Johnstone P.T. Sketches of an Elephant: A Topos Theory Compendium. Vol. 1-2

Oxford: Oxford University Press, 2002. — 1016 p. + 1089 p. — (Oxford Logic Guides, 43 & 44). — ISBN: 0 19 852496 X (2 Vol. set).Topos Theory is a subject that stands at the junction of geometry, mathematical logic and theoretical computer science, and it derives much of its power from the interplay of ideas drawn from these different areas. Because of this, an account of topos theory which approaches the subject from one particular direction can only hope to give a partial picture; the aim of this compendium is to present as comprehensive an account as possible of all the main approaches and thereby to demonstrate the overall unity of the subject. The material is organized in such a way that readers interested in following a particular line of approach may do so by starting at an appropriate point in the text.Toposes as Categories.
Regular and cartesian closed categories.
Toposes - basic theory.
Allegories.
Geometric morphisms - basic theory.
2-categorical aspects of topos theory.
Indexed categories and fibrations.
Internal and locally internal categories.
Toposes over a base.
BTop/S as a 2-Category.Toposes as spaces.
Sheaves on a locale.
Sheaves on a site.
Classes of geometric morphisms.
Local compactness and exponentiability.
Toposes as groupoids.
Toposes as theories.
Ferst-order categorical logic.
Sketches.
Classifying toposes.
Higher-order logic.
Aspects of funiteness.
Bibliography (1262 publ).