Summary: Logic in topoi: Functorial semantics for higher­order logic
Dissertation abstract
Steven M. Awodey
The University of Chicago
Supervised by: Professors Saunders Mac Lane and William W. Tait
The category­theoretic notion of a topos is called upon to study the syntax
and semantics of higher­order logic. Syntactical systems of logic are replaced
by topoi, and models by functors on those topoi, as per the general scheme
of functorial semantics. Each (possibly higher­order) logical theory T gives
rise to a syntactic topos I[U T ] of polynomial­like objects. The chief result
is the universal characterization of I[U T ] as a so­called classifying topos: for
any topos E, the category Log(I[U T ]; E) of logical morphisms I[U T ] ! E is
naturally equivalent to the category Mod T (E) of models of T in E,
Log(I[U T ]; E) ' Mod T (E):
In particular, there is a T­model U T in I[U T ] such that any T­model in any
topos is an image of U under an essentially unique logical morphism. In this
sense, I[U T ] is freely generated by this ``universal'' model U T of T .
Having cast the principal logical notions in familiar algebraic terms, it be­
comes possible to apply standard algebraic and functorial techniques to some
classical logical topics, such as interpolation, definability, and completeness.

Source: Andrews, Peter B. - Department of Mathematical Sciences, Carnegie Mellon University

Collections: Mathematics