# nLab Elephant

topos 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

## B 2-Categorical Aspects of Topos Theory

### B3 Toposes over a base

#### 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

(…)

### C2 Sheaves on a site

#### C2.1 Sites and coverages

• coverage

• Grothendieck topology

• 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.

### C3 Classes of geometric morphisms

#### 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.

## F Toposes as Mathematical Universes

### F4 Topos theory and set theory

#### F4.4 Independence of the axiom of choice

category: reference

Revised on July 19, 2016 06:56:14 by David Corfield (129.12.18.218)