nLab Elephant

Contents

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.

C2.2 The topos of sheaves

• sheaf

• 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.

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

Last revised on December 17, 2019 at 21:05:52. See the history of this page for a list of all contributions to it.