continuous maps


C1.5 Some topological properties of toposes

C2 Sheaves on a site

C2.1 Sites and coverages

C2.2 The topos of sheaves

C2.3 Morphisms of sites

C2.4 Internal sites and pullbacks

C2.5 Fibrations of sites

C3 Classes of geometric morphisms

C3.1 Open maps

C3.2 Proper maps

C3.3 Locally connected morphisms

C3.4 Tidy morphisms

C3.5 Atomic morphisms

C3.6 Local maps

  • local geometric morphism

  • local topos

  • Example C3.6.15(e) says that (E/l(M)) ce(E/l(M))_{ce} is equivalent to SetSet, but the referred-to paper “Local maps of toposes” seems to say that it should be equivalent to EE instead.

D Toposes as theories

D1 First-order categorical logic

D1.1 First-order languages

D1.2 Categorical semantics

D1.3 First-order logic

D1.4 Syntactic categories

D1.5 Classical completeness

D2 Sketches

D2.1 The concept of sketch

D2.2 Sketches and theories

D2.3 Sketchable and accessible categories

D2.4 Properties of model categories

D3 Classifying toposes

D3.1 Classifying toposes via syntactic sites

D3.2 The object classifier

D3.3 Coherent toposes

D3.4 Boolean classifying toposes

D3.5 Conceptual completeness

D4 Higher-order logic

D4.1 Interpreting higher-order logic in a topos

D4.2 λ\lambda-Calculus and cartesian closed categories

D4.3 Toposes as type theories

D4.4 Predicative type theories

D4.5 Axioms of choice and Booleanness

D4.6 De Morgan’s law and the Gleason cover

D4.7 Real numbers in a topos

D5 Aspects of finiteness

D5.1 Natural number objects revisited

D5.2 Finite cardinals

D5.3 Finitary algebraic theories

D5.4 Kuratowski-finiteness

D5.5 Orbitals and numerals

E Homotopy and Cohomology

E1 Homotopy theory for toposes

E1.1 Path-connectedness for locales

E1.2 The fundamental groupoid via paths

E1.3 The fundamental groupoid via coverings

E1.4 Natural homotopy

E2 Algebraic homotopy theory

E2.1 Quillen model structures

E2.2 Model structure for simplicial sets

E2.3 Model structures for sheaves

E2.4 Axiomatic theory of open maps

E3 Cohomology theory

E3.1 Abelian groups and modules in a topos

E3.2 Cech cohomology

E3.3 Torsors and non-abelian cohomology

E3.4 Classifying toposes and classifying spaces

E3.5 Cohomological applications of descent theory

F Toposes as Mathematical Universes

F1 Synthetic differential geometry

F1.1 Properties of the generic ring

F1.2 Rings of line type

F1.3 Well-adapted models

F1.4 Tiny objects

F1.5 Synthetic integration theory

F1.6 Intrinsic infinitesimal

F2 Realizability toposes

F2.1 Schönfinkel algebras and assemblies

F2.2 Realizability toposes

F2.3 Modified Realizability

F2.4 Synthetic domain theory

F3 The free topos

F3.1 The free topos as a mathematical universe

F3.2 Disjunction and existence properties

F3.3 Doing without the natural numbers

F3.4 Recursive functions in the free topos

F4 Topos theory and set theory

F4.1 Internal sets in a topos

F4.2 Algebraic set theory

F4.3 Independence proofs via classifying toposes

F4.4 Independence of the axiom of choice

category: reference

Last revised on March 23, 2021 at 05:46:28. See the history of this page for a list of all contributions to it.