# Homotopy Type Theory: A synthetic approach to higher equalities

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2 For helpful conversations and feedback, I would like to thank (in random order) Emily Riehl, David Corfield, Dimitris Tsementzis, Richard Williamson, Martín Escardó, Andrei Rodin, Urs Schreiber, John Baez, and Steve Awodey; as well as numerous other contributors at the n-category Café and the hott list. 2 -groupoids The word -groupoid looks complicated, but the underlying idea is extremely simple, arising naturally from a careful consideration of what it means for two things to be the same. Specifically, it happens frequently in mathematics that we want to define a collection of objects that are determined by some kind of presentation, but where the same object may have more than one presentation. As a simple example, if we try to define a real number to be an infinite decimal expansion such as π = , we encounter the problem that (for instance) 0.5 = and 0.49 = are distinct decimal expansions but ought to represent the same real number. Therefore, the collection of infinite decimal expansions is not a correct way to define the collection of real numbers. If by collection we mean set in the sense of zfc, then we can handle this by defining a real number to be a set of decimal expansions that all define the same number, and which is maximal in that there are no other expansions that define the same number. Thus, one such set is {0.5, 0.49}, and another is {0.3}. These sets are equivalence classes, and the information about which expansions define the same number is an equivalence relation (a binary relation such that x x, if x y then y x, and if x y and y z then x z). The set of equivalence classes is the quotient of the equivalence relation. Similarly, Frege [10, 68] defined the cardinality of a set X to be (roughly, in modern language) the set of all sets that are related to X by a bijection. Thus for instance 0 is the set of all sets with no elements, 1 is the set of all singleton sets, and so on. These are exactly the equivalence classes for the equivalence relation of bijectiveness. That is, we consider a cardinal number to be presented by a set having that cardinality, with two sets presenting the same cardinal number just when they are bijective. An example outside of pure mathematics involves Einstein s theory of general relativity, in which the universe is represented by a differentiable manifold with a metric structure. General covariance implies that if two manifolds are isomorphic respecting their metric structure (i.e. isometric ), then they represent the same physical reality. Thus we find for instance in [26, 1.3] that A general relativistic gravitational field [(M, g)] is an equivalence class of spacetimes [manifolds M with metrics g] where the equivalence is defined by... isometries. In physics more generally, the phrase gauge invariance refers to this sort of phenomenon, where distinct mathematical entities represent the same physical reality. Definitions by equivalence classes are thus very common in mathematics and its applications, but they are not the only game in town. A different approach to the problem of presentations was proposed by Bishop [6, 1.1]: A set is defined by describing exactly what must be done in order to construct an element of the set and what must be done in order to show that two elements are equal. In other words, according to Bishop, a set is a collection of things together with the information of when two of those things are equal (which must be an equivalence relation). Thus the real numbers would be infinite decimal expansions, but the set of real numbers would include the information that (for instance) 0.5 and 0.49 are the same real number. One advantage of this is that if we are given a real number, we never need to worry about choosing a decimal expansion to represent it. (Of course, for decimal expansions there are canonical ways to make such a choice, but in other examples there are not.) As a much older example of this style of definition, in Euclid s Elements we find: 2

3 Definition 4. Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another. Definition 5. Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order. That is, Euclid first defined how to construct a ratio, and then secondly he defined when two ratios are equal, exactly as Bishop says he ought. On its own, Bishop s conception of set is not a very radical change. But it paves the way for our crucial next step, which is to recognize that frequently there may be more than one reason why two presentations define the same object. For example, there are two bijections between {a, b} and {c, d}, and likewise a pair of manifolds may be isometric in more than one way. This fact lies at the root of Einstein s famous hole argument, which can be explained most clearly as follows. Suppose M and N are manifolds with spacetime metrics g and h respectively, and φ is an isometry between them. Then any point x M corresponds to a unique point φ(x) N, both of which represent the same event in spacetime. Since φ is an isometry, the gravitational field around x in M is identical to that around φ(x) in N. However, if ψ is a different isomorphism from M to N which does not respect the metrics, then the gravitational field around x in M may be quite different from that around ψ(x) in N. So far, this should seem fairly obvious. But Einstein originally considered only the special case where M and N happened to be the same manifold (though not with the same metric), where ψ was the identity map, and where φ was the identity id M outside of a small hole. In this case, it seemed wrong that two metrics could be the same outside the hole but different inside of it. The solution is clear from the more general situation in the previous paragraph: the fact that the two metrics represent the same reality is witnessed by the isomorphism φ, not ψ. Thus, even for a point x inside the hole, we should be comparing g at x with h at φ(x), not with h at id M (x) = x. 2 This and other examples show that it is often essential to remember which isomorphism we are using to treat two objects as the same. The set-theoretic notion of equivalence classes is unable to do this, but Bishop s approach can be generalized to handle it. Indeed, such a generalization is arguably already latent in Bishop s constructive phrasing: both the construction of elements and the proofs of equality are described in terms of what must be done, so it seems evident that just as there may be more than one way to construct an element of a set, there may be more than one way to show that two elements are equal. The laws of an equivalence relation then become algebraic structure on these reasons for equality : given a way in which x = y and a way in which y = z, we must have an induced way in which x = z, and so on, satisfying natural axioms. The resulting structure is called a groupoid. Thus, for instance, spacetime manifolds form a groupoid, in which the ways that M = N are the isometries from M to N (if any exist). If it should happen that for every x and y in some groupoid, there is at most one reason why x = y, then our groupoid is essentially just a set in Bishop s sense; thus the universe of sets is properly included in that of groupoids. This is what happens with decimal expansions: there is only one way in which 0.5 and 0.49 represent the same real number. This should not be confused with the question of whether there is more than one proof that 0.5 = 0.49; the point is rather that in any statement or construction involving 0.5, there is only one way to replace 0.5 by This is in contrast to the situation with manifolds, where using a different isomorphism φ or ψ from M to N can result in different statements, e.g. one which speaks about φ(x) N and another about ψ(x) N. The final step of generalization is to notice that we introduced sets (and generalized them to groupoids) to formalize the idea of collection ; but we have now introduced, for each pair of things x and y in a groupoid, an additional collection, namely the ways in which x and y are equal. Thus, it seems quite that this collection should itself be a set, or more generally a groupoid; so that two ways in which x = y could themselves be equal or not, and perhaps in more than one way. Taken to its logical conclusion, this observation demands an infinite tower consisting of elements, ways in which they are equal, ways in which those are equal, ways in 2 While this description in modern language makes it clear why there is no paradox, it does obscure the reasons why for many years people thought there was a paradox! I will return to this in 7. 3

4 which those are equal, and so on. Together with all the necessary operations that generalize the laws of an equivalence relation, this structure is what we call an -groupoid. This notion may seem very abstruse, but over the past few decades -groupoids have risen to a central role in mathematics and even physics, starting from algebraic topology and metastasizing outwards into commutative algebra, algebraic geometry, differential geometry, gauge field theory, computer science, logic, and even combinatorics. It turns out to be very common that two things can be equal in more than one way. 3 Foundations for mathematics In 2 I introduced the notion of -groupoid informally. At this point a modern mathematician would probably try to give a definition of -groupoid, such as an -groupoid consists of a collection of elements, together with for any two elements x, y a collection of ways in which x = y, and for any two such ways f, g a collection of ways in which f = g, and so on, plus operations... Clearly, any such definition must refer to a prior notion of collection, which a modern mathematician would probably interpret as set. Such definitions of -groupoids are commonly used, although they are quite combinatorially complicated. However, in 2 we considered -groupoids not as defined in terms of sets, but as substitutes or rather generalizations of them. Thus, we should instead seek a theory at roughly the same ontological level as zfc, whose basic objects are -groupoids. This is exactly what hott/uf is: a synthetic theory of -groupoids. The word synthetic here is, as usual, used in opposition to analytic. In modern mathematics, an analytic theory is one whose basic objects are defined in some other theory, whereas a synthetic theory is one whose basic objects are undefined terms given meaning by rules and axioms. For example, analytic geometry defines points and lines in terms of numbers; whereas synthetic geometry is like Euclid s with point and line essentially undefined. Thus, our first step to understanding hott/uf is that it is an axiomatic system in which -groupoid is essentially an undefined term. One advantage of this can already be appreciated: it allows us to say simply that for any two elements x and y of an -groupoid, the ways in which x = y form another -groupoid, so that -groupoids are really the only notion of collection that we need consider. As part of a definition of -groupoid, this would appear circular; but as an axiom, it is unobjectionable. So far, this description of hott/uf could also be applied (with different terminology) to the field of abstract homotopy theory. However, although hott/uf is strongly influenced by homotopy theory, there is more to it: as suggested above, its -groupoids can substitute for sets as a foundation for mathematics. When I say that a synthetic theory can be a foundation for mathematics, I mean simply that we can encode the rest of mathematics into it somehow. 3 This definition of foundation is reasonably precise and objective, and agrees with its common usage by most mathematicians. A computer scientist might describe such a theory as mathematics-complete, by analogy with Turing-complete programming languages (that can simulate all other languages) and NP-complete problems (that can solve all other NP problems). For example, it is commonly accepted that zfc set theory has this property. On the other hand, category theory in its role as an organizing principle for mathematics, though of undoubted philosophical interest, is not foundational in this sense (although a synthetic form of category theory like [16] could be). In particular, a synthetic theory cannot fail to be foundational because some analytic theory describes similar objects. The fact that we can define and study -groupoids inside of set theory says nothing about whether a synthetic theory of -groupoids can be foundational. To the contrary, in fact, it is highly desirable of a new foundational theory that we can translate back and forth to previously existing foundations; among other things it ensures the relative consistency of the new theory. Similarly, we cannot dismiss a new foundational theory because it requires some pre-existing notions : the simple fact of being synthetic means that it does not. Of course, humans always try first to understand new ideas in terms of old ones, but that doesn t make the new ideas intrinsically dependent on the old. A student may learn that dinosaurs are like big lizards, but that doesn t make lizards logically, historically, or genetically prior to dinosaurs. 3 Or into some natural variant or extension of it, such as by making the logic intuitionistic or adding stronger axioms. 4

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