Abstract
In this short note we give a glimpse of homotopy type theory, a new field of mathematics at the intersection of algebraic topology and mathematical logic, and we explain Vladimir Voevodsky’s univalent interpretation of it. This interpretation has given rise to the univalent foundations program, which is the topic of the current special year at the Institute for Advanced Study. The Institute for Advanced Study in Princeton is hosting a special program during the academic year 2012-2013 on a new research theme that is based on recently discovered connections between homotopy theory, a branch of algebraic topology, and type theory, a branch of mathematical logic and theoretical computer science. In this brief paper our goal is to take a glance at these developments. For those readers who would like to learn more about them, we recommend a number of references throughout. Type theory was invented by Bertrand Russell [20], but it was first developed as a rigorous formal system by Alonzo Church [3, 4, 5]. It now has numerous applications in computer science, especially in the theory of programming languages [19]. Per Martin-Lof [15, 11, 13, 14], among others, developed a generalization of Church’s system which is now usually called dependent, constructive, or simply Martin-Lof type theory; this is the system that we consider here. It was originally intended as a rigorous framework for constructive mathematics. In type theory objects are classified using a primitive notion of type, similar to the data-types used in programming languages. And as in programming languages, these elaborately structured types can be used to express detailed specifications of the objects classified, giving rise to principles of reasoning about them. To take a simple example, the objects of a product type A × B are known to be of the form 〈a, b〉, and so one automatically knows how to form them and how to decompose them. This aspect of type
Highlights
The Institute for Advanced Study in Princeton is hosting a special program during the academic year 2012–2013 on a new research theme that is based on recently discovered connections between homotopy theory, a branch of algebraic topology, and type theory, a branch of mathematical logic and theoretical computer science
Type theory was invented by Bertrand Russell [20], but it was first developed as a rigorous formal system by Alonzo Church [3], [4], [5]
Per Martin-Löf [15], [11], [13], [14], among others, developed a generalization of Church’s system which is usually called dependent, constructive, or Martin–Löf type theory; this is the system that we consider here. It was originally intended as a rigorous framework for constructive mathematics
Summary
This difficulty has been solved by the idea of regarding types not as strange sets (perhaps constructed without using classical logic) but as spaces, regarded from the perspective of homotopy theory. The key new idea of the homotopy interpretation is that the logical notion of identity a = b of two objects a, b : A of the same type A can be understood as the existence of a path p : a b from point a to point b in the space A.
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