nLab Sketches of an Elephant

This page collects material related to the book

• Peter Johnstone:

  
  

  Sketches of an Elephant – A Topos Theory Compendium

  
  

  Oxford University Press (2002)

  Volume 1: ISBN:9780198534259

  Volume 2: ISBN:9780198515982

  Volume 3: (long announced, yet to be published)

on topos theory.

A Toposes as Categories

A1 Regular and cartesian closed categories

A1.1 Preliminary assumptions

• category theory

• reflective subcategory

• monadicity theorem

• adjoint lifting theorem

• Grothendieck construction

A1.2 Cartesian categories

• finitely complete category / cartesian category

• cartesian functor

A1.3 Regular categories

• strong epimorphism

• conservative functor

• regular category

  • regular epimorphism

• exact category

• regular and exact completion

A1.4 Coherent categories

• coherent category

• disjoint coproduct

• extensive category

• pretopos

• Heyting category

• Boolean category

A1.5 Cartesian closed categories

• cartesian closed category

• locally cartesian closed category

• cartesian closed functor

• Frobenius reciprocity

• dependent product, dependent sum

A1.6 Subobject classifiers

• subobject classifier

A2 Toposes - basic theory

A2.1 Definition and examples

• topos

• Examples

  • Set, FinSet

  • category of sheaves, presheaf

A2.2 The monadicity theorem

• monadicity theorem

A2.3 The Fundamental Theorem

A2.4 Effectiveness, positivity and partial maps

A2.5 Natural number objects

• natural number object

A2.6 Quasitoposes

• quasitopos

A3 Allegories

A3.1 Relations in regular categories

• relation
• Rel

A3.2 Allegories and tabulations

• allegory
• bicategory of relations
• 1-category equipped with relations

A3.3 Splitting symmetric idempotents

• idempotent

A3.4 Division allegories and power allegories

A4 Geometric morphisms - basic theory

A4.1 Definition and examples

• geometric morphism

  • direct image

  • inverse image

  • geometric transformation

• essential geometric morphism

B 2-Categorical Aspects of Topos Theory

B1 Indexed categories and fibrations

B1.1 Review of 2-categories

• 2-category

• 2-limit

  • inverter

B1.2 Indexed categories

• indexed category

• indexed functor

• pseudofunctor

B1.3 Fibrations

• Grothendieck fibration

• bifibration

• Grothendieck construction

B1.4 Limits and colimits

• limit

• dependent product, dependent sum

B1.5 Descent conditions and stacks

• descent

• stack

B2 Internal and locally internal categories

B2.1 Review of enriched categories

• enriched category

• enriched category theory

B2.2 Locally internal categories

• locally internal category

B2.3 Internal categories and diagram categories

• internal category, internal diagram

• internal (co-)limit

B2.4 The indexed adjoint functor theorem

• adjoint functor theorem

• indexed adjoint functor theorem

B2.5 Discrete opfibrations

• discrete opfibration

B2.6 Filtered colimits

• filtered colimit

B2.7 Internal profunctors

• profunctor

• internal profunctor

B3 Toposes over a base

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

• indexed topos

• bounded geometric morphism

• bounded topos

B3.2 Diaconescu’s theorem

• torsor, principal bundle

• Diaconescu's theorem

• Sierpinski topos

B3.3 Giraud’s theorem

• 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

• frame
• Heyting algebra

C1.2 Locales and spaces

• locale

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

• sheaf
• local homeomorphism

C1.4 Continuous maps

(…)

C1.5 Some topological properties of toposes

• locally connected topos

C2 Sheaves on a site

C2.1 Sites and coverages

• coverage

  • regular coverage

• Grothendieck topology

  • Grothendieck pretopology

• site

  • standard site

• Lemma 2.1.7 is incorrect as stated: one should assume that $A$ is a sheaf for a (sifted) coverage. See here for details.

C2.2 The topos of sheaves

• sheaf

• category of sheaves,

• Grothendieck topos, sheaf topos

  • Giraud's theorem

  • sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes

• subcanonical topology, canonical topology

• dense sub-site

• The statement that the induced coverage inherits closure properties requires assumptions on either the coverage or the subcategory. See the discussion here.

C2.3 Morphisms of sites

• morphism 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

• proper geometric morphism

• separated geometric morphism

• compact topos, Hausdorff topos

C3.3 Locally connected morphisms

• locally connected topos

• locally connected geometric morphism

C3.4 Tidy morphisms

C3.5 Atomic morphisms

• atomic geometric morphism

C3.6 Local maps

• local geometric morphism

• local topos

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

D Toposes as theories

D1 First-order categorical logic

D1.1 First-order languages

• signature

• term

• formula

• context

• sequent

• axiom

• theory

  • algebraic theory

  • propositional theory

  • Horn theory

  • regular theory

  • coherent theory

  • first-order theory

  • geometric theory

  • infinitary first-order theory

D1.2 Categorical semantics

• categorical semantics

• internal logic

• models in presheaf toposes

D1.3 First-order logic

D1.4 Syntactic categories

• syntactic category

D1.5 Classical completeness

D2 Sketches

D2.1 The concept of sketch

• sketch

D2.2 Sketches and theories

• theory

• Lawvere theory

D2.3 Sketchable and accessible categories

• locally presentable category

• accessible category

D2.4 Properties of model categories

D3 Classifying toposes

D3.1 Classifying toposes via syntactic sites

• syntactic site

  • regular coverage

  • coherent coverage

• classifying topos

D3.2 The object classifier

• theory of objects

• classifying topos for the theory of objects

D3.3 Coherent toposes

• coherent topos

D3.4 Boolean classifying toposes

D3.5 Conceptual completeness

• conceptual completeness

D4 Higher-order logic

D4.1 Interpreting higher-order logic in a topos

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

• cartesian closed category

• lambda-calculus

D4.3 Toposes as type theories

D4.4 Predicative type theories

• predicative mathematics

D4.5 Axioms of choice and Booleanness

• axiom of choice
• internally projective object— note that Lemma 4.5.3(iii) in the Elephant is not quite strong enough to imply (i) and (ii).
• excluded middle

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

D4.7 Real numbers in a topos

• real numbers object

D5 Aspects of finiteness

D5.1 Natural number objects revisited

D5.2 Finite cardinals

• finite set
• finite object

D5.3 Finitary algebraic theories

• algebraic theory

• Lawvere theory

D5.4 Kuratowski-finiteness

• finite set
• finite object

D5.5 Orbitals and numerals

E Homotopy and Cohomology

E1 Homotopy theory for toposes

• homotopy groups in an (∞,1)-topos

E1.1 Path-connectedness for locales

• locale

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

• model category

E2.2 Model structure for simplicial sets

• model structure on simplicial sets

E2.3 Model structures for sheaves

• model structure on simplicial sheaves

E2.4 Axiomatic theory of open maps

E3 Cohomology theory

• cohomology

• abelian sheaf cohomology

E3.1 Abelian groups and modules in a topos

• abelian group, abelian sheaf

• module

E3.2 Cech cohomology

• Cech cohomology

E3.3 Torsors and non-abelian cohomology

• nonabelian cohomology

• principal bundle

• principal ∞-bundle

E3.4 Classifying toposes and classifying spaces

• classifying topos

E3.5 Cohomological applications of descent theory

• descent

F Toposes as Mathematical Universes

F1 Synthetic differential geometry

• synthetic differential geometry

F1.1 Properties of the generic ring

• classifying topos

F1.2 Rings of line type

• line object

F1.3 Well-adapted models

• Models for Smooth Infinitesimal Analysis

F1.4 Tiny objects

• tiny object

F1.5 Synthetic integration theory

• integration axiom

F1.6 Intrinsic infinitesimal

• infinitesimal object

• amazing right adjoint

F2 Realizability toposes

F2.1 Schönfinkel algebras and assemblies

F2.2 Realizability toposes

• realizability topos

F2.3 Modified Realizability

F2.4 Synthetic domain theory

• 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

• algebraic set theory

F4.3 Independence proofs via classifying toposes

F4.4 Independence of the axiom of choice

• axiom of choice

category: reference