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Propositions as [Types] Steve Awodey Andrej Bauer y

Summary: Propositions as [Types]
Steve Awodey  Andrej Bauer y
Institut Mittag-Leer
The Royal Swedish Academy of Sciences
June 2001
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.


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


Collections: Mathematics