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Theories, Sites, Toposes - Relating and studying mathematical theories through topos-theoretic 'bridges' (Hardcover)
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Theories, Sites, Toposes - Relating and studying mathematical theories through topos-theoretic 'bridges' (Hardcover)
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According to Grothendieck, the notion of topos is "the bed or deep
river where come to be married geometry and algebra, topology and
arithmetic, mathematical logic and category theory, the world of
the continuous and that of discontinuous or discrete structures".
It is what he had "conceived of most broad to perceive with
finesse, by the same language rich of geometric resonances, an
"essence" which is common to situations most distant from each
other, coming from one region or another of the vast universe of
mathematical things". The aim of this book is to present a theory
and a number of techniques which allow to give substance to
Grothendieck's vision by building on the notion of classifying
topos educed by categorical logicians. Mathematical theories
(formalized within first-order logic) give rise to geometric
objects called sites; the passage from sites to their associated
toposes embodies the passage from the logical presentation of
theories to their mathematical content, i.e. from syntax to
semantics. The essential ambiguity given by the fact that any topos
is associated in general with an infinite number of theories or
different sites allows to study the relations between different
theories, and hence the theories themselves, by using toposes as
'bridges' between these different presentations. The expression or
calculation of invariants of toposes in terms of the theories
associated with them or their sites of definition generates a great
number of results and notions varying according to the different
types of presentation, giving rise to a veritable mathematical
morphogenesis.
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