nLab
Elephant
Context
Topos Theory
topos theory

Background
Toposes
Internal Logic
Topos morphisms
Cohomology and homotopy
In higher category theory
Theorems
The Elephant is a book on topos theory by Peter Johnstone .

The full title is Sketches of an Elephant: A Topos Theory Compendium . 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.)

Contents
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 localy 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 $\mathcal{S}$ -Toposes as $\mathcal{S}$ -indexed categories
B3.2 Diaconescu’s theorem
B3.3 Giraud’s theorem
B3.4 Colimits in Top
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
C3.2 Proper maps
C3.3 Locally connected morphisms
C3.5 Atomic morphisms
C3.6 Local maps
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
D4 Higher-order logic
D4.4 Predicative type theories
D4.5 Axioms of choice and Booleanness
D4.7 Real numbers in a topos
D5 Aspects of finiteness
D5.2 Finite cardinals
D5.3 Finitary algebraic theories
D5.4 Kuratowski-finiteness
E Homotopy and Cohomology
E1 Homotopy theory for toposes
E1.1 Homotopy theory of toposes
E1.3 The fundamental groupoid via paths
E1.4 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
E3 Cohomology theory
E3.1 Abelian groups and modules in a topos
E3.2 Cech cohomology
E3.3 Torsors and non-abelian cohomology
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
F1.5 Intrinsic infinitesimal
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