Homotopy Type Theory (HoTT) is a putative new foundation for mathematics grounded in constructive intensional type theory that offers an alternative to the foundations provided by ZFC set theory and category theory. This article explains and motivates an account of how to define, justify, and think about HoTT in a way that is self-contained, and argues that, so construed, it is a candidate for being an autonomous foundation for mathematics. We first consider various (...) questions that a foundation for mathematics might be expected to answer, and find that many of them are not answered by the standard formulation of HoTT as presented in the ‘HoTT Book’. More importantly, the presentation of HoTT given in the HoTT Book is not autonomous since it explicitly depends upon other fields of mathematics, in particular homotopy theory. We give an alternative presentation of HoTT that does not depend upon ideas from other parts of mathematics, and in particular makes no reference to homotopy theory (but is compatible with the homotopy interpretation), and argue that it is a candidate autonomous foundation for mathematics. Our elaboration of HoTT is based on a new interpretation of types as mathematical concepts, which accords with the intensional nature of the type theory. 1 Introduction2 What Is a Foundation for Mathematics? 2.1 A characterization of a foundation for mathematics 2.2 Autonomy3 The Basic Features of Homotopy Type Theory 3.1 The rules 3.2 The basic ways to construct types 3.3 Types as propositions and propositions as types 3.4 Identity 3.5 The homotopy interpretation4 Autonomy of the Standard Presentation?5 The Interpretation of Tokens and Types 5.1 Tokens as mathematical objects? 5.2 Tokens and types as concepts6 Justifying the Elimination Rule for Identity7 The Foundations of Homotopy Type Theory without Homotopy 7.1 Framework 7.2 Semantics 7.3 Metaphysics 7.4 Epistemology 7.5 Methodology8 Possible Objections to this Account 8.1 A constructive foundation for mathematics? 8.2 What are concepts? 8.3 Isn’t this just Brouwerian intuitionism? 8.4 Duplicated objects 8.5 Intensionality and substitution salva veritate9 Conclusion 9.1 Advantages of this foundation. (shrink)

Homotopy Type Theory is a putative new foundation for mathematics grounded in constructive intensional type theory that offers an alternative to the foundations provided by ZFC set theory and category theory. This article explains and motivates an account of how to define, justify, and think about HoTT in a way that is self-contained, and argues that, so construed, it is a candidate for being an autonomous foundation for mathematics. We first consider various questions (...) that a foundation for mathematics might be expected to answer, and find that many of them are not answered by the standard formulation of HoTT as presented in the ‘HoTT Book’. More importantly, the presentation of HoTT given in the HoTT Book is not autonomous since it explicitly depends upon other fields of mathematics, in particular homotopy theory. We give an alternative presentation of HoTT that does not depend upon ideas from other parts of mathematics, and in particular makes no reference to homotopy theory, and argue that it is a candidate autonomous foundation for mathematics. Our elaboration of HoTT is based on a new interpretation of types as mathematical concepts, which accords with the intensional nature of the type theory. 1 Introduction2 What Is a Foundation for Mathematics?2.1 A characterization of a foundation for mathematics2.2 Autonomy3 The Basic Features of Homotopy Type Theory3.1 The rules3.2 The basic ways to construct types3.3 Types as propositions and propositions as types3.4 Identity3.5 The homotopy interpretation4 Autonomy of the Standard Presentation?5 The Interpretation of Tokens and Types5.1 Tokens as mathematical objects?5.2 Tokens and types as concepts6 Justifying the Elimination Rule for Identity7 The Foundations of Homotopy Type Theory without Homotopy7.1 Framework7.2 Semantics7.3 Metaphysics7.4 Epistemology7.5 Methodology8 Possible Objections to this Account8.1 A constructive foundation for mathematics?8.2 What are concepts?8.3 Isn’t this just Brouwerian intuitionism?8.4 Duplicated objects8.5 Intensionality and substitution salva veritate9 Conclusion9.1 Advantages of this foundation. (shrink)

Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy provides a reasonably gentle introduction to this new logic, thoroughly motivated by intuitive explanations of the need for all of its component parts, and illustrated through innovative applications of the calculus.

I explore the possibility of a structuralist interpretation of homotopy type theory (HoTT) as a foundation for mathematics. There are two main aspects to HoTT's structuralist credentials. First, it builds on categorical set theory (CST), of which the best-known variant is Lawvere's ETCS. I argue that CST has merit as a structuralist foundation, in that it ascribes only structural properties to typical mathematical objects. However, I also argue that this success depends on the adoption of a (...) strict typing system which undermines the metaphysical seriousness of this structuralism. Homotopy type theory adds to CST a distinctive theory of identity between sets, which arguably allows its objects to be seen as ante rem structures. I examine the prospects for such a view, and address many other interpretive problems as they arise. (shrink)

Homotopy Type Theory is a proposed new language and foundation for mathematics, combining algebraic topology with logic. An important rule for the treatment of identity in HoTT is path induction, which is commonly explained by appeal to the homotopy interpretation of the theory's types, tokens, and identities as spaces, points, and paths. However, if HoTT is to be an autonomous foundation then such an interpretation cannot play a fundamental role. In this paper we give a (...) derivation of path induction, motivated from pre-mathematical considerations, without recourse to homotopy theory. (shrink)

The Hole Argument is primarily about the meaning of general covariance in general relativity. As such it raises many deep issues about identity in mathematics and physics, the ontology of space–time, and how scientific representation works. This paper is about the application of a new foundational programme in mathematics, namely homotopy type theory, to the Hole Argument. It is argued that the framework of HoTT provides a natural resolution of the Hole Argument. The role of the Univalence (...) Axiom in the treatment of the Hole Argument in HoTT is clarified. (shrink)

Among the most interesting features of Homotopy Type Theory is the way it treats identity, which has various unusual characteristics. We examine the formal features of “identity types” in HoTT, and how they relate to its other features including intensionality, constructive logic, the interpretation of types as concepts, and the Univalence Axiom. The unusual behaviour of identity types might suggest that they be reinterpreted as representing indiscernibility. We explore this by defining indiscernibility in HoTT and examine its (...) relationship with identity. We argue that identity types are a primitive component of HoTT and cannot be reduced to indiscernibility. (shrink)

The Univalence axiom, due to Vladimir Voevodsky, is often taken to be one of the most important discoveries arising from the Homotopy Type Theory research programme. It is said by Steve Awodey that Univalence embodies mathematical structuralism, and that Univalence may be regarded as ‘expanding the notion of identity to that of equivalence’. This article explores the conceptual, foundational and philosophical status of Univalence in Homotopy Type Theory. It extends our Types-as-Concepts interpretation of HoTT (...) to Universes, and offers an account of the Univalence axiom in such terms. We consider Awodey’s informal argument that Univalence is motivated by the principle that reasoning should be invariant under isomorphism, and we examine whether an autonomous and rigorous justification along these lines can be given. We consider two problems facing such a justification. First, there is a difference between equivalence and isomorphism and Univalence must be formulated in terms of the former. Second, the argument as presented cannot establish Univalence itself but only a weaker version of it, and must be supplemented by an additional principle. The article argues that the prospects for an autonomous justification of Univalence are promising. (shrink)

The notion of a natural model of type theory is defined in terms of that of a representable natural transfomation of presheaves. It is shown that such models agree exactly with the concept of a category with families in the sense of Dybjer, which can be regarded as an algebraic formulation of type theory. We determine conditions for such models to satisfy the inference rules for dependent sums Σ, dependent products Π, and intensional identity types (...) Id, as used in homotopy type theory. It is then shown that a category admits such a model if it has a class of maps that behave like the abstract fibrations in axiomatic homotopy theory: they should be stable under pullback, closed under composition and relative products, and there should be weakly orthogonal factorizations into the class. It follows that many familiar settings for homotopy theory also admit natural models of the basic system of homotopy type theory. (shrink)

1. The theory referred to by the—perhaps intimidating—main title of this book is an extension of Per Martin-Löf's dependent type theory. Much philosophical work pertaining to dependent type theory...

As a new foundational language for mathematics with its very different idea as to the status of logic, we should expect homotopy type theory to shed new light on some of the problems of philosophy which have been treated by logic. In this article, definite description, and in particular its employment within mathematics, is formulated within the type theory. Homotopy type theory has been proposed as an inherently structuralist foundational language for mathematics. (...) Using the new formulation of definite descriptions, opportunities to express ‘the structure of’ within homotopy type theory are explored, and it is shown there is little or no need for this expression. (shrink)

In this short note we give a glimpse of homotopy type theory, a new field of mathematics at the intersection of algebraic topology and mathematical logic, and we explain Vladimir Voevodsky’s univalent interpretation of it. This interpretation has given rise to the univalent foundations program, which is the topic of the current special year at the Institute for Advanced Study.

of type theory has been used successfully to formalize large parts of constructive mathematics, such as the theory of generalized recursive definitions [NPS90, ML79]. Moreover, it is also employed extensively as a framework for the development of high-level programming languages, in virtue of its combination of expressive strength and desirable proof-theoretic properties [NPS90, Str91]. In addition to simple types A, B, . . . and their terms x : A b(x) : B, the theory also has (...) dependent types x : A B(x), which are regarded as indexed families of types. There are simple type forming operations A × B and A → B, as well as operations on dependent types, including in particular the sum x:A B(x) and product x:A B(x) types (see the appendix for details). The Curry-Howard interpretation of the operations A × B and A → B is as propositional conjunction and.. (shrink)

The original article has been corrected. The article is published with Open Access but was missing Open Access information. This has been added.

Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed by this approach to the formalizing homotopy-theoretic material. We also compare our constructions with the more classical approach to homotopy limits via fibration categories.

Drawing on the analogy between any unary first-order quantifier and a "face operator," this paper establishes several connections between model theory and homotopy theory. The concept of simplicial set is brought into play to describe the formulae of any first-order language L, the definable subsets of any L-structure, as well as the type spaces of any theory expressed in L. An adjunction result is then proved between the category of o-minimal structures and a subcategory of (...) the category of linearly ordered simplicial sets with distinguished vertices. (shrink)

This article proposes a way of doing type theory informally, assuming a cubical style of reasoning. It can thus be viewed as a first step toward a cubical alternative to the program of informalization of type theory carried out in the homotopy type theory book for dependent type theory augmented with axioms for univalence and higher inductive types. We adopt a cartesian cubical type theory proposed by Angiuli, Brunerie, Coquand, (...) Favonia, Harper, and Licata as the implicit foundation, confining our presentation to elementary results such as function extensionality, the derivation of weak connections and path induction, the groupoid structure of types, and the Eckmman–Hilton duality. (shrink)

Mathematicians often consider Zermelo-Fraenkel Set Theory with Choice (ZFC) as the only foundation of Mathematics, and frequently don’t actually want to think much about foundations. We argue here that modern Type Theory, i.e. Homotopy Type Theory (HoTT), is a preferable and should be considered as an alternative.

We introduce type-theoretic algebraic weak factorisation systems and show how they give rise to homotopy-theoretic models of Martin-Löf type theory. This is done by showing that the comprehension category associated with a type-theoretic algebraic weak factorisation system satisfies the assumptions necessary to apply a right adjoint method for splitting comprehension categories. We then provide methods for constructing several examples of type-theoretic algebraic weak factorisation systems, encompassing the existing groupoid and cubical sets models, as well (...) as new models based on normal fibrations. (shrink)

Quillen [17] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success and—as, e.g., the work of Voevodsky on the homotopy theory of schemes [15] or the work of Joyal [11, 12] and Lurie [13] on quasicategories seem to indicate—it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories (...) and mathematical logic, inspired by the groupoid model of (intensional) Martin–Löf type theory [14] due to Hofmann and Streicher [9]. In particular, we show that a form of Martin–Löf type theory can be soundly modelled in any model category. This result indicates moreover that any model category has an associated “internal language” which is itself a form of Martin-Löf type theory. This suggests applications both to type theory and to homotopy theory. Because Martin–Löf type theory is, in one form or another, the theoretical basis for many of the computer proof assistants currently in use, such as Coq and Agda (cf. [3] and [5]), this promise of applications is of a practical, as well as theoretical, nature. This paper provides a precise indication of this connection between homotopy theory and logic; a more detailed discussion of these and further results will be given in [20]. (shrink)

We introduce a new model construction for Martin-Löf intensional type theory, which is sound and complete for the 1-truncated version of the theory. The model formally combines, by gluing along the functor from the category of contexts to the category of groupoids, the syntactic model with a notion of realizability. As our main application, we use the model to analyse the syntactic groupoid associated to the type theory generated by a graph G, showing that it (...) has the same homotopy type as the free groupoid generated by G. (shrink)

In a series of papers Ladyman and Presnell raise an interesting challenge of providing a pre-mathematical justification for homotopy type theory. In response, they propose what they claim to be an informal semantics for homotopy type theory where types and terms are regarded as mathematical concepts. The aim of this paper is to raise some issues which need to be resolved for the successful development of their types-as-concepts interpretation.

Let T be the theory of dense cyclically ordered sets with at least two elements. We determine the classifying space of $\mathsf {Mod}(T)$ to be homotopically equivalent to $\mathbb {CP}^\infty $. In particular, $\pi _2(\lvert \mathsf {Mod}(T)\rvert )=\mathbb {Z}$, which answers a question in our previous work. The computation is based on Connes’ cycle category $\Lambda $.

The aim of the present thesis is twofold. First we propose a constructive solution to Frege's puzzle using an approach based on homotopy type theory, a newly proposed foundation of mathematics that possesses a higher-dimensional treatment of equality. We claim that, from the viewpoint of constructivism, Frege's solution is unable to explain the so-called ‘cognitive significance' of equality statements, since, as we shall argue, not only statements of the form 'a = b', but also 'a = a' (...) may contribute to an extension of knowledge. Second, we study this higher-dimensional account of equality from a constructive (computational) standpoint and, based on these considerations, we offer a new perspective to the project proposed by Peter Aczel of developing conventions and notations for an informal style of doing mathematics in type theory. To that end, we adopt the cubical type theory of Angiuli, Favonia and Harper, a framework that can be seen as constructive refinement of the ideas of homotopy type theory. (shrink)

One takes advantage of some basic properties of every homotopic \(\lambda\)-model (e.g. extensional Kan complex) to explore the higher \(\beta\eta\)-conversions, which would correspond to proofs of equality between terms of a theory of equality of any extensional Kan complex. Besides, Identity types based on computational paths are adapted to a type-free theory with higher \(\lambda\)-terms, whose equality rules would be contained in the theory of any \(\lambda\)-homotopic model.

This edited work presents contemporary mathematical practice in the foundational mathematical theories, in particular set theory and the univalent foundations. It shares the work of significant scholars across the disciplines of mathematics, philosophy and computer science. Readers will discover systematic thought on criteria for a suitable foundation in mathematics and philosophical reflections around the mathematical perspectives. The first two sections focus on the two most prominent candidate theories for a foundation of mathematics. Readers may trace current research in set (...) theory, which has widely been assumed to serve as a framework for foundational issues, as well as new material elaborating on the univalent foundations, considering an approach based on homotopy type theory (HoTT). The further sections then build on this and are centred on philosophical questions connected to the foundations of mathematics. Here, the authors contribute to discussions on foundational criteria with more general thoughts on the foundations of mathematics which are not connected to particular theories. This book shares the work of some of the most important scholars in the fields of set theory (S. Friedman), non-classical logic (G. Priest) and the philosophy of mathematics (P. Maddy). The reader will become aware of the advantages of each theory and objections to it as a foundation, following the latest and best work across the disciplines and it is therefore a valuable read for anyone working on the foundations of mathematics or in the philosophy of mathematics. (shrink)

We introduce a homotopy-theoretic interpretation of intuitionistic first-order logic based on ideas from Homotopy Type Theory. We provide a categorical formulation of this interpretation using the framework of Grothendieck fibrations. We then use this formulation to prove the central property of this interpretation, namely homotopy invariance. To do this, we use the result from [8] that any Grothendieck fibration of the kind being considered can automatically be upgraded to a two-dimensional fibration, after which the invariance (...) property is reduced to an abstract theorem concerning pseudonatural transformations of morphisms into two-dimensional fibrations. (shrink)

I apply homotopy type theory to the hole argument as formulated by Earman and Norton. I argue that HoTT gives a precise sense in which diffeomorphism-related Lorentzian manifolds represent the same spacetime, undermining Earman and Norton’s verificationist dilemma and common formulations of the hole argument. However, adopting this account does not alleviate worries about determinism: general relativity formulated on Lorentzian manifolds is indeterministic using this standard of sameness and the natural formalization of determinism in HoTT. Fixing this (...) indeterminism results in a more faithful mathematical representation of general relativity as used by physicists. It also gives a substantive notion of general covariance. (shrink)

Homotopy type theory is a new branch of mathematics that connects algebraic topology with logic and computer science, and which has been proposed as a new language and conceptual framework for math- ematical practice. Much of the power of HoTT lies in the correspondence between the formal type theory and ideas from homotopy theory, in par- ticular the interpretation of types, tokens, and equalities as spaces, points, and paths. Fundamental to the use of (...) identity and equality in HoTT is the powerful proof technique of path induction. In the ‘HoTT Book’ this principle is justified through the homotopy interpretation of type theory, by treating identifications as paths and the induction step as a homotopy between paths. This is incompatible with HoTT being an au- tonomous foundation for mathematics, since any such foundation must be able to justify its principles without recourse to existing areas of mathemat- ics. In this paper it is shown that path induction can be motivated from pre-mathematical considerations, and in particular without recourse to ho- motopy theory. This makes HoTT a candidate for being an autonomous foundation for mathematics. (shrink)

Higher-order logic, with its type-theoretic apparatus known as the simple theory of types (STT), has increasingly come to be employed in theorizing about properties, relations, and states of affairs—or ‘intensional entities’ for short. This paper argues against this employment of STT and offers an alternative: ordinal type theory (OTT). Very roughly, STT and OTT can be regarded as complementary simplifications of the ‘ramified theory of types’ outlined in the Introduction to Principia Mathematica (on a realist (...) reading). While STT, understood as a theory of intensional entities, retains the Fregean division of properties and relations into a multiplicity of categories according to their adicities and ‘input types’ and discards the division of intensional entities into different ‘orders’, OTT takes the opposite approach: it retains the hierarchy of orders (though with some modifications) and discards the categorisation of properties and relations according to their adicities and input types. In contrast to STT, this latter approach avoids intensional counterparts of the Epimenides and related paradoxes. Fundamental intensional entities lie at the base of the proposed hierarchy and are also given a prominent part to play in the individuation of non-fundamental intensional entities. (shrink)

Recent debates in metaphysics have highlighted the significance of type theories, such as Simple Type Theory (STT), for our philosophical analysis. In this chapter, I present the salient features of a constructive type theory in the style of Martin-Löf, termed CTT. My principal aim is to convey the flavour of this rich, flexible and sophisticated theory and compare it with STT. I especially focus on the forms of quantification which are available in CTT. A (...) further aim is to argue that a comparison between a plurality of theories is beneficial to the philosophical analysis. We may, for example, discover helpful features of one theory that we may want to import into another context, thus enriching our repertoire of formal tools. Or, through comparison with a less well-known theory, we may gain a better understanding of the characteristics of the theories we are familiar with. As argued in this chapter, CTT has much to offer in all of these respects. (shrink)

An appropriate framework is put forward for the construction of $$\lambda $$ -models with $$\infty $$ -groupoid structure, which we call homotopic $$\lambda $$ -models, through the use of an $$\infty $$ -category with cartesian closure and enough points. With this, we establish the start of a project of generalization of Domain Theory and $$\lambda $$ -calculus, in the sense that the concept of proof (path) of equality of $$\lambda $$ -terms is raised to higher proof (homotopy).

How does one formalize the structure of structures necessary for the foundations of physics? This work is an attempt at conceptualizing the metaphysics of pregeometric structures, upon which new and existing notions of quantum geometry may find a foundation. We discuss the philosophy of pregeometric structures due to Wheeler, Leibniz as well as modern manifestations in topos theory. We draw attention to evidence suggesting that the framework of formal language, in particular, homotopy type theory, provides the (...) conceptual building blocks for a theory of pregeometry. This work is largely a synthesis of ideas that serve as a precursor for conceptualizing the notion of space in physical theories. In particular, the approach we espouse is based on a constructivist philosophy, wherein ``structureless structures'' are syntactic types realizing formal proofs and programs. Spaces and algebras relevant to physical theories are modeled as type-theoretic routines constructed from compositional rules of a formal language. This offers the remarkable possibility of taxonomizing distinct notions of geometry using a common theoretical framework. In particular, this perspective addresses the crucial issue of how spatiality may be realized in models that link formal computation to physics, such as the Wolfram model. (shrink)

Type theory is a fast-evolving field at the crossroads of logic, computer science and mathematics. This gentle step-by-step introduction is ideal for graduate students and researchers who need to understand the ins and outs of the mathematical machinery, the role of logical rules therein, the essential contribution of definitions and the decisive nature of well-structured proofs. The authors begin with untyped lambda calculus and proceed to several fundamental type systems culminating in the well-known and powerful Calculus of (...) Constructions. The book also covers the essence of proof checking and proof development, and the use of dependent type theory to formalize mathematics. The only prerequisites are a good knowledge of undergraduate algebra and analysis. Carefully chosen examples illustrate the theory throughout. Each chapter ends with a summary of the content, some historical context, suggestions for further reading and a selection of exercises to help readers familiarize themselves with the material. (shrink)

Selection type theories solve adaptation problems. Natural selection, clonal selection for antibody production, and selective theories of higher brain function are examples. An abstract characterization of typical selection processes is generated by analyzing and extending previous work on the nature of natural selection. Once constructed, this abstraction provides a useful tool for analyzing the nature of other selection theories and may be of use in new instances of theory construction. This suggests the potential fruitfulness of research to find (...) other theory types and construct their abstractions. (shrink)

Many philosophers believe in things, propositions, which are the things that we believe, assert etc., and which are the contents of sentences. The act-type theory of propositions is an attempt to say what propositions are, to explain how we stand in relations to them, and to explain why they are true or false. The core idea of the act-type theory is that propositions are types of acts of predication. The theory is developed in various ways (...) to offer explanations of the important properties of propositions. I present the core idea of the theory, and some developments of it. I discuss the relationship between the theory and the content–force distinction. I also present an important type of objection that has been raised to the explanations offered by the act-type theory. (shrink)

Standard Type Theory, STT, tells us that b^n(a^m) is well-formed iff n=m+1. However, Linnebo and Rayo have advocated the use of Cumulative Type Theory, CTT, has more relaxed type-restrictions: according to CTT, b^β(a^α) is well-formed iff β > α. In this paper, we set ourselves against CTT. We begin our case by arguing against Linnebo and Rayo’s claim that CTT sheds new philosophical light on set theory. We then argue that, while CTT ’s (...) class='Hi'>type-restrictions are unjustifiable, the type-restrictions imposed by STT are justified by a Fregean semantics. What is more, this Fregean semantics provides us with a principled way to resist Linnebo and Rayo’s Semantic Argument for CTT. We end by examining an alternative approach to cumulative types due to Florio and Jones; we argue that their theory is best seen as a misleadingly formulated version of STT. (shrink)

Neil Tennant’s core logic is a type of bilateralist natural deduction system based on proofs and refutations. We present a proof system for propositional core logic, explain its connections to bilateralism, and explore the possibility of using it as a type theory, in the same kind of way intuitionistic logic is often used as a type theory. Our proof system is not Tennant’s own, but it is very closely related, and determines the same consequence relation. (...) The difference, however, matters for our purposes, and we discuss this. We then turn to the question of strong normalization, showing that although Tennant’s proof system for core logic is not strongly normalizing, our modified system is. (shrink)

The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the Univalence Axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. While it is incompatible with conventional foundations, it is a (...) powerful addition to homotopy type theory. It also gives the new system of foundations a distinctly structural character. (shrink)