 
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.
