Homotopy Type Theory Research Papers (original) (raw)

- by Paul Green
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- Mathematical Physics, Quantum Physics, Calabi-Yau, Homotopy Type Theory

On the basis of Martin-Löf's meaning explanations for his type theory a detailed justification is offered of the rule of identity elimination. Brief discussions are thereafter offered of how the univalence axiom fares with respect to... more

- by Ansten Klev
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- Type Theory, Martin-Löf's Type Theory, Homotopy Type Theory

in S. Centrone, D. Kant, and D. Sarikaya, eds, Reflections on the Foundations of Mathematics

Homotopy type theory (HoTT) is a proposal for a new foundational framework for mathematics, officially launched by a pool of mathematicians, philosophers, and computer scientists, who worked together during the year 2012-2013 IAS dedicated to the project.There are three the key features that distinguish HoTT from traditional foundational systems. The first concerns its logic, which, being based on a dependent type theory, facilitates a direct implementation of computer-assisted mathematics. The second aspect is its essentially geometrical nature, for it regards abstract spaces as fundamental objects, namely homotopy types. The third point is the bridge HoTT builds between the latter and higher category theory, following what was Grothendieck’s dream. As a result, by extracting an ∞-groupoid from a shape, the structural character of mathematics is revealed. The aim of this dissertation is to provide a fairly extensive survey of the project, albeit striving to keep it approachable by practising mathematicians without a background in the subject. My hope is to open a discussion on HoTT as a foundation by assessing its aptness to the extent of replacing the traditional system, as well as to shed a light on the philosophically interesting issues concerning the foundations of mathematics.

The aim of this paper is to present a different reading of the structuralist thesis in philosophy of mathematics. Accordingly, this thesis is not regarded as providing an account of the ontological and metaphysical nature of mathematical objects and their properties; in fact, such a view has elicited several objections difficult to dismiss. On the contrary, the core insight of structuralism could be regarded as prescribing what mathematics should be about and on how such mathematics should be expressed. As a consequence, the tenets of structuralism become normative constraints which act at the level of foundations, specifically on the formal language and framework. We shall show that, once faced within the appropriate conception of mathematics, the former criticisms could be countered. Finally, since such a shift in foundations is motivated by independent mathematical reasons as well, we shall argue that going “normative” and trying to change mathematics in order to save the structuralist insight is not an ad hoc move.

- by Michael Hopkins
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- Algebra, Structures, Normalization, Hochschild Cohomology

Resumen Lo que distingue a los diálogos formales de los materiales es que en estos últimos la formulación de la Regla Socrática prescribe una forma de interacción que permite al Proponente basar la afirmación de una proposición elemental en identidades específicas a la proposición en cuestión. El objetivo del presente artículo es describir sucintamente la forma de producir diálogos materiales y al mismo tiempo discutir sus vínculos con el desarrollo de lenguajes totalmente interpretados en la Teoría Constructiva de Tipos (TCT). A modo de ilustración discutiremos brevemente la formulación de la Regla Socrática para el conjunto de números naturales, para la noción de identidad predicativa y para el conjunto de Booleanos.

- by Shahid Rahman
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- Type Theory, Game Theory, Philosophy, Philosophy Of Language

We introduce a new algebraic concept of an algebra which is "almost" commutative (more precisely "quasi-commutative differential graded algebra" or ADGQ, in French). We associate to any simplicial set X an ADGQ - called D(X) - and show how we can recover the homotopy type of the topological realization of X from this algebraic structure (assuming some finiteness conditions). The

This paper responds to recent work in the philosophy of Homotopy Type Theory by James Ladyman and Stuart Presnell. They consider one of the rules for identity, path induction, and justify it along 'pre-mathematical' lines. I give an alternate justification based on the philosophical framework of in-ferentialism. Accordingly, I construct a notion of harmony that allows the inferentialist to say when a connective or concept is meaning-bearing and this conception unifies most of the prominent conceptions of harmony through category theory. This categorical harmony is stated in terms of adjoints and so any concept definable by iterated adjoints from general categorical operations is harmonious. Moreover, it has been shown that identity in a categorical setting is determined by an adjoint in the relevant way. Furthermore, path induction as a rule comes from this definition. Thus we arrive at an account of how path induction, as a rule of inference governing identity, can be justified on mathematically motivated grounds.

Homotopy Type Theory (HoTT) has recently been challenged as to its foundational status. James Ladyman and Stuart Presnell have contested that HoTT's presentation is not sufficiently `pre-mathematical'. I argue that indeed the presentation is not pre-mathematical and that's a good thing. I give a \emph{mathematical} justification for the rule of inference in HoTT they find particularly problematic: Path Induction. I provide a categorical justification of both the internal structure and external representation of Path Induction. This justification, although mathematical, has its philosophic roots in philosophy of logic and language. Particularly, I draw from Prawitz style \emph{inferentialism} and the notion of \emph{Harmony} championed for example by Michael Dummett. This constellation of ideas is unified through categorical logic. In the end, I hope to have justified Path Induction in response to valid philosophical concerns.

Let M:=(M^{4},\om) be a 4-dimensional rational ruled symplectic manifold and denote by w_{M} its Gromov width. Let Emb_{\omega}(B^{4}(c),M) be the space of symplectic embeddings of the standard ball B^4(c) \subset \R^4 of radius r and of capacity c:= \pi r^2 into (M,\om). By the work of Lalonde and Pinsonnault, we know that there exists a critical capacity \ccrit \in (0,w_{M}] such that, for all c\in(0,\ccrit), the embedding space Emb_{\omega}(B^{4}(c),M) is homotopy equivalent to the space of symplectic frames \SFr(M). We also know that the homotopy type of Emb_{\omega}(B^{4}(c),M) changes when c reaches \ccrit and that it remains constant for all c \in [\ccrit,w_{M}). In this paper, we compute the rational homotopy type, the minimal model, and the cohomology with rational coefficients of \Emb_{\omega}(B^{4}(c),M) in the remaining case c \in [\ccrit,w_{M}). In particular, we show that it does not have the homotopy type of a finite CW-complex.

- by Martin Pinsonnault
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- Geometry And Topology, Pure Mathematics, Symplectic geometry, Dimensional

We introduce a new algebraic concept of a difierential graded al- gebra which is "almost" commutative (ADGQ, in French). We associate to any simplicial set X an ADGQ - called D(X) - and show how we can recover the homotopy type of the topological realization of X from this algebraic structure (assuming some flniteness conditions). The theory is su-ciently general

- by Max Karoubi
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- Tensor product semigroups, Hochschild Homology, Homotopy Type Theory

We outline the main features of the definitions and applications of crossed complexes and cubical omega\omegaomega-groupoids with connections. These give forms of higher homotopy groupoids, and new views of basic algebraic topology and the... more

- by Ronald Brown
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- Category Theory, Non-commutative Geometry, Homotopy Type Theory

This paper aims to help the development of new models of homotopy type theory, in particular with models that are based on realizability toposes. For this purpose it develops the foundations of an internal simplicial homotopy that does... more

- by Wouter W P Stekelenburg
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- Homotopy Theory, Category Theory, Homotopy Type Theory

Статья посвящена рассмотрению текущего состояния конструктивистского и структуралистского направлений в философии математики. Обосновывается взаимосвязь указанных направлений в рамках теоретико-типового подхода к основаниям математики через рассмотрение понятий изоморфизма, инварианта и структуры. Особо обсуждаются аксиома унивалентности и современное теоретико-типовое понятие равенства.

- by Lev Lamberov
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- Type Theory, Philosophy Of Mathematics, Constructivism, Structure

This paper shows constructive alternatives for a couple of theorems that make simplicial sets a model of homotopy type theory. The constructive theories are valid in the exact completion of the category of assemblies. Ultimately this... more

- by Wouter W P Stekelenburg
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- Categorical Logic, Homotopy Type Theory

The concept of " change of perspective " , where a single object is viewed from different angles, each providing a different image (that is a different formal description) but each lacking nothing of the ipseity of the object nor of its essential qualities, has received no formalization so far. In this paper we formalize this concept using identity types in pure intentional type theory and we briefly give the sketch of two of the applications of this formalization in natural language processing and programming languages.

Let PsubsetRdP\subset\R^dPsubsetRd be a ddd-dimensional polytope. The {\em realization space} of~$P$ is the space of all polytopes P′subsetRdP'\subset\R^dP′subsetRd that are combinatorially equivalent to~$P$, modulo affine transformations. We report on work by the first author, which shows that realization spaces of \mbox{4-dimensional} polytopes can be ``arbitrarily bad'': namely, for every primary semialgebraic set~$V$ defined over~$\Z$, there is a 444-polytope P(V)P(V)P(V) whose realization space is ``stably equivalent'' to~$V$. This implies that the realization space of a 444-polytope can have the homotopy type of an arbitrary finite simplicial complex, and that all algebraic numbers are needed to realize all 444- polytopes. The proof is constructive. These results sharply contrast the 333-dimensional case, where realization spaces are contractible and all polytopes are realizable with integral coordinates (Steinitz's Theorem). No similar universality result was previously known in any fixed dimension.

- by Jürgen Richter-Gebert
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- Pure Mathematics, Dimensional, Homotopy Type Theory, Affine Transformation

In this paper we show that fundamental groups of complements of curves are \small" in the sense that they are \almost solvable". Thus we can start to compute …2 as a module over …1 in order to produce new invariants of surfaces that might distinguish difierent components of a moduli space. 0. Applications of the calculations of fundamental groups to algebraic surfaces. Our study of fundamental groups of branch curves is aimed towards understand- ing algebraic surfaces. Algebraic surfaces are classifled by discrete and continuous invariants. Fixing the discrete invariants (of homotopy type), one gets a family of algebraic surfaces parametrized by an algebraic variety which is called the moduli space (the word \moduli" stands for \continuous invariants"). Very little is known about the struc- ture of moduli space of surfaces of general type. We study algebraic surfaces in order to understand the structure of their mod- uli spaces. We intend to construct new invaria...

- by Mina Teicher
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- Homotopy Type Theory, Algebraic Variety
- by Sergey Yuzvinsky
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- Pure Mathematics, Homotopy Type Theory, Topological Space

We show that the extension types occurring in Riehl—Shulman's work on synthetic (∞, 1)-categories can be interpreted in the intended semantics in a way so that they are strictly stable under substitution. The splitting method used here is... more

- by Jonathan Weinberger
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- Type Theory, Category Theory, Homotopy Type Theory
- by Emmanuel Farjoun
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- Pure Mathematics, Homotopy Type Theory

For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy perturbation transform method(HPTM). The solution is compared with the exact solution. The comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height of water table. The results resemble well with the physical phenomena.

- by Hamidreza Bahramian
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- Type Theory, Formal Semantics, Homotopy Type Theory, Copredication
- by Grbic Jelena
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- Applied Mathematics, Mathematical Physics, Topology, Pure Mathematics
- by Haynes Miller
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- Spectrum, Homotopy Type Theory, Contemporary Mathematics

Within the framework of Riehl-Shulman's synthetic (∞, 1)-category theory, we present a theory of two-sided cartesian fibrations. Central results are several characterizations of the two-sidedness conditionà la Chevalley, Gray, Street, and Riehl-Verity, a two-sided Yoneda Lemma, as well as the proof of several closure properties. Along the way, we also define and investigate a notion of fibered or sliced fibration which is used later to develop the two-sided case in a modular fashion. We also briefly discuss discrete two-sided cartesian fibrations in this setting, corresponding to (∞, 1)-distributors. The systematics of our definitions and results closely follows Riehl-Verity's ∞-cosmos theory, but formulated internally to Riehl-Shulman's simplicial extension of homotopy type theory. All the constructions and proofs in this framework are by design invariant under homotopy equivalence. Semantically, the synthetic (∞, 1)-categories correspond to internal (∞, 1)-categories implemented as Rezk objects in an arbitrary given (∞, 1)-topos.

- by Jonathan Weinberger
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- Higher Category Theory, Homotopy Type Theory, Fibered Category
- by Grbic Jelena
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- Applied Mathematics, Mathematical Physics, Topology, Pure Mathematics

Abstract Preserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. This paper constitutes an introduction to the study of non-trivial simple sets in the framework of cubical 3-D complexes. A simple set has the property that the homotopy type of the object in which it lies is not changed when the set is removed. The main contribution of this paper is a characterisation of the non-trivial simple sets composed of exactly two voxels, such sets being called minimal simple pairs.

- by Michel Couprie
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- Applied Mathematics, Image Analysis, Homotopy Type Theory, Topological Properties

Weak ω-groupoids are the higher dimensional generalisation of setoids and are an essential ingredient of the constructive semantics of Homotopy Type Theory [13]. Following up on our previous formalisation [3] and Brunerie's notes [6], we present a new formalisation of the syntax of weak ω-groupoids in Agda using heterogeneous equality. We show how to recover basic constructions on ω-groupoids using suspension and replacement. In particular we show that any type forms a groupoid and we outline how to derive higher dimensional composition. We present a possible semantics using globular sets and discuss the issues which arise when using globular types instead.

- by Ondrej Rypacek
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- Type Theory, Category Theory, Homotopy Type Theory

Рассматривается понятие доказательства в связи с соответствием Карри-Говарда. Исследуются особенности этого понятия, а также различия классической доктрины «высказывания как типы» и современной доктрины «высказывания как некоторые типы». Особое внимание уделяется проблеме статуса логического в математике. Настоящая статья — третья в серии о понятии теоретико-типового доказательства.

- by Lev Lamberov
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- Mathematics, Type Theory, Logic, Intuitionism
- by Daniel Licata
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- Cognitive Science, Type Theory, Homotopy Theory, Category Theory
- by Alexandru Ion
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- Geometric Topology, Algebraic Geometry, Combinatorics, Algebraic Topology
- by Boris Shapiro
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- Stratification, Homotopy Type Theory
- by Emzar Khmaladze
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- Mathematics, Pure Mathematics, Homotopy Type Theory, Equivariant Cohomology
- by Ralph Cohen
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- Pure Mathematics, Loop Groups, Homotopy Type Theory
- by Martin Pinsonnault
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- Differential Geometry, Pure Mathematics, Stratification, Symplectic geometry
- by Thomas Schick
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- Mathematical Sciences, Physical sciences, Homotopy Type Theory

- by Ralph Cohen
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- Mathematics, Applied Mathematics, Homotopy Theory, Pure Mathematics

The Hom complex of homomorphisms between two graphs was originally introduced to provide topological lower bounds on the chromatic number of graphs. In this paper we introduce new methods for understanding the topology of Hom complexes, mostly in the context of Γ-actions on graphs and posets (for some group Γ). We view the Hom(T, •) and Hom(•, G) as functors from graphs to posets, and introduce a functor (•) 1 from posets to graphs obtained by taking the atoms as vertices. Our main structural results establish useful interpretations of the equivariant homotopy type of Hom complexes in terms of spaces of equivariant poset maps and Γ-twisted products of spaces. When P: = F(X) is the face poset of a simplicial complex X, this provides a useful way to control the topology of Hom complexes. These constructions generalize those of the second author from [18] and well as the calculation of the homotopy groups of Hom complexes from [9]. Our foremost application of these results is the const...

- by Anton Dochtermann
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- Mathematics, Pure Mathematics, Symmetric group, Lower Bound

We study some of the combinatorial structures related to the signature of G-symmetric products of (open) surfaces SP m G (M) = M m /G where G ⊂ Sm. The attention is focused on the question what information about a surface M can be recovered from a symmetric product SP n (M). The problem is motivated in part by the study of locally Euclidean topological commutative (m + k, m)-groups, [16]. Emphasizing a combinatorial point of view we express the signature Sign(SP m G (M)) in terms of the cycle index Z(G; ¯x) of G, a polynomial which originally appeared in Pólya enumeration theory of graphs, trees, chemical structures etc. The computations are used to show that there exist punctured Riemann surfaces Mg,k, Mg ′,k ′ such that ′) are often not homeomorphic, the manifolds SP m (Mg,k) and SP m (Mg′,k although they always have the same homotopy type provided 2g + k = 2g ′ + k ′ and k, k ′ ≥ 1. 1

- by Pavle Blagojević
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- Pure Mathematics, Homotopy Type Theory, Indexation, Point of View
- by Indranil Biswas
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- Mathematics, Pure Mathematics, Homotopy Type Theory