Toposes, Triples and Theories
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As its title suggests, this book is an introduction to three ideas and the connections between them.
A topos is a special kind of category defined by axioms saying roughly that certain constructions one can make with sets can be done in the category.
The concept of triple originated (under the name "standard constructions") in Godement's book on sheaf theory for the purpose of computing sheaf cohomology. Chapter 1 is an introduction to category theory which develops the basic constructions in categories needed for the rest of the book. Chapters 2, 3, 5, 6 and 7 form a fairly thorough introduction to the theory of toposes, covering topologies and the representation theorems but omitting the connections with algebraic geometry and logic. Adding chapters 4 and 8 provides an introduction to the concept of categorical theory, again without the connection to logic.