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.

The title refers to the strikingly different aspects of topos theory (such as functorial geometry versus mathematical logic) by alluding to the Indian folklore story of the blind men and the elephant (cf. E. J. Robinson’s Tales and Poems of South India), recalled in Johnstone’s preface like this:

“Four men, who had been blind from birth, wanted to know what an elephant was like; so they asked an elephant-driver for information. He led them to an elephant, and invited them to examine it; so one man felt the elephant’s leg, another its trunk, another its tail and the fourth its ear. Then they attempted to describe the elephant to one another. The first man said ”The elephant is like a tree“. ”No,“ said the second, ”the elephant is like a snake“. ”Nonsense!“ said the third, ”the elephant is like a broom“. ”You are all wrong,“ said the fourth, ”the elephant is like a fan“. And so they went on arguing amongst themselves, while the elephant stood watching them quietly.”

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