Voevodsky’s Univalence Axiom in Homotopy Type Theory Steve Awodey, Álvaro Pelayo, and Michael A. Warren he Institute for Advanced Study in used in programming languages. As in program- Princeton is hosting a special program ming languages, these elaborately structured types during the academic year 2012–2013 can be used to express detailed specifications of on a new research theme that is based the objects classified, giving rise to principles of on recently discovered connections be- reasoning about them. To take a simple example, Ttween homotopy theory, a branch of algebraic the objects of a product type A × B are known to topology, and type theory, a branch of mathemati- be of the form ha; bi, and so one automatically cal logic and theoretical computer science. In this knows how to form them and how to decompose brief paper our goal is to take a glance at these them. This aspect of type theory has led to its developments. For those readers who would like to extensive use in verifying the correctness of com- learn more about them, we recommend a number puter programs. Type theories also form the basis of references throughout. of modern computer proof assistants, which are Type theory was invented by Bertrand Russell used for formalizing mathematics and verifying [20], but it was first developed as a rigorous the correctness of formalized proofs. For example, formal system by Alonzo Church [3], [4], [5]. It now the powerful Coq proof assistant [6] has recently has numerous applications in computer science, been used to formalize and verify the correctness especially in the theory of programming languages of the proof of the celebrated Feit-Thompson odd [19]. Per Martin-Löf [15], [11], [13], [14], among order theorem [7]. others, developed a generalization of Church’s One problem with understanding type theory system which is now usually called dependent, from a mathematical point of view, however, has constructive, or simply Martin–Löf type theory; always been that the basic concept of type is unlike this is the system that we consider here. It was that of set in ways that have been hard to make originally intended as a rigorous framework for precise. This difficulty has now been solved by constructive mathematics. the idea of regarding types not as strange sets In type theory objects are classified using a (perhaps constructed without using classical logic) primitive notion of type, similar to the data types but as spaces, regarded from the perspective of homotopy theory. Steve Awodey is professor of philosophy at Carnegie Mel- In homotopy theory one is concerned with lon University and a member of the Institute for Advanced spaces and continuous mappings between them, Study School of Mathematics (2012–2013). His email ad- dress is [email protected]. up to homotopy; a homotopy between a pair of continuous maps f : X ! Y and g : X ! Y is Álvaro Pelayo is assistant professor of mathematics at Wash- a continuous map H : X × [0; 1] ! Y satisfying ington University in St. Louis and a member of the Institute for Advanced Study School of Mathematics (December 2010– H(x, 0) = f (x) and H(x, 1) = g(x). The homotopy August 2013). His email address is [email protected]. H may be thought of as a “continuous deformation” edu. of f into g. The spaces X and Y are said to be Michael A. Warren is professor of mathematics at the homotopy equivalent, X ' Y , if there are continuous Institute for Advanced Study School of Mathematics (2011– maps going back and forth, the composites of 2013). His email address is [email protected]. which are homotopical to the respective identity DOI: http://dx.doi.org/10.1090/noti1043 mappings, i.e., if they are isomorphic “up to 1164 Notices of the AMS Volume 60, Number 9 homotopy". Homotopy equivalent spaces have the property, which he termed univalence and which same algebraic invariants (e.g., homology, or the is not usually assumed in type theory. Adding fundamental group) and are said to have the same univalence to type theory in the form of a new homotopy type. axiom has far-reaching consequences, many of Homotopy type theory is a new field of math- which are natural, simplifying, and compelling. The ematics which interprets type theory from a Univalence Axiom thus further strengthens the homotopical perspective. In homotopy type theory, homotopical view of type theory since it holds in one regards the types as spaces, or homotopy types, the simplicial model but fails in the view of types and the logical constructions (such as the product as sets. A × B) as homotopy-invariant constructions on The basic idea of the Univalence Axiom can be spaces. In this way, one is able to manipulate explained as follows. In type theory, one can have spaces directly, without first having to develop a universe U, the terms of which are themselves point-set topology or even define the real numbers. types, A : U, etc. Of course, we do not have U : U, Homotopy type theory is connected to several so only some types are terms of U—call these topics of interest in modern algebraic topology, the small types. Like any type, U has an identity such as 1-groupoids and Quillen model structures type IdU, which expresses the identity relation (see [18]); we will only mention one simple example A = B among small types. Thinking of types as below, namely the homotopy groups of spheres. spaces, U is a space, the points of which are To briefly explain the homotopical perspective spaces. To understand its identity type, we must of types, consider the basic concept of type theory, ask, “What is a path p : A B between spaces in namely that the term a is of type A, which is U?” The Univalence Axiom says that such paths written correspond to homotopy equivalences A ' B, as a : A: explained above (the actual notion of equivalence This expression is traditionally thought of as akin required is slightly different). A bit more precisely, to “a is an element of the set A”. However, in given any (small) types A and B, in addition to homotopy type theory we think of it instead as “a the type IdU(A, B) of identities between A and B is a point of the space A”. Similarly, every term there is the type Eq(A, B) of equivalences from A f : A ! B is regarded as a continuous function to B. Since the identity map on any object is an from the space A to the space B. equivalence, there is a canonical map, This perspective clarifies features of type theory IdU(A, B) ! Eq(A, B). which were puzzling from the perspective of types The Univalence Axiom states that this map is itself as sets, for instance, that one can have nontrivial an equivalence. At the risk of oversimplifying, we types X such that (X ! X) X + 1. But the key new idea of the homotopy interpretation is that can state this succinctly as the logical notion of identity a = b of two objects Univalence Axiom: (A = B) ' (A ' B). a; b : A of the same type A can be understood as In other words, identity is equivalent to equivalence. the existence of a path p : a b from point a From the homotopical point of view, this says to point b in the space A. This also means that that the universe U is something like a classifying two functions f ; g : A ! B are identical just in space for (small) homotopy types, which is a practi- case they are homotopic, since a homotopy is just cal and natural assumption. From the logical point a family of paths px : f (x) g(x) in B, one for of view, however, it is revolutionary: it says that each x : A. In type theory, for every type A there isomorphic things can be identified! Mathemati- is a (formerly somewhat mysterious) type IdA cians are, of course, used to identifying isomorphic of identities between objects of A; in homotopy structures in practice, but they generally do so type theory, this is just the path space AI of all with a wink, knowing that the identification is not continuous maps I ! A from the unit interval. (See “officially” justified by foundations. But in this new [2], [1], [18].) foundational scheme, not only are such structures At around the same time that Awodey and formally identified, but the different ways in which Warren advanced the idea of homotopy type theory, such identifications may be made themselves form Voevodsky showed how to model type theory using a structure that one can (and should!) take into Kan simplicial sets, a familiar setting for classical account. homotopy theory, thus arriving independently Part of the appeal of homotopy type theory at essentially the same idea around 2005. Both with the Univalence Axiom is the many interesting were inspired by the prior work of Hofmann and connections it reveals between logic and homotopy. Streicher, who had constructed a model of type Another remarkable aspect is that it can be carried theory using groupoids [9]. out in a computer proof assistant since type theory Voevodsky, moreover, recognized that this exhibits such good computational properties (see simplicial interpretation satisfies a further crucial [21], [8] on the use of computer proof assistants October 2013 Notices of the AMS 1165 in general). In practical terms, this means that it is univalent setting. A preliminary treatment in the possible to use the powerful, currently available construction of the p–adic numbers is given in [17].