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Summary: TYPE THEORY AND HOMOTOPY
STEVE AWODEY
1. Introduction
The purpose of this informal survey article is to introduce the reader
to a new and surprising connection between Geometry, Algebra, and
Logic, which has recently come to light in the form of an interpreta-
tion of the constructive type theory of Per Martin-L¨of into homotopy
theory, resulting in new examples of certain algebraic structures which
are important in topology. This connection was discovered quite re-
cently, and various aspects of it are now under active investigation by
several researchers. (See [AW09, AHW09, War08, BG09, GG08, Garar,
GvdB08, Lum09, BG10, Voe06].)
1.1. Type theory. Martin-L¨of type theory is a formal system origi-
nally intended to provide a rigorous framework for constructive mathe-
matics [ML75, ML98, ML84]. It is an extension of the typed -calculus
admitting dependent types and terms. Under the Curry-Howard cor-
respondence [How80], one identifies types with propositions, and terms
with proofs; viewed thus, the system is at least as strong as second-
order logic, and it is known to interpret constructive set theory [Acz74].
Indeed, Martin-L¨of type theory has been used successfully to formalize
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