 
Summary: Propositions as [Types]
Steve Awodey Andrej Bauer y
Institut MittagLeer
The Royal Swedish Academy of Sciences
June 2001
Abstract
Image factorizations in regular categories are stable under pull
backs, so they model a natural modal operator in dependent type the
ory. This unary type constructor [A] has turned up previously in a
syntactic form as a way of erasing computational content, and formal
izing a notion of proof irrelevance. Indeed, semantically, the notion of
a support is sometimes used as surrogate proposition asserting inhab
itation of an indexed family.
We give rules for bracket types in dependent type theory and pro
vide complete semantics using regular categories. We show that depen
dent type theory with the unit type, strong extensional equality types,
strong dependent sums, and bracket types is the internal type theory
of regular categories, in the same way that the usual dependent type
theory with dependent sums and products is the internal type theory
of locally cartesian closed categories.
