Univalence and Ontic Structuralism

Abstract

The persistent challenge of formulating ontic structuralism in a rigorous manner, which prioritizes structures over the entities they contain, calls for a transformation of traditional logical frameworks. I argue that Univalent Foundations (UF), which feature the axiom that all isomorphic structures are identical, offer such a foundation and are more attractive than other proposed structuralist frameworks. Furthermore, I delve into the significance in the case of the hole argument and, very briefly, the nature of symmetries.

1 Introduction

Homotopy Type Theory (HoTT) with the univalence axiom, together known as ‘Univalent Foundations’ (UF), is a novel contender in the ongoing search for the best logico-mathematical foundations for philosophy and science, which is developed by the Univalent Foundations Program [44].¹ The exploration of its relevance to philosophy is currently limited to mathematical structuralism and its application to the hole argument (see [2, 13, 24, 42]). I believe that it is worth delving deeper into the significance of UF, including in the case of the hole argument. But it is particularly important to examine UF’s relationship with ontic structuralism, a metaphysical doctrine that roughly says that objects do not have identities over and beyond the structure they occupy. This would help clarify what UF can uniquely contribute to many topics in philosophy of physics, including the hole argument and symmetries.

But first, what is UF and its univalence axiom? UF is a radical departure from first-order predicate logic and standard set theory ZF(C) that largely define the landscape of metaphysics. It is a type theory where every entity has a type. Unlike simple type theory, types are part of the object language.² They are like sets in some respects such as consisting of elements (or otherwise being empty). But they are unlike sets in other respects. For one thing, they can have additional structures in the form of ‘higher-order’ elements (technically, they are conceptualized as homotopy spaces or $\infty$-groupoids, hence the name ‘homotopy type theory’; see Sect. 2). The univalence axiom further says that equivalent types are identical. This can be shown to entail that isomorphic structures are identical (here, the notions of ‘structure’ and ‘isomorphism’ are formally defined in HoTT in a way that captures their more familiar meanings).

The relation between the univalence axiom with mathematical structuralism is clear, since the latter is often defined by that isomorphic mathematical structures are identical. [42] further argues that UF is the only feasible structuralist foundation for mathematics among the existent ones. A natural further step is to argue that UF is also a structuralist foundation for science, where structuralism here concerns physical entities. Thus understood, UF is a foundational implementation of ontic structuralism. This is desirable because ontic structuralism has faced criticisms that it is either incoherent or indistinguishable from its traditional rivals (see [26]). The standard framework’s limitations in formulating ontic structuralism as a coherent and distinct doctrine necessitate a transformation in the very foundation. As [37] puts it:

What basic notions is the structural realist proposing? What are the proposed rules governing those notions? And how can those notions then be used in a foundational account of scientific theories?... You need to properly specify a replacement framework, some replacement inventory of basic notions, rules governing those notions, and methods for using those notions in foundational contexts. (p.64)

The core task of this paper is to point out that UF is an answer to all these questions, namely that it is a rigorous logical framework that implements the idea of ontic structuralism and promises to found science. In Sect. 3, I will compare this approach with two other noteworthy attempts, which are respectively generalism proposed by [10] and algebraic structuralism by [11]. I will argue that they face similar problems. The problems concern their inadequacy to dispense with distinct isomorphic structures and higher-order individuals. UF does not face these problems.³ Note that I advance UF specifically as an answer to Sider’s questions above, not as a solution to all problems of ontic structuralism. In particular, I am not concerned with the question of what structures we should commit ourselves to given our best scientific theories.⁴

In Sect. 4, I will consider and reply to a few objections. One objection is about how we should interpret the notion of identity in the univalence axiom. The other two objections are against ontic structuralism in general: the collapse problem of mathematical and physical structures, and undesirable holism mentioned by [37]. The upshot is that UF is at least as resourceful as other structuralist approaches in response to these problems.

Once UF is viewed as a viable structuralist foundation, its relevance to significant issues in philosophy of physics including the hole argument and symmetries in general becomes clearer (Sect. 5). The hole argument, as commonly formulated, relies on the premise that the model $\langle M, g\rangle$ is distinct from $\langle M, g'\rangle$, where M is a manifold, g is a metric field on M, and $g'$ is g carried by a non-identity diffeomorphism from M to itself. [24] have noted that the univalence axiom implies that the diffeomorphically related models are identical, which undermines the hole argument. However, one may ask how this differs from other ‘mathematical solutions’ that also reject this premise. For example, [47] argues that isomorphic models have the same representational capacity. As he puts it, the hole argument arises from ‘a misleading use of the mathematical formalism of general relativity’ ([47], p. 2). However, various authors have pointed out that this does not solve the hole argument, because the latter is about metaphysical possibilities rather than mathematical models [32, 40]. To address this, we should generalize mathematical structuralism in the mathematical solution to ontic structuralism. UF is precisely a structuralist framework for science that blocks the hole argument (which is different from the semantic stipulation imposed by fiat from outside of a formal system in Weatherall’s approach).

Additionally, the univalence axiom can be employed to address symmetry-related models more broadly. Various authors have argued that such models should be reformulated as isomorphic ones (see [12]; see also [46] and [45]).⁵ Under the univalence axiom, these isomorphic models are identified, which enforces that symmetry-related models represent the same physical situation. This sheds further light on symmetries as result of representational redundacies, albeit very useful ones.

However, given how exotic UF is in comparison with the standard framework, questions naturally arise about whether the benefits of UF are worth the tradeoff with its unfamiliarity.⁶ Nevertheless, the exoticness is rather proportional to its significance as an answer to the longstanding tension between our standard logico-mathematical foundation and increasingly structuralist approach to mathematics, science and philosophy. Given that UF is the only serious structuralist foundation so far, it is not an overtreatment for a small ailment.

2 An Intuitive Guide to Univalence

In this section, I will explain the univalence axiom featured by Univalent Foundations (UF) in preparation for the upcoming discussions. My purpose is to help readers understand the univalence axiom (and the involved notions) as efficiently as possible. My strategy is to primarily focus on the semantics based on homotopy theory as a heuristic guide, rather than the syntax. (The syntax can be rather technical in comparison. For other expositions on HoTT and UF, see for example [23, 39], and the main reference book [44].)

As its name indicates, HoTT not only is a type theory where every entity has a type but also features homotopy types. To understand this, let’s start with the familiar notion of a topological space, which is invariant under certain continuous deformation such as stretching, shrinking, and bending, but not gluing and tearing. For example, an ordinary cup is topologically equivalent (namely, ‘homeomorphic’) to a simple wedding ring. Every topological space is associated with an algebraic structure called ‘$\infty$-groupoid’, which contains elements of different levels: its 0-elements are points of the space, its 1-elements are paths between the points, the 2-elements are paths between paths (called ‘homotopies’), the next level consists of homotopies between homotopies, and so on ad infinitum, without distinguishing elements connected by paths. If we only keep this information, we obtain a ‘homotopy space’. Intuitively, two spaces are homotopically equivalent if they can be continuously deformed into each other. For instance, a disk is homotopically equivalent to a point. The notion of a homotopy space may sound the same as that of a topological space. Indeed they are closely related, but they are not the same: a homotopical structure typically has less information than the corresponding topological structure; the deformations that the latter permits are more restricted. A disk is not homeomorphic to a point (because there is no bijection between the underlying point sets) though they are homotopically equivalent. As another often-cited example, a Möbius strip is topologically distinct from a cylindrical strip, but homotopically equivalent to the latter (both can be deformed into a circle).

Standardly, a homotopy space or a groupoid is defined set-theoretically: a path between two points in a topological space is a continuous map from interval [0,1] to the space that maps 0 and 1 to the two points respectively, and similarly for homotopies. But this does not have to be the case. Instead, we can treat these notions (i.e., paths, homotopies, etc.) synthetically: they can be basic, unanalysable notions directly governed by axioms and inference rules. (As an analogy, analytic geometry reduces a line to a set of points satisfying various conditions, while synthetic geometry such as Euclid’s original formulation of Euclidean geometry treats the notion of line as primitive governed by his basic postulates.) HoTT (or UF) is exactly an axiomatic deductive system where homotopy spaces or groupoids and their high-order elements are basic objects, irreducible to their 0-elements.⁷ The trick to achieve this involves the identity types in HoTT. Suppose a, b are elements of the same type A. Then the expression ‘$a=_Ab$’ itself refers to a type, which can have more than one elements, each of which can be thought as a way of identification. These elements can further form identity types, which in turn have elements that can be identified, and so on ad infinitum. This precisely corresponds to the structure of a homotopy space or a $\infty$-groupoid, where ‘path-connectedness’ is reconceptualized as ‘identification’, paths become ways of identification, and higher homotopies become higher-order identifications. In this sense, every type is modelled by a homotopy space—hence the name ‘homotopy type’.

In other words, homotopy spaces constitute an interpretation for HoTT (or UF).⁸ Are these interpretations literal or heuristic? That is, are statements in UF really statements about homotopy spaces so that UF is in fact a peculiarly axiomatized mathematical theory? Or, is the homotopy-theoretic interpretation just a tool to prove the consistency of UF and connect UF with classical results in homotopy theory? While I don’t have a decisive answer, I can at least say that we are not obligated to think UF as just a heuristic reasoning tool for classical homotopy theory. It is perfectly intelligible as an autonomous foundation without appealing to the homotopical interpretations (see [23]; also see [4, 43] for different perspectives). Moreover, the identity types and the univalence axiom are not exclusively motivated by the homotopy-theoretic interpretation. It is worth mentioning that historically, HoTT did not originate as an axiomatization of $\infty$-groupoids—on the contrary, groupoids are used to informally conceptualize identity types in type theory with more than one elements, which are independently conceived.

HoTT (or UF) can serve as a logical foundation for science. An important underlying observation is that a version of set theory can be modeled in HoTT (see Appendix A for more detail). A set is a type that contains no higher homotopical information, called a ‘0-truncated’ type, which has all information beyond its immediate elements discarded. In HoTT, logical axioms and rules are also subsumed under the rules for type constructions rather than being imposed externally, unlike in set theory. Mere propositions are considered ‘-1-truncated’ type, meaning that it is a set with at most one element (when a proposition is true, it has exactly one element, which is sometimes called its ‘(truth) certificate’, ‘proof’ or ‘witness’, and none if it is false). Standard propositional deduction rules can be derived from type construction rules when the types in question are propositions. In this sense, UF is a unified foundation aiming to replace both first-order predicate logic and set theory.

Finally, let’s turn to the univalence axiom featured by UF. Formally, the axiom says:

Univalence. For any two types $A,B: \mathcal {U}$, $(A=_\mathcal {U} B)\simeq (A\simeq B)$.⁹

In ordinary English, this says that the identity between A and B is equivalent to the equivalence between them, or in short, equivalent types are identical. This may sound nonsensical, but we can make sense of it using the homotopy-theoretic interpretation. The notion of equivalence between types is precisely that of homotopical equivalence between spaces (except that, of course, it is defined in the language of HoTT rather than classically), and it is a natural idea to identify equivalent spaces.¹⁰

3 Foundation of Structuralism

Ontic structuralism can be informally characterized as the view that structure is more fundamental than objects. But it is unclear how the view can be made more precise. Various attempts at it have been criticized as ‘incoherent’ or ‘indistinguishable from its supposed rivals’ [26]. The incoherence charge is largely against eliminativism, an approach to ontic structuralism which says that there are no individuals or objects but only structures at the fundamental level (see alse Footnote 3). In contrast, the UF approach that I will propose is non-eliminative. I propose that the following principle holds, which is understood in the framework of UF:

The Identity Thesis (Identity). Fundamentally, all isomorphic structures are identical.

This approach does not eliminate objects at the fundamental level: a structure consists of objects and their structural relations (see Appendix B).¹¹ Rather, objects are intrinsically characterless—they are exhaustively characterized by the structure they are in. Merely permuting which objects instantiate what relations in a structure would not lead to any real change, but merely notational. So we can consider this approach as a version of priority-based structuralism, according to which both structure and objects exist but the former is more fundamental than the latter.¹² Indeed, we can think of types as structures of a minimal kind, and they are more fundamental than their elements.¹³

I will advance UF-implemented Identity as a response to [37] complaint that ontic structuralism lacks a rigorous logical framework. In particular, I will compare it with generalism by [10] and algebraic structuralism by [11], both of which are explicitly proposed as formal frameworks for ontic structuralism, and both are eliminativist. I argue that UF avoids the problems with these frameworks.

3.1 Generalism

Generalism is based on predicate functor logic developed by [33] and others, the syntactical categories of which do not include variables or constants for individuals. Instead, predicates are considered atomic terms. For example, $L^2$ replaces the standard formula Lxy (‘x loves y’), where the superscript indicates the number of arguments of its standard counterpart. Furthermore, we have a handful of predicate functors that operate on predicates (including standard logical connectives $\lnot$ and $\wedge$). For example, if we want to say $\exists x\exists yLxy$ in standard logic, we simply say in predicate functor logic ‘ccL’, where c is a functor that behaves as if it binds the first free variable in the formula in its scope. As another example, the standard formula $\forall x(Cx\rightarrow \exists yLxy)$ (‘every cat loves something’) can be expressed as $\lnot c(C^1\wedge \lnot c\lnot \sigma L^2)$, where the operator $\sigma$ permutes the arguments of a predicate (for otherwise the expression would mean ‘every cat is loved by something’). Predicates and functors are the only syntactical categories other than logical ones.

Importantly, we can show that algebraic generalism is expressively equivalent to first-order predicate logic without constants, which is called quantifier generalism (see Appendix in [10]. This is not surprising, since predicate functor logic is designed to be an algebraization of first-order predicate logic. The translation scheme between the two systems is laid out in [33] and other authors. Axiomatic set theory ZFC, the standard foundation for mathematics, is expressible in first-order predicate logic, and is therefore expressible in predicate functor logic or generalism. In this sense, generalism can serve as a foundational framework for science (albeit not convenient to use—the interested readers can try translating the axiom of extensionality).

Consider a model $\langle D, v\rangle$, which is a structure, where v is a set of relations defined on elements of D. We say that $\langle D, v\rangle$ and $\langle D', v'\rangle$ are isomorphic if there is a bijective map between D and $D'$ that preserves all the relations.¹⁴ Under generalism, we can still define distinct isomorphic models without constants (even as part of normal practice). For instance, let D consist of natural numbers 0 and 1 and v consists of a single relation holds of 0 and itself alone. We can easily define a distinct isomorphic structure $\langle D, v'\rangle$ where $v'$ consists of the same relation holding of 1 and itself alone. Note that we do not need constants to express natural numbers, since they are definable in standard set theory as different sets (such as through von Neumann reduction). Such examples are abundant. In general, the apparatus of set theory allows us to represent the elements of a structure in many different ways. Algebraic generalism has the expressible power of set theory in standard logic, thus it is capable of distinguishing isomorphic structures that generalists aspire to identify.

3.2 Algebraic Structuralism

[11] complains that generalism does not really dispense with individuals because the semantics for the framework is the same as that for the first-order predicate logic.¹⁵ That is, we use models $\mathcal {M}=\langle D, v\rangle$ to interpret sentences formulated in generalism, where D is the domain of individuals.

As an alternative, Dewar proposes algebraic structuralism. Instead of eliminating syntactical categories of variables and constants for individuals, Dewar seeks to eliminate the individuals in the semantics of theories (I’ll come back to comment on this transition from syntax to semantics). Again, consider a standard first-order model $\langle D, v\rangle$. In the set-theoretic framework, all the relations in v are definable as set-theoretic constructions based on D. Now, D is a domain of individuals, which we want to get rid of. Can we reconceptualize the relations in v without resorting to elements of D? Yes. For simplicity, suppose v has only monadic properties, which are construed as subsets of D. Now, we can reconceptualize those properties as elements of a Boolean algebra defined by algebraic operators including disjunction (or ‘addition’), conjunction (or ‘multiplication’), and negation. These operators are governed by axioms of Boolean algebras rather than defined through set-theoretic constructions. From such a Boolean algebra, we can recover the set-theoretic definitions of monadic properties, so we can ensure that this reconceptualization does not lose information. This idea can generalize to arbitrary first-order model $\langle D, v\rangle$ where v has more than monadic properties. The resulting algebras are called ‘cylindrical algebras’. Shortly put, Dewar’s algebraic structuralism amounts to an algebraization of standard first-order models.

In philosophy of physics, this idea is implemented more directly through algebraizing relevant models, such as spacetime models. Instead of taking manifolds as fundamental and defining physical fields on them (which can be considered as distributions of qualitative features over spacetime points), we can take fields as primitive entities without an underlying manifold—call this algebraicism (see [7, 17, 34]). This works because the fields can encode all necessary information about manifolds for doing physics up to general relativity and arguably also quantum field theory.

However, adopting algebraic structuralism or algebraicism that dispenses with spacetime points does not solve the problem of individuals in principle, nor do they satisfy Identity. Under algebraic structuralism, models are now in the form of $\langle R, O\rangle$, where R is a set of properties and relations and O is a set of operators on them. But this is formally the same as $\langle D, v\rangle$, and the difference mainly concerns whether D contains ordinary individuals or ordinary properties and relations (or rather tropes since we are really talking about instances of properties and relations). The problem is most salient in algebraicism where physical fields are the new individuals. It is long observed in the literature that we can just as easily come up with isomorphic models consisting of physical fields (see [35]).¹⁶ As another example, suppose being negatively charged and positively charged are completely symmetric in their nomic roles so that exchanging them has no empirical consequences. Since they are expressed as distinct members of the domain in $\langle R, O\rangle$, the algebraic formalism still entails that exchanging them results in a distinct physical situation. Thus, while there are no ordinary individuals in the algebraic models, the problem of individuals still arises.

Of course, we can try to remedy this problem by algebraicizing models at the second order. But even if this attempt is technically feasible, it does not guarantee that the problem does not arise for the higher-order entities and thus cannot solve the problem once and for all.¹⁷ Indeed, this problem is recognized and discussed by Dewar himself, who concedes that there are potential higher-order problems, but adds that eliminating first-order individuals is still a progress for ontic structuralists.

Note that algebraic structuralism and generalism are in the same boat regarding my criticisms. The criticism of generalism also applies here, namely that as long as the models are formulated in standard set-theoretic foundation, we cannot rule out isomorphic but distinct models. The problem of higher-order individuals also applies to generalism, since it does not allow invariant permutations of predicates. In contrast, UF does not face these problems since UF-implemented Identity applies to structures of any order.

Comments on Syntax vs Semantics When criticising Dasgupta’s approach, Dewar writes:

What is needed is a semantics for G which explains how the world could make sentences of G true, even if the world is [...] not fundamentally constituted by individuals standing in properties and relations[.] [M]etaphysical proposals should address themselves to semantics not to syntax. ([11], p. 1845)

I agree with Dewar that we should be able to say how the world is such that the sentences in our best theory are true. This, in Dewar’s context, amounts to a model theory without individuals. Since UF is advanced as our fundamental framework, it is natural to use it to not only regiment our physical theories but also explain how the world makes them true. It is adequate for this purpose (recall that a version of set theory that satisfies Univalence can be recovered from HoTT, which is powerful enough for such a model theory; see Appendix A). Indeed, the main application of the UF-implemented ontic structuralism, as we will see in more detail in Sect. 5, is to clarify the relation between symmetry-related models for our physical theories.

At the meantime, I want to point out that there is no clear distinction between a model-theoretic representation and a sentential one in the UF formalism unlike in standard logic, so we should not assume such a dichotomy when we switch to the UF approach. In standard logic or simple type theory, a model is an individual that can be predicated of while a sentence is not. So there is a clear syntactical distinction between representations of a worldly structure and a proposition. In contast, in UF (or HoTT), sets and propositions are both types and belong to the same universe of types. Indeed, a proposition is a set with at most one element. But even this formal difference is not conceptually important for us. Indeed, an ordinary existential statement can be directly formalized as a set with more than one element in HoTT, where the elements are its truth witnesses, and only then is truncated to a proposition by identifying all the elements by fiat.

Let’s briefly consider a concrete example. The theory of general relativity can be written out as the Einstein field equations, which are statements about how the metric field on the spacetime manifold is correlated with the matter distribution—denote them by ‘EFEs’. In HoTT, since propositions are types, we formalize them in the form of ‘a :  EFEs’ (see Sect. 4 for more details of the syntax). Now, consider solutions to EFEs, which are set-theoretic models. If we assume that these solutions are fully characterized by the field equations, they can be formalized in HoTT in the form of ‘z :  SolvesEFEs’, which gives the models that satisfy EFEs, with ‘SolvesEFEs’ denoting the type of such models (see also Section  5).¹⁸ We can see that there is little difference between these two ways of describing our general-relativistic world (again, the only formal distinction between them is the number of elements, which—as I have contended—plays no important conceptual role).¹⁹

4 Objections and Replies

I will turn to some possible objections to UF-implemented ontic structuralism, some of which do not specifically target at the UF approach but are helpful for clarifying features of UF.

4.1 Interpretation of Univalence

Although UF formally implements Identity, there is a glaring interpretative issue of the identity type. One may object that the identity in UF really means indiscernibility instead, and therefore we haven’t said anything new by the univalence axiom—surely, isomorphic structures that preserve all observable structures are indiscernible. This reading is especially consistent with the homotopy-theoretic interpretation. Recall that when we say $a=_A b$ (a, b are elements of type A), a, b can be interpreted as two points in a homotopy space A that are connected by a path. These two points are topologically indistinguishable but need not be one and the same point. Nevertheless, as I have commented, the homotopy-theoretic interpretation is useful to provide intuitions for UF but the latter can be perfectly understood without appealing to the former. Indeed, I shall argue that identity should be understood literally as identity. Assuming that the notion of indiscernibility between structures is captured by the notion of isomorphism, we can say that Univalence successfully implements the principle of the identity of indiscernibles. My reason is simply that identity in UF is defined in the exact same way as in standard logic, barring syntactical differences. In standard logic, identity is defined as follows:

A binary relation $=$ is an identity relation if (1) $(\forall x) x=x$ is an axiom or a theorem; (2) for any well-formed formula $\mathcal {F}$, we have as an axiom or theorem that

$$\begin{aligned} \forall x\forall y(x=y\rightarrow (\mathcal {F}[x,x]\rightarrow \mathcal {F}[x,y])) \end{aligned}$$

where $\mathcal {F}[x,y]$ is obtained from $\mathcal {F}[x,x]$ by substituting zero or more free occurrences of x by y.

The definition in HoTT is completely analogous:

For any type A, the type $=_A$ is a binary relation that satisfies: for any x : A, (1) $x=_Ax$; (2) for any type P, we have

$$\begin{aligned} ind: \prod _{(y:A)} (x=_Ay\rightarrow (P[x,x]\rightarrow P[x,y])) \end{aligned}$$

where P[x, y] is obtained from P[x, x] by replacing any number of occurrences of x by y.

Here, $`\prod '$ behaves like a universal quantifier. (2) is directly entailed by path induction.²⁰ Barring technical details that are irrelevant to the current discussion, these two definitions are exactly the same. One may think that the univalence axiom alters the meaning of the identity type, but that should not be the case, since the above conditions are sufficient for identity in standard logic.²¹

Note that Identity does not lead to an identification of merely weakly discernible individuals, which are qualitatively identical. Two things are weakly discernible if there is an irreflexive relation that holds between them (see [21]). A typical example is a toy universe where the only two existents are two symmetrically-arranged half-disks—they are weakly discernible because they are ‘next to’ each other. What’s more, Identity does not lead to an identification of two identical particles which have no qualitative or structural difference from each other (in particular they don’t have distinct spatiotemporal positions, unlike the half-disks). A type that contains two elements can be distinguished from that with one element based on the number of its self-identifications (see Appendix A). To emphasize: the permutation invariance of the elements of a type does not entail the collapse of all elements into one. Endorsing Identity does not lead to the conflation between strict identity and qualitative identity.

It might be worth mentioning—though this has been relatively well discussed—that in UF or HoTT, in addition to the identity, we also have a notion of judgmental equality (denoted by ‘$\equiv$’). If two terms or types are judgmentally equal, say $A\equiv B$, then they are interchangeable in any circumstances within HoTT. But if we just have $A=B$, then we cannot replace A with B in an expression like $`a:A'$.²² Indeed, if we could exchange A with B freely, then substituting A with B in the trivial judgment $A\equiv A$ would immediately give us $A\equiv B$, and it would follow that identity entails judgmental equality.²³ (If such an substitution is possible, the system is called extensional. UF, which denies this, is intensional.) However, this does not mean that judgmental equality is a better candidate for expressing strict identity in HoTT than the identity type. Indeed, I endorse the standard interpretation of the judgmental equality as indicating the synonimity of two expressions.²⁴ It is a peculiar but important feature of UF that we can keep a record of distinct expressions within the formal system that refer to the same entity (see Footnote 22).

4.2 Undesirable Holism?

[37] objects to generalism (recall: formalism without constants for individuals) that it leads to holism in the sense that we need to use a world-sentence that describes the world, which cannot be broken down to smaller units (since without constants, we cannot form atomic sentences like Rab, and the whole world structure can be relevant to any entity in it). This strikes him as undesirable since it seems alien comparing to our cognition that focuses on localized bits of our world and is in this way economical. But this crticism assumes a distinction between how we use variables and constants that does not apply to formalisms that use sequent calculus including UF. I shall use this as an opportunity to explain how variables in UF behave like constants and yet are not constants in the ordinary sense. For example, an expression that contains unbound variables can be a complete unit of information. A free variable can also refer to the same thing across different formulas.

Let’s look at the relevant details of the syntax of UF to see how this works. A judgment is a building block of inferences in HoTT and takes the form $\Gamma \vdash a:A$ where $\Gamma$ is called a context, which contains conditions for a : A. An inference is of the form:

$\Gamma \vdash a:A$
$\Gamma ' \vdash b:B$

An inference can also have more than one premise.²⁵ As an example, let’s regiment the famous syllogism ‘Socrates is a man, and all men are mortal; therefore Socrates is mortal’ in UF. Let $\Gamma$ be a context including ‘Socrates: Human, Mortal: Human$\rightarrow$ Proposition, z: $\prod _{y:Human}$ Mortal(y)’. Note that ‘Socrates’ is a variable (see the discussion below).²⁶

$\Gamma \vdash Socrates : \text {Human}$ $\qquad \qquad \qquad \Gamma \vdash z:\prod _{y:Human} \text {Mortal}(y)$
$\Gamma \vdash z(Socrates): \text {Mortal}(Socrates)$

The first premise says that Socrates is a human in the context $\Gamma$, and the second premise says that all humans are mortal (in $\Gamma$). These premises are very simple valid inferences, since their right-hand sides are part of the left-hand sides. The conclusion is that Socrates is a human (in $\Gamma$), which follows from the premises by $\prod$-elimination, which is like universal instantiation (UFP, A2.4).

What is the relevance of all this? We observe that a statement (i.e., the right side of a judgment) can have unbound terms and variables as long as they are declared in the context. For example, Socrates, z and even Human are all variables (only variables can be introduced by a context). Also, these variables can appear across different statements within the same proof without changing their references and meanings. Thus, there is no need for a constant ‘Socrates’. Indeed, it would be very unnatural to introduce ‘Socrates’ as a constant in UF.²⁷ Now, one might argue that these variables behave exactly like constants in standard logic, and therefore they are constants. To respond, these variables, while like constants in the familiar sense in some respects, are still importantly different from them because ‘a : A’ refers to anything that satisfies A and does not presume primitive identity. If we declare a, b : A, then exchanging them in any statement preserves its truth value by the inference rule in UF (or HoTT). We cannot say the same about the constants in standard logic: substituting constants by each other is not allowed unless they are explicitly identified. Univalence further reinforces that the different notations we use for isomorphic types and terms have no ontological significance. In this sense, we can say that variables in UF are constants for entities without haecceities.

4.3 The Collapse Problem

Let’s start with the notion of ‘physical entities’ as referring to concrete things that we can be in contact with (typically understood as having spatiotemporal locations or being parts of spacetime) and ‘mathematical entities’ as abstract things that exist in some Platonic heaven. Accordingly, mathematical structures are abstract while physical structures are instantiated by concrete things, e.g., the physical structure of two particles next to each other exist if and only if there are two particles next to each other. Now, there is an objection that, if we assume Identity, then the isomorphic mathematical and physical structures will be identified. This is sometimes called ‘the collapse problem’ (see [15]). Given the abundance of mathematical structures, for any physical structure, we can find an isomorphic mathematical one. So it seems that ontic structuralists who adopt Identity must systematically conflate these two kinds. I think this worry is not as formidable as one might think, but a thorough solution to this problem requires a detailed account of how we formulate our physical theories and model physical systems within UF, which exceeds the scope of this paper.²⁸ But I will very briefly sketch some toy models and strategies just to illustrate why the UF approach does not need to lead to a problematic collapse of physical structures into mathematical ones.

Let’s first recall how isomorphism is usually used in standard mathematics: an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping (see also Footnote 14). For example, we talk about isometries between metric spaces, and homeomorphisms between topological spaces, but not between different kinds of spaces. In UF, this is not only true but highlighted. Like identity claims, the notion of isomorphism is only defined relative to certain types. Consider a, b :  Met, where Met is the type of metric spaces. If a, b are isometric, then they are identified through Univalence, namely $a=_{Met}b$. Here, isometry is precisely defined relative to the type of metric spaces.²⁹ This is not surprising, for otherwise it would lead to an ill-defined identity claim.

How do we distinguish between (say) a mathematical manifold and a physical one that are allegedly diffeomorphic, provided that we want to? The answer is to be expected: we can do so by distinguishing between the mathematical type and the physical type, in which case there can be no isomorphism between their elements. Of course, from the point of view of structuralists, the distinction had better not be primitive but based on structural differences. The merit of UF is that it has ample resources to implement whatever structuralists come up with. For example, we can let the physical type incorporate an observer.³⁰ Or we can let the physical type incorporate a causal structure [15]. We can even make a more systematic distinction by delineating between a physical ‘universe’ (or ‘kind’) and mathematical one (here, ‘universe’ or ‘kind’ refers to higher-order types; see [31], §29).

Moreover, even if we do not use any of these strategies, we can still incorporate both physical and mathematical structures into the same global model, and straightforwardly distinguish between them by singling out what actually obtains.³¹

To illustrate, let’s consider the approach of algebraicism (see [7]; also mentioned in Sect. 3), and suppose that the universe is so simple that it contains just one scalar field. Using the apparatus of Einstein algebras, we can model such a universe by an algebra $C^\infty (M)$ that consists of all smooth functions on a manifold M defined by their smooth operators (see [17]). Each element represents a scalar field configuration. Now, regarding the ontological inventory of the world, it is intuitive to think of the actual field configuration as physical and all the other configurations as mathematical. In such a model, we can single out the physical field configuration by stipulating which obtains. The resulting model looks like $\langle C^\infty (M), \{\psi \}\rangle$, where $\psi$ is an algebraic 0-ary operator (i.e., a constant) representing the physical field configuration in $C^\infty (M)$.

Of course, it may not even come to that. Perhaps we should simply reject the need to be realists about both physical and mathematical entities that are structural identical. For example, we may ‘collapse’ physical and mathematical structures by endorsing physicalism about mathematical entities (see, for example, [25]; for related strategies, see [14, 41]). At the meantime, we may endorse nominalism (or anti-realism) about mathematical entities that cannot be plausibly construed in this way. Arguably, both physicalism and nominalism are attractive independently of ontic structuralism. In a sense, the aforementioned algebraicism can be interpreted as adopting this strategy. Since $\langle C^\infty (M), \{\psi \}\rangle$ is a model of our world, there is a sense that the whole model is physical and every constituent of the model is also physical. A alleged mathematical configuration can be considered as an actual but uninstantiated state or property of a physical field. In this sense, we effectively rebrand what we originally considered mathematical (namely unactualized field configurations) as physical.

The upshot is that, once we look at the technical details of how we would like to model our physical world in UF together with various structuralists’ strategies, we would see that there is no special difficulty regarding the collapse problem at all for the UF advocates. We can implement numerous strategies structuralists have advanced and can advance.

5 Revisiting the Hole Argument

Let’s put UF in action. It would be instructive to see how UF applies to the hole argument, since spacetime points are considered exemplary structural entities, and how this approach fares comparing to other solutions to the hole argument.

First, let’s see how UF tackles the hole argument—this is not straightforward in its technical aspect. The hole argument (as it is standardly formulated) relies on the premise that diffeomorphically related models $\langle M,g\rangle$ and $\langle M,g'\rangle$ represent distinct physical possibilities, where M is a smooth manifold, and g and $g'$ are two metrics on M related by a diffeomorphism from M to itself. One might expect that, since these two models are isomorphic, Univalence directly entails that they are identical, which blocks the hole argument. But this is not the case. First of all, Univalence only applies to types, but $\langle M,g\rangle$ is not a type, but has the type of Lorentzian manifolds (it cannot be a homotopy type itself since it includes more structure than its elements and their homotopical information). Also, while I have informally explained the notion of equivalence between types, the notion of equivalence between structures is not even defined.

Fortunately, this can be solved satisfactorily via ‘the structure identity principle’, which identifies isomorphic structures given Univalence ([44], §9.8; Appendix B). Recall that we can recover a version of set theory from UF, so defining a manifold as a set-theoretic structure is not a problem in UF. We can thereby recover the standard definition of manifold in UF, namely that a manifold is a set of points equipped with topological and differential structures (technically, an ‘atlas’). The structure identity principle entails that we can identify such set-theoretic structures that are isomorphic (Appendix B). While the full technical detail is quite involved, this result is rather intuitive. Recall that a set is a 0-truncated type (Sect. 2). Univalence entails that any two equinumerous sets are identical, with every bijection corresponding to an identification. Given two structures based on a set, it is intuitive that every self-identification of the set that preserves all the structure is an identification between the two structured sets (or rather, a self-identification of the structured set). Thus, Univalence entails that two isomorphic structures are identical. With this in mind, we can indeed easily show that the diffeomorphically related models are identical, as long as we model the metric g as a structure dependent on the type of manifolds M in $\langle M,g\rangle$, which is only natural (see [24] for more details). The same kind of reasoning can show that more complicated diffeomorphically related models with additional matter fields are also identical.³²

Now that I have laid out the univalence solution to the hole argument, we can ask what exactly this solution offers to the existent literature. It falls under the camp called ‘mathematical solution’ which generally argues that there are no distinct diffeomorphically related models in the hole argument (in contrast, metaphysical solutions are those that argue distinct diffeomorphically related models represent the same physical situation). In particular, [47] argues that isomorphic models have the same representational capacity. Does the univalence approach offer anything new?

Let’s start by reviewing Weatherall’s rebuttal of the hole argument, which says that the mathematics of general relativity ‘does not force one to confront a metaphysical dilemma’ contrary to what the hole argument purports to show ([47], p. 16). Once again, consider two diffeomorphically related models $\langle M,g\rangle$, $\langle M, g'\rangle$, which are isometric. It is clear that they are isomorphic with respect to an isometry $\phi$. (To be more precise, Weatherall distinguishes between the automorphism $\phi$ and the isomorphism $\phi '$ induced by $\phi$, with the former a map on M and the latter between the two models. But I will gloss over this distinction.) However, when we further infer that the two models are distinct, namely that g and $g'$ assign different metric properties to some spacetime points in M, we implicitly appeal to the identity map on M. But the identity map does not induce any isometry: it does not take g to $g'$. So the two models are not isometric or isomorphic relative to the identity map. That is, if we stick with the identity map as the relevant comparison, then the two models are neither mathematically equivalent (nor empirically indistinguishable, if we let an observer be located at the same point with respect to the identity map). The models are only equivalent and indistinguishable with respect to isometries such as $\phi$. So when we invoke the hole argument, we no longer consider the identity map as the relevant comparison on M. Thus, from the mathematical point of view, it is not true that g and $g'$ assign different properties to the same spacetime points.

As Weatherall adds, if we want to keep to the identity map while comparing the isometric models, we are in fact claiming that there is some additional structure that is not preserved by an isometry, which amounts to saying that a model of general relativity is not a Lorentzian manifold.

There are various authors objecting to responses along this line. For example, [32] argue that Weatherall’s arguments are faulty and leave the hole argument largely untouched. Contrary to what Weatherall claims, the hole argument does not involve an ‘illegitimate equivocation’ between the identity map and isometry $\phi$—both are invoked for their legitimate purposes. In particular, when we compare two models $\langle M,g\rangle$ and $\langle M, g'\rangle$ relative to the identity map, we do have two distinct physical situations (different properties are assigned to the same points) that are empirically indistinguishable (since the models are isomorphic simpliciter). To illustrate this point, Pooley and Read give a simple analogy. Alice and Barbara are twins who can look identical if they want. In one situation, Alice wears a red hat and Barbara blue, and they are otherwise indistinguishable. In another situation, they have exchanged their hats and positions so that it looks exactly like the first situation. In this case, it would be absurd to say that we can only either claim that the two situations are compared relative to the identity map and are empirically distinguishable, or that they are compared relative to an isomorphism (which assigns Alice to Barbara and Barbara to Alice) and are not physically distinct. But this is analogous to Weatherall’s claim.

Putting the objection differently, when we compare $\langle M,g\rangle$ and $\langle M, g'\rangle$ with respect to isometry maps, the self-identity of M or of spacetime points exists regardless. Hence there is no equivocation involved in the hole argument on whether the identity map plays a role.

Weatherall is correct that, as far as mathematical practice is concerned, whenever we consider $\langle M,g\rangle$ and $\langle M, g'\rangle$ as isomorphic models, we do not assume any prior identity map on M. If we assume a prior identity map on M, then we are either not considering the two models isomorphic, or we are using the mathematical language improperly. The moral of his response is restricted to the mathematical part of the hole argument, namely that the mathematics does not force us to confront the metaphysical choice. If we want to infer a metaphysical dilemma from the formalism of general relativity, we must assume some metaphysical thesis to begin with.³³ But as the objection to Weatherall’s approach goes, instead of solving the hole argument, it seems to only draw out the irrelevance of mathematical formalism to the argument. (This irrelevance is expounded by [40], who reconstructs the hole argument in terms of modality so that we can ‘directly discuss the phenomena we use this formalism to represent’ (p.29).)

I very much agree with Weatherall’s diagnosis of the misuse of mathematical formalism in the hole argument (as it is standardly formulated) from the point of view of contemporary mathematical practice, which already incorporates the spirit of mathematical structuralism. I also agree with him that ‘our interpretations of our physical theories should be guided by the formalism of those theories’ ([2018], p.2). I object to, however, the dichotomy between mathematics and physics (or metaphysics) implicitly assumed by authors on both sides. Instead of focusing on mathematical formalism in its narrow sense, we should focus on the entire logical framework that we use for expressing our worldviews, including defining concepts, formulating principles, constructing models, making deductions and so on. Accordingly, instead of using mathematical structuralism as a response to the hole argument, we should generalize the spirit to this whole framework in which scientific or fundamental discourse about our reality is best formulated. In particular, it ought not be the case that our best model in this framework is indeterminate between distinct physical situations, the difference of which is independently captured by a discourse external to this framework. In the case of Alice and Barbara, the apparent absurdity arises not because of a conflation of structural mathematical representations with physical situations, but because we are still using a meta-language that distinguishes between the two situations that are indistinguishable by structural features.

This foundational approach, however, is not available to Weatherall because our standard formal framework lacks the technical means to express how isomorphic models ought to be interpreted and whether they refer to the same physical situation. The claim that isomorphic models have the same representation capability is a meta-mathematical or meta-formal characterization of our mathematical practice. In particular, it is neither an axiom nor derivable from the axioms of our standard foundation. (In fact, it is not only unprovable in our standard set-theoretic foundation but also in category theory, which is worth mentioning because many consider the latter as the backdrop of contemporary mathematical practice. Although it is often said that category theory does not ‘care’ about how many isomorphic copies of an object there are, it is nevertheless entirely possible to define multiple isomorphic copies. As a simple example, we can define a category with two objects, each of which has an arrow pointing towards the other. The two objects, though structurally identical, are not the same object. This particular category is also not the same as the category of only one object that only has an arrow pointing towards itself, even though their only difference is how many isomorphic copies of an object they contain. Thus, it is not internal to the framework that isomorphic models have the same representational capacity.)

This is where UF comes to the rescue. As a foundation that implements ontic structuralism (or so I have argued), UF generalizes the spirit of mathematical structuralism to the description of our physical world. Since we can develop everything we need in the foundational framework, there does not need to be a split between a mathematical structuralist language and a scientific one. Moreover, insofar as modal claims (and more generally, metaphysical claims) are driven by science and used to express our worldview, they should also be rooted in our best foundational framework. For example, upon adopting UF, if we consider certain possibilities as structurally isomorphic, then Univalence is applicable and would entail that these possibilities are one and the same. Thus, we cannot carve out the space of metaphysical possibilities differently from how we view the space of best mathematical representations.

To put it in another way, the UF-implemented ontic structuralism is both a metaphysical solution to the hole argument in denying the primitive identity of spacetime points, and also a mathematical solution in denying that there are distinct isomorphic models,³⁴ This, I think, is a unification between formal treatment and ontic considerations that we need for solving the hole argument satisfactorily. Without the latter, a formal apparatus risks being irrelevant to the problem at hand, and without the former, a metaphysical position lacks clarity and risks being irrelevant to systematic sciences.

Generalization to (Gauge) Symmetries

Diffeomorphism invariance concerned by the hole argument is usually subsumed under gauge invariance. It is standardly considered as a formal symmetry, namely it relates models representing the same physical situation, as opposed to empirical symmetries that relate distinct but indistinguishable situations. The univalence solution to the hole argument can be applied to gauge symmetries in general, identifying all symmetry-related models. A well-known obstacle for this generalization is that symmetries are often not isomorphisms.³⁵ Nevertheless, I have argued elsewhere that we can reformulate any symmetry-related models as isomorphic (see also [12, 46], and [45]).³⁶ As a result, we can straightforwardly apply the univalence approach to gauge symmetries and conclude that all symmetry-related models are identical. Thus, in UF, the view about symmetries is completely clear-cut: all (perfect) symmetries are formal rather than empirical, since they represent the very same physical situation as enforced by Univalence.³⁷ It is intriguing to contemplate the similar spirit of UF and gauge theory (which introduces fields with gauge degrees of freedom): gauge theory takes advantage of the gauge degrees of freedom as representational ‘redundancy’ while UF legitimizes this freedom (see, for example, [18] and [28] for related discussions of gauge theory).

6 Conclusion

I have shown that UF, featuring Univalence, is a rigorous foundational implementation of ontic structuralism with unique advantages over alternative proposals, and can shed light on important issues in philosophy of physics including the hole argument and promisingly the nature of symmetries in general.

Appendices

Appendix A: Set Theory in UF

Thinking in terms of sets is deeply entrenched in philosophers in the analytic tradition. It is therefore helpful to see how set theory can be modeled in HoTT/UF, and how it is different from standard set theory. But this is a large topic, so I will only sketch some very basic formal features here that help illustrate their structuralist characteristics. The presentation is based on [44])

Let us start with the definition of sets in HoTT.

Definition 1

(Set) A type A is a set iff for all x, y : A, and for all $p,q:x=y$, we have $p=q$. More formally:

$$\begin{aligned}{} {\textbf {isSet}}(A):\equiv \Pi _{x,y:A}\Pi _{p,q:x=y}p=q.\end{aligned}$$

We also need the basic notion of inductive types:

Definition 2

(Inductive Type) An inductive type X is a type that is generated by a finite set of constructors, each of which is a function with codomain X. This includes functions of zero arguments, which are simply elements of X.

In other words, an inductive type is defined through positing constant elements of the type and functions that map between elements of the type.

Example 1

The empty type 0, which is an inductive type with no constructor, is a set.

Since there is no constructor, the proof is trivial.

Example 2

The unit type 1, which has one constant $\star :$ 1, is a set.

Proof (sketch). We need to show that isSet(1) $:\equiv$ $\Pi _{x,y:{\textbf {1}}}\Pi _{p,q:x=y}p=q$. It may be tempting to first reduce the desired theorem to $\Pi _{p,q:\star =\star }p=q$, and then prove it by induction on p and q. But this is not possible because the type $\star =\star$ is not inductive and we would be stuck. Instead, we directly show that $\Pi _{x,y:{\textbf {1}}}x=y$ has only one element. To establish this, we need to construct a bijection between it and 1. Since 1 has only one element, let the function f from $\Pi _{x,y:{\textbf {1}}}x=y$ to 1 send everything to $\star$, and let the function g from 1 send $\star$ to Refl $_\star$ (note: Refl is the trivial self-identity that holds of every element). To check that the two functions are inverses of each other, we need to show that $f(g(\star ))=\star$ and $g(f(p))=p$ for all $p:\Pi _{x,y:{\textbf {1}}}x=y$. The first is trivial. The second is also straightforward because $g(f(p))=g(\star )={\textbf {Refl}}_\star$, and we have that $p={\textbf {Refl}}_\star$ for all $p:\Pi _{x,y:{\textbf {1}}}x=y$ by path induction. So we have shown that $\Pi _{x,y:{\textbf {1}}}x=y$ has only one element, and hence 1 is a set.

Now that we see sets are types whose elements have at most one way of identification, we can ask how many ways of identification a set has with itself. The answer is to be expected: for an n-element set, it has n! ways of identification with itself which is the number of permutations of the elements.

Example 3

The boolean type 2, an inductive type with two constants $0_2,1_2:$ 2, is a set and has two ways of identification with itself.

Proof omitted. Informally, the two ways of self-identification are the identity and the swap, which can be shown to be distinct. That is, they map each element to different ones in 2.

Sets in HoTT/UF are different from those in standard set theory in that their elements do not have primitive identity. To appreciate the difference, let’s consider how union is defined in HoTT/UF. For example, consider two n-element sets A, B that are not judgmentally equal. They are identified in UF since they are equinumerous. If the notion of union were understood exactly as in standard set theory, the union of A with itself would be distinct from the union of A with B, which would violate path induction (which implements the indiscernibility of identicals). To avoid this problem, the set-theoretic union in HoTT/UF is only definable for sets that are considered as subsets of a larger set. When we take set-theoretic union of two sets, we must specify how these sets are included in a larger set. For a formal definition, we first need the notion of coproduct, which corresponds to the notion of disjoint union in set theory.

Definition 3

(coproduct) For any types A and B, the coproduct $A+B$ is the inductive type with two generating elements: left inclusion $i_1:A\rightarrow A+B$ and right inclusion $i_2:B\rightarrow A+B$.

The induction principle for $A+B$ says that for any type C, if we have $f:A\rightarrow C$ and $g:B\rightarrow C$, then there is a function $h:A+B\rightarrow C$ such that $h\circ i_1=f$ and $h\circ i_2=g$. For an illuminating example, consider $A+A$. In set theory, we first force the two copies of A to be disjoint, and then take the union of them (e.g., we can define $A+B$ to be $A\times \{0\}\cup B\times \{1\}$). The difference is important because it illustrates the structuralist spirit of HoTT/UF where the individuality of set elements is not primitive but rather derived from the structure of the whole. To continue:

Definition 4

(Subset) $\{x:A \mid P(x)\}:\equiv \Sigma _{x:A} P(x)$ is a subset of A if A is a set and P(x) is a proposition for all x : A.

Definition 5

(Disjunction) For propositions P, Q, their disjunction is $P\vee Q:\equiv ||P+Q||$, where $||P+Q||$ is the propositional truncation of $P+Q$ (a propositional truncation of a type adds identification between all elements of the type).

Definition 6

(Union) $\{x:A \mid P(x)\}\cup \{x:A \mid Q(x)\}:\equiv \{x:A \mid P(x)\vee Q(x)\}$

It is easy to check that the union of a subset with itself is just itself, and that two sets that are equinumerous are not necessarily identical as subsets of a larger set. Thus we avoid the violation of indiscernibility of identicals.

I hope this very brief introduction to set theory in HoTT/UF has given a flavor of how we can model set theory without assuming the primitive individuality of set elements as well as shown some important differences between the two.

Appendix B: Structure Identity Principle

Given its centrality for implementing Identity, it would be beneficial to look at how exactly the structure identity principle is formulated as a theorem of UF, even if we do not go into its proof. We will make use of notions from category theory, which is in turn formulated in HoTT/UF.

We start with the notion of precategory in HoTT/UF, which corresponds to the notion of category in the standard approach. We will see soon why the term ‘precategory’ instead of ‘category’ is used. For brevity, I do not include the full definitions, which are lengthy and can be easily found in [44]).

Definition 7

(Precategory) A precategory A consists of the following:

• A type $\text {A}_0$ whose elements are called objects. We write a : A for $a:A_0$.

• For each pair of objects $a, b: \text {A}$, a type $\text {Hom}_A(x, y)$ whose elements are called morphisms from $a$ to $b$.

They satisfy the usual conditions, such as the composition rule for morphisms and the existence of identity morphisms. We write the identity morphism for a : A as $1_a:Hom_A(a,a)$.

The notion of isomorphism between objects is defined similarly as in standard category theory:

Definition 8

(Isomorphism) A morphism f: hom$_A$(a, b) is an isomorphism if there is a morphism g: hom$_A$(b, a) such that g $\circ$ f = 1$_a$ and f $\circ$ g = 1$_b$. We write $f\simeq g$.

Now, a category is a precategory that satisfies the univalence axiom:

Definition 9

(Category) A precategory A is a category if for any a, b : A, $a\simeq b$ implies $a=b$.

It is attractive to formally identify isomorphic objects in a category, since it is common to talk about objects up to isomorphism in category theory. Thus we replace the old notion of category (now ‘precategory’) with this notion.

Let’s turn to structures. Let X be a precategory.

Definition 10

((P,H)-structure) A notion of structure (P, H) over X consists of the following.

• A type family P: X$_0$ $\rightarrow$ U. For each x: X$_0$ the elements of Px are called (P, H)-structures on x.

• For x, y: X$_0$, f: Hom$_X$(x, y) and $\alpha$: Px, $\beta$: Py, a mere proposition H$_{\alpha \beta }$( f ). If H$_{\alpha \beta }$( f ) is true, we say that f is a (P, H)-homomorphism from $\alpha$ to $\beta$.

H satisfies further conditions that amount to the usual properties of homomorphisms.

Definition 11

A precategory of (P, H)-structures, A = Str$_{(P,H)}$(X), consists of the following:

• A$_0:\equiv \Sigma (x:X_0) Px$.

• For $(x, \alpha ), (y, \beta ): A_0$, we define $Hom_A((x, \alpha ), (y, \beta )):\equiv \{f: x \rightarrow y\mid H_{\alpha \beta }( f )\}$

For example, X can be the precategory of sets, and A can be the precategory of groups, or rings, or topological spaces that are built on sets.

Finally let’s turn to the structure identity principle:

Theorem 1

(The structure identity principle) If X is a category, then $Str_{(P,H)}(X)$ is a category.

As a special case, X can be the category of sets, and $Str_{(P,H)}(X)$ can be the category of manifolds defined set-theoretically. This gives us what we need for identifying set-theoretic structures in UF.