 
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 MartinL¨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. MartinL¨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 CurryHoward 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, MartinL¨of type theory has been used successfully to formalize
