# Toposes, Triples and Theories by M. Barr and C. Wells

Toposes – As its title suggests, this book is an introduction to three ideas and the connections between them. Before describing the content of the book in detail, we describe each concept briefly. More extensive introductory descriptions of each concept are in the introductions and notes to Chapters 2, 3 and 4. 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.

In that sense, a topos is a generalized set theory. However, it originated with Grothendieck and Giraud as an abstraction of the of the category of sheaves of sets on a topological space. Later, properties Lawvere and Tierney introduced a more general id~a which they called “elementary topos” (because their axioms did not quantify over sets), and they and other mathematicians developed the idea that a theory in the sense of mathematical logic can be regarded as a topos, perhaps after a process of completion.

The concept of triple originated (under the name “standard construc­ in Godement’s book on sheaf theory for the purpose of computing tions”) sheaf cohomology. Then Peter Huber discovered that triples capture much of the information of adjoint pairs. Later Linton discovered that triples gave an equivalent approach to Lawverc’s theory of equational theories (or rather the infinite generalizations of that theory). Finally, triples have turned out to be a very important tool for deriving various properties of toposes.

From the reviews:

“Michael Barr, Charles Wells uses the most efficient tools to get at the central theorems. This is a very readable introduction to the subject … . no other book goes as concisely and comprehensively to all the aspects of toposes as this one. Category theory, algebra, logic, arithmetic, geometry, and cohomology all come in, in a well chosen perspective. … it remains the best single text on the subject.” (Wordtrade, 2008)

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