Anagram & Information om | Engelska ordet TOPOSES

For differentiable manifolds, we obtain the local diffeomorphisms; for schemes, we have the formally étale morphisms and the étale morphisms; and for toposes, we get the étale geometric morphisms.
Back in Zürich for 1968 and 1969 he proposed elementary (first-order) axioms for toposes generalizing the concept of the Grothendieck topos (see History of topos theory) and worked with the algebraic topologist Myles Tierney to clarify and apply this theory.
This has been most successful in the theory of toposes, where the internal language of a topos together with the semantics of intuitionistic higher-order logic in a topos enables one to reason about the objects and morphisms of a topos as if they were sets and functions.
Extending the ideas to toposes (as generalised spaces) he found himself channelled into constructive mathematics in a geometric form and in Topical Categories of Domains he set out a geometrisation programme of, where possible, using this geometric mathematics as a tool for treating point-free spaces (and toposes) as though they had "enough points".
Caramello's methodology of toposes as bridges has been qualified by André Joyal as a "vast extension of Felix Klein's Erlangen Programme" and has been endorsed by Fields Medalists Alain Connes and Laurent Lafforgue.

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