Univalent foundations and the equivalence principle Benedikt Ahrens Paige North May 31, 2018 In this paper, we explore the `equivalence principle' (EP): roughly, state- ments about mathematical objects should be invariant under an appropriate notion of equivalence of the kinds of objects under consideration. In set theoretic foundations, EP may not always hold: for instance, the statement `1 2 N' is not invariant under isomorphism of sets. In univalent foundations, on the other hand, EP has been proven for many mathematical structures. We first give an overview of earlier attempts at designing foundations that satisfy EP. We then describe how univalent foundations validates EP. Contents 1 The equivalence principle2 2 History 4 3 Univalent foundations and transport of structures along equivalences5 3.1 Indiscernability of identicals in type theory . .6 3.2 From equality to equivalence . .7 3.3 The univalence axiom . .7 4 The equivalence principle for set-level structures8 4.1 Propositions . .8 4.2 Sets . .9 4.3 Monoids . .9 4.4 Univalent categories . 10 5 The equivalence principle for (higher) categorical structures 12 1 1 The equivalence principle What should it mean for two objects x and y to be equal? One proposal by Leibniz [10], known as the \identity of indiscernables", states if x and y have the same properties, then they must be equal: 8 properties P; (P (x) $ P (y)) ! x = y: For this proposal to be reasonable, then the converse, the \indiscernability of identi- cals," should hold incontrovertibly. That is, if x and y are equal, then they must have the same properties: x = y ! 8 properties P; (P (x) $ P (y)) : (1) Indeed, one would be hard-pressed to find a mathematician who disagreed with this principle. However, as with most truths that are taken for granted, this principle is of limited usefulness: in classical mathematics based on set theory, too few objects are equal. A group theorist, for example, would have little interest in a principle which required them to suppose that two groups are equal. Instead, mathematicians are interested in weaker notions of sameness and properties that are invariant under them. A group theorist, for example, would have more interest in an analogous principle that described the properties of any pair of isomorphic groups G and H: G =∼ H ! 8 group theoretic properties P; (P (G) $ P (H)) : Similarly, category theorists would be more interested in a principle that described the properties of any pair of equivalent categories A and B: A ' B ! 8 category theoretic properties P; (P (A) $ P (B)) : To generalize: working mathematicians are interested in a stronger variant of the principle given in line (1) above, namely the equivalence principle for some domain D of mathematics and all objects x and y of study in D: x ∼D y ! 8 D-properties P; (P (x) $ P (y)) ; (2) where ∼D denotes a suitable notion of sameness for the domain D. We might also consider a stronger variant of the equivalence principle. A group the- orist, for example, might not only want properties of groups to be invariant under iso- morphism, but they might also want structures on groups to also be invariant under isomorphism. For example, if the equivalence principle (1) holds and if two groups G and H are isomorphic, then the statements "G has a representation on V " and "H has a representation on V " are equivalent (for some vector space V ). However, it is actually the case that the isomorphism G =∼ H induces a bijection between the set of representations of G on V and the set of representations of H on V . Such a variant 2 of the equivalence principle has become known as the Structure Identity Principle (see [5],[14, Section 9.8]). Our goal in this paper is to describe how one finds the right notion ∼D of sameness and the right class of `D-properties and D-structures' for some specific domains D. This right notion of sameness is not uniformly defined across different mathematical objects. However, we usually use the one already present in mathematical practice since we hope the equivalence principle to capture mathematical practice. As a rule of thumb, it is usually considered to be • equality when the objects naturally form a set|numbers, functions, etc. • isomorphism when the objects naturally form a category|sets, groups, etc. • equivalence when the objects naturally form a bicategory|e.g., categories. The hard part will be in determining the right class of D-properties and D-structures for some specific domain D. In usual mathematical practice, we can state properties which break the equivalence principle; that is, we can state properties of mathematical objects that are not invariant under sameness. We will seek to exclude such properties from our class of D-properties and D-structures. Exercise 1. We denote by 2N the set of even natural numbers. Find a property of sets ∼ that is not invariant under the isomorphism N = 2N given by multiplying and dividing by 2, respectively. n uhsaeeti ie nteabstract. the in given is statement such : One Answer Exercise 2. Find a property of categories that is true for one, but not for the other of these two, equivalent, categories. # • c •'• a xcl n bet"i uhastatement. a such is object." one exactly has C category \The statement : The Answer Thus, to assert an equivalence principle for sets or categories, we need to exclude these properties from our collection of `set theoretic properties' and `category theoretic properties'. M. Makkai [12] says The basic character of the Principle of Isomorphism is that of a constraint on the language of Abstract Mathematics; a welcome one, since it provides for the separation of sense from nonsense. Put differently, establishing an equivalence principle means establishing a syntactic cri- terion for properties and constructions that are invariant under sameness. 3 2 History Look again at Example 2. There, we violated the equivalence principle for categories by referring to equality of objects. This might lead one to conjecture (correctly) that categorical properties which obey the equivalence principle cannot mention equality of objects. However, the traditional definition of category mentions equality of objects. It usually includes the following axiom: for any two morphisms f and g such that the codomain of f equals the domain of g, there is a morphism gf such that domain of gf equals the domain of f and the codomain of gf equals the codomain of g. To avoid mentioning equality of objects, one can express the composability of mor- phisms of that category via different means, specifically by having not one collection of morphisms but many \hom-sets": one for each pair of objects. This idea, for instance explained in [11, Section I.8] usually requires asking the hom-sets to be disjoint. This last requirement is automatic if we work instead in a multi-sorted language, where sorts are automatically disjoint. A category is then given by • a sort O of objects, • for each x; y 2 O, a sort A(x; y) of arrows from x to y, • for each x; y; z 2 O and f 2 A(x; y), g 2 A(y; z), a composite arrow g ◦f 2 A(x; z), and • for each x 2 O, an identity arrow idx 2 A(x; x) such that • for each w; x; y; z 2 O and f 2 A(w; x), g 2 A(x; y), h 2 A(y; z), there is an equality h ◦ (g ◦ f) = (h ◦ g) ◦ f) in A(w; z), • for each x; y 2 O and f 2 A(x; y), there is an equality f ◦ idx = f in A(x; y), and • for each x; y 2 O and f 2 A(x; y), there is an equality idy ◦ f = f in A(x; y). Note that when stating axioms, the only equality that is mentioned is the equality within a homset of the form A(x; y), that is, between arrows of the same sort. By adding quantifiers, ranging over one sort at a time, to this many-sorted language, we obtain a language for stating properties of, and constructions on, categories. It turns out that the statements of that language are invariant under equivalence of categories: Theorem 3 (Th´eor`emede pr´eservationpar ´equivalence [4]). A property of categories (expressed in 2-sorted first order logic) is invariant under equivalence iff it can be ex- pressed in the many-sorted languages sketched above, and without referring to equality of objects. 4 We do not give here the precise form of the many-sorted language, but refer instead to Blanc's article for details. Type theory will provide us with a language very similar to the one sketched here. Note that Freyd [8] states a similar result to Blanc's above, in terms of \diagrammatic properties". Makkai [12] develops a notion of signature and theory, to specify mathematical struc- tures. A theory is a pair (L; Σ) consisting of a signature L (specifying the data of the structure) and a set Σ of \axioms" over L (specifying the axioms of the structure). A theory determines a notion of \model"|which is an L-structure satisfying the properties specified by Σ—and and of \equivalence" of such models, called L-equivalence. His Invariance Theorem gives a result similar to Theorem 3 for models of a theory: Given an interpretation T = (L; Σ) ! S of a FOLDS theory in a FOL theory, an S- sentence φ is invariant under L-equivalence if and only if it is expressible in First Order Logic with Dependent Sorts (FOLDS) over L.