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Home , Axiom, Homotopy type theory, Type theory

Voevodsky’s Univalence in Type

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 of on recently discovered connections be- reasoning about them. To take a simple example, Ttween , a branch of algebraic the objects of a type A × B are known to , and , a branch of mathemati- be of the form ha, bi, and so one automatically cal and theoretical computer . 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 also form the basis of throughout. of modern computer assistants, which are Type theory was invented by used for formalizing and verifying [20], but it was first developed as a rigorous the correctness of formalized proofs. For example, by [3], [4], [5]. It now the powerful [6] has recently has numerous applications in , 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 [7]. others, developed a generalization of Church’s One problem with understanding type theory system which is now usually called dependent, from a mathematical 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 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 of regarding types not as strange sets In type theory objects are classified using a (perhaps constructed without using ) primitive of type, similar to the data types but as , regarded from the perspective of homotopy theory. Steve Awodey is professor of 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 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 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 , which he termed univalence and which same algebraic invariants (e.g., , or the is not usually assumed in type theory. Adding fundamental ) 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 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 U, the terms of which are themselves point-set topology or even define 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 , the small types. Like any type, U has an identity such as ∞- 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 . spaces, U is a , 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 p : A Ž B between spaces in namely that the 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 more precisely, to “a is an 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 to B. Since the identity map on any 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 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 of a path p : a Ž 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) Ž 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 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]. proof assistants based on type theory, like the One of Voevodsky’s goals (as we understand it) is Coq system, to develop mathematics involving that, in a not too distant future, homotopy theory, to verify the correctness of will be able to verify the correctness of their own proofs, and even to provide some degree of papers by working within the system of univa- automation of proofs. lent foundations formalized in a proof assistant To give just one example, in homotopy type and that doing so will become natural even for theory one can directly define the n-dimensional pure mathematicians (the same way that most Sn as a type, with its associated principles mathematicians now typeset their own papers in of reasoning. Moreover, for any type A one can TEX). We believe that this aspect of the univalent define the homotopy groups πn(A), again in a foundations program distinguishes it from other very direct way in terms of the identity type IdA approaches to foundations by providing a practical explained above. One can then reason directly in utility for the working . type theory, using the principles associated with Our goal in this announcement has been to give these constructions, and prove, for example, that a brief and intentionally superficial glimpse of two n πn(S ) = Z for n ≥ 1 (as has recently been done closely related recent developments: homotopy by G. Brunerie and D. Licata at the Institute for type theory and Voevodsky’s Advanced Study, using the Univalence Axiom in an program. Since these subjects are still developing essential way). Finally, the proof can be formalized quite rapidly, the current literature tends to be in a proof assistant and verified by a computer. In rather specialized and accessible mainly to those this way, one not only has new methods of proof in with prior knowledge of homotopy theory and classical homotopy theory, but indeed ones which logic. One exception is the survey article [18], provide associated computational tools. which goes into much greater depth than the Voevodsky has christened this combination present article, while still intended for a of homotopy type theory with the Univalence general mathematical readership; it also contains Axiom, implemented on a computer proof assistant, an introduction to the use of the Coq proof assistant the Univalent Foundations program. It can be in the univalent setting. A complete exposition regarded as a new foundation for mathematics of the current state of the art in homotopy type in general, not just for homotopy theory, as theory is available in the form of a book which Voevodsky has shown by developing an extensive was jointly authored by the participants of the IAS code of formalized mathematics in this special year and is freely available at [10]. See also setting. Moreover, he is promoting more interaction [1], and [23]. between pure mathematicians and the developers of such proof assistants, as is occurring in the Acknowledgments special year on Univalent Foundations at the We thank the Institute for Advanced Study for the Institute for Advanced Study. excellent resources which have been made available For those interested in contributing to this to the authors during the preparation of this article. new of mathematics, it may be encouraging We thank Thierry Coquand, Dan Grayson, and to know that there are many interesting open for useful discussions on the questions. The most pressing of them is perhaps topic of this paper, and we thank the referees for the “constructivity” of the Univalence Axiom itself, helpful suggestions. Awodey is partly supported conjectured by Voevodsky in [23]. It concerns the by NSF Grant DMS-1001191 and AFOSR Grant effect of adding the Univalence Axiom on the 11NL035, and was supported by the Friends of computational behavior of the system of type the Institute for Advanced Study and the Charles theory and thus on the existing proof assistants. Simonyi Endowment. Pelayo is partly supported Another major direction, of course, is the further by NSF CAREER Grant DMS-1055897, NSF Grant formalization of classical results and current DMS-0635607, and Spain Ministry of Science Grant mathematical research in the univalent setting. Sev-2011-0087. Warren is supported by the Oswald We expect that it will eventually be possible to Veblen Fund. formalize large amounts of modern mathematics in this setting and that doing so will give rise to both theoretical insights and good numerical References (extracted from code in a proof assistant). [1] S. Awodey, Type theory and homotopy, on the arXiv as arXiv:1010:1810v1, 2010. In this direction, together with Voevodsky, the [2] S. Awodey and M. A. Warren, Homotopy theoretic last two authors are working on an approach to the models of identity types, Mathematical Proceedings theory of integrable systems (using the new notion of the Cambridge Philosophical Society 146 (2009), of p-adic integrable system as a test case) in the no. 1, 45–55.

1166 Notices of the AMS Volume 60, Number 9 [3] A. Church, A set of postulates for the foundation of logic, Annals of Mathematics. Second Series 34 (1933), no. 4, 839–864. [4] , A formulation of the simple theory of types, Journal of Symbolic Logic 5 (1940), 56–68. [5] , The Calculi of Lambda-Conversion, Annals of Mathematics Studies, no. 6, Princeton University Press, Princeton, N. J., 1941. [6] http://coq.inria.fr. [7] http://research.microsoft.com/en-us/news/ features/gonthierproof-101112.aspx. Call for [8] T. C. Hales, , Notices of the American Mathematical Society 55 (2008), no. 11, 1370–1380. [9] M. Hofmann and T. Streicher, The in- terpretation of type theory, Twenty-five Years of Professors and Constructive Type Theory (Venice, 1995), Oxford Univ. Press, New York, 1998, pp. 83–111. Assistant Professors [10] http://homotopytypetheory.org/book. [11] P. Martin-Löf, An intuitionistic theory of types: Pred- icative part, Logic Colloquium 1973 (Bristol, 1973), North-Holland, Amsterdam, 1975, pp. 73–118. Stud- IST Austria invites applications for tenured and ies in Logic and the Foundations of Mathematics, Vol. tenure-track leaders of independent research 80. [12] , Constructive mathematics and computer pro- groups in following fields: gramming. In Proceedings of the 6th International Congress for Logic, and Philosophy of I I Science, North-Holland, Amsterdam, 1979. Mathematics Computer Science [13] , Constructive mathematics and computer Physics I Chemistry I Biology I programming. In Logic, Methodology and Philoso- phy of Science, VI (Hannover, 1979), North-Holland, Neuroscience I Earth Science I Amsterdam, 1982, pp. 153–175. Interdisciplinary [14] , Intuitionistic Type Theory, Studies in . Lecture Notes, vol. 1, Bibliopolis, Naples, 1984. The Institute is dedicated to basic research and [15] , An intuitionistic theory of types, Twenty-five graduate education in the natural and formal Years of Constructive Type Theory (Venice, 1995), . The successful candidates will receive a Oxford Univ. Press, New York, 1998, originally a 1972 substantial annual research budget, are expected preprint from the Department of Mathematics at the to apply for external research grants and to partici- University of Stockholm, pp. 127–172. [16] B. Nordström, K. Petersson, and J. M. Smith, pate in the Graduate School. Programming in Martin-Löf’s Type Theory, An Introduction, Oxford University Press, 1990. [17] Á. Pelayo, V. Voevodsky, M. A. Warren, A prelimi- Deadline for receiving Assistant Professor nary univalent formalization of the p-adic numbers, submitted, on the arXiv as arXiv:math/1302.1207, applications: November 15, 2013 2013. [18] Á. Pelayo and M. A. Warren, Homotopy type theory Open call for Professor applications and Voevodsky’s univalent foundations, submitted, on the arXiv as arXiv:math/1210.5658, 2012. Further and online application: [19] B. C. Pierce, Types and Programming Languages, MIT Press, Cambridge, MA, 2002. www.ist.ac.at/professor-applications [20] B. Russell, as based on the theory of types, American Journal of Mathematics 30 (1908), 222–262. IST Austria values diversity and is committed to equality. [21] C. Simpson, Computer theorem proving in mathe- Female researchers are encouraged to apply. matics, Letters in 69 (2004), 287–315. [22] V. Voevodsky, Notes on type systems, unpub- lished notes, http://www.math.ias.edu/~vladimir/ Site3/Univalent_Foundations.html, 2009. [23] , Univalent foundations project, modified version of an NSF grant application, http://www. math.ias.edu / ~vladimir / Site3 / Univalent_ Foundations.html, 2010.

October 2013 Notices of the AMS 1167