Homotopical patch theory | ACM SIGPLAN Notices

Homotopy Type Theory: Univalent Foundations of Mathematics

Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and beautiful "univalence axiom" implies that isomorphic structures can be identified. On the other hand, "higher inductive types" provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory. Both are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of "logic of homotopy types". This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an "invariant" conception of the objects of mathematics -- and convenient machine implementations, which can serve as a practical aid to the working mathematician. This book is intended as a first systematic exposition of the basics of the resulting "Univalent Foundations" program, and a collection of examples of this new style of reasoning -- but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant.

Internal type theory

We introduce categories with families as a new notion of model for a basic framework of dependent types. This notion is close to ordinary syntax and yet has a clean categorical description. We also present categories with families as a generalized algebraic theory. Then we define categories with families formally in Martin-Löf's intensional intuitionistic type theory. Finally, we discuss the coherence problem for these internal categories with families.

The HoTT Book

Homotopy Type Theory: Univalent Foundations of Mathematics The Univalent Foundations Program Institute for Advanced Study Buy a hardcover copy for $21.00. [620 pages, 6″ × 9″ size, hard…

Categorical logic and type theory

Categorical logic and type theory by Bart Jacobs, 1999, Elsevier Science edition, in English - 1st ed.

The Principal Type-Scheme of an Object in Combinatory Logic

R. Hindley, The Principal Type-Scheme of an Object in Combinatory Logic, Transactions of the American Mathematical Society, Vol. 146 (Dec., 1969), pp. 29-60

generalized algebraic theory in nLab

A generalized algebraic theory (GAT, Cartmell 1978/86) is a dependent type theory in the syntactic sense: There are judgments declaring types and terms of types, where, critically, a type AA declared in context Γ\Gamma is allowed to depend on terms of Γ\Gamma. The only other judgments allowed are of equality (judgemental equality) of pairs of types or terms.

Syntax and Semantics of Linear Dependent Types

A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax...

Principal type-schemes for functional programs | Proceedings of the 9th ACM SIGPLAN-SIGACT symposium on Principles of programming languages

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Categories of Containers

We introduce the notion of containers as a mathematical formalisation of the idea that many important datatypes consist of templates where data is stored. We show that containers have good closure properties under a variety of constructions including the formation of initial algebras and final coalgebras. We also show that containers include strictly positive types and shapely types but that there are containers which do not correspond to either of these. Further, we derive a representation result classifying the nature of polymorphic functions between containers. We finish this paper with an application to the theory of shapely types and refer to a forthcoming paper which applies this theory to differentiable types.

General Theory of Natural Equivalences

Samuel Eilenberg, Saunders MacLane, General Theory of Natural Equivalences, Transactions of the American Mathematical Society, Vol. 58, No. 2 (Sep., 1945), pp. 231-294

A formulation of the simple theory of types

The purpose of the present paper is to give a formulation of the simple theory of types which incorporates certain features of the calculus of λ-conversion. A complete incorporation of the calculus of λ-conversion into the theory of types is impossible if we require that λx and juxtaposition shall retain their respective meanings as an abstraction operator and as denoting the application of function to argument. But the present partial incorporation has certain advantages from the point of view of type theory and is offered as being of interest on this basis (whatever may be thought of the finally satisfactory character of the theory of types as a foundation for logic and mathematics).For features of the formulation which are not immediately connected with the incorporation of λ-conversion, we are heavily indebted to Whitehead and Russell, Hilbert and Ackermann, Hilbert and Bernays, and to forerunners of these, as the reader familiar with the works in question will recognize.The class of type symbols is described by the rules that ı and o are each type symbols and that if α and β are type symbols then (αβ) is a type symbol: it is the least class of symbols which contains the symbols ı and o and is closed under the operation of forming the symbol (αβ) from the symbols α and β.

Elements of ∞-Category Theory in nLab

on (∞,1)-category theory formulated via ∞-cosmoi and the homotopy 2-category of (∞,1)-categories (formal ( ∞ , 1 ) (\infty,1) -category theory).

Formalizing the Real Numbers in Homotopy Type Theory with Cubical Agda

Real numbers in constructive mathematics have always seemed to require compromises of one form or another. Classical proofs of Cauchy completeness require countable choice, Bishop's setoid construction introduces persistent bookkeeping overhead on every definition and theorem, and Dedekind cuts force cumbersome universe-level tracking in predicative type theory. The Homotopy Type Theory (HoTT) book presents an alternative construction of the Cauchy real numbers as a higher inductive-inductive type family, avoiding all three compromises. We formalize the HoTT book reals in Cubical Agda, a proof assistant whose native support for higher inductive types allows the construction to be expressed directly. The code type-checks without postulates or holes, providing a foundation for further machine-assisted work in constructive analysis.