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\tableofcontents
\section{Idea}
Homotopy type theory is a framework of dependent type theories which additionally consists of
identity typesidentity types
dependent product types?dependent product types
dependent sum types?dependent sum types
univalent universesunivalent universes
inductive types?inductive types, higher inductive types, inductive type families, et cetera., higher inductive types, inductive type families?, et cetera.
\section{Presentation}
The model of homotopy type theory we shall be presenting here is the objective type theory version of homotopy type theory. There are multiple reasons for this:
Since objective type theory lacks definitional equality,
The ruleset is simpler in the objective type theory model of homotopy type theory than other models such as Martin-Löf type theory, cubical type theory, or higher observational type theory
The results in objective type theory are more general than in models which use definitional equality
It is similar to other non-type theory foundations such as the various flavors of set theory, since it also only has one notion of equality, which is propositional equality in both objective type theory and set theory, and uses propositional equality to define terms and types.
From a more practical standpoint, objective type theory not only has decidable type checking, it has polynomial (quadratic) time type checking, which makes it efficient from a computational standpoint.
\subsection{Types In a similar manner, for simplicity and terms} ease of presentation, we shall also follow the HoTT book in not including a separate judgment and rather stipulating that every type is an element of a Russell universe.
As \subsection{Judgments with and any contexts} type theory, the basic judgments of this model are judging to be a type, , and judging to be a term of , , and judging to be a context, . Contexts are lists of type and term judgments.
\subsection{Dependent We types introduce and two sections} judgments in the model: typing judgments, where we judge to be an element of , , and context judgments, where we judge to be a context, . Contexts are lists of typing judgments , , , et cetera, and are formalized by the rules for the empty context and extending the context by a typing judgment
A dependent type is a type in the context of the variable judgment , . The dependent type is usually written as to indicate the dependence upon .
A \subsection{Structural section rules} is a term in the context of the variable judgment , . Similarly for the case for dependent types, sections are usually written as to indicate its dependence upon .
\subsection{Variable, The Substitution, variable and rule Weakening states rules} that we may derive a typing judgment if the typing judgment is in the context already:
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Let be any arbitrary judgment. Then we have the following rules:
The weakening rule:
The substitution rule:
The weakening and substitution rules are admissible rules: they do not need to be explicitly included in the type theory as they could be proven by induction on the structure of all possible derivations.
\subsection{Russell universes}
Universes are the types of types in type theory.
We introduce a universe hierarchy of universes , with indexed rules by for natural universe numbers formation and type reflection in the metatheory, with rules for universe formation and cumulativity.
\subsection{Sections and dependent types}
A section is a term in the context of the variable judgment , . Sections are usually written as to indicate its dependence upon .
A dependent type is a section of a universe in the context of the variable judgment , , and since they are sections, they are usually written as .
\subsection{Equality}
Equality in type theory is represented by the identity type, which is also called the path type or identification type. The terms of the identity type could be called paths or identifications.
… Equality comes with a formation rule, an introduction rule, an elimination rule, and a computation rule:
Formation rule for identity types:
Introduction rule for identity types:
Elimination rule for identity types:
Conversion rules for identity types:
\subsection{Definitions}
In mathematics, many times one would want to define an element to be another element of a type. In order to define an element to be an element , one says that element comes with an identification . Since types are elements of universes, one defines a type to be a type if the type comes with an identification .
Sometimes, for ease of simplicity, the identity type is simply written for elements and . The type argument for the identity type then becomes an implicit argument. Thus, in order to define to be , one says that comes with an identification .
In a proof assistant or some other program, elaboration is needed to expand out all the implicit arguments to get the right type .
\subsection{Functions}
\subsection{Dependent products}
\subsection{Products}
\subsection{Dependent sums}
\subsection{Sums}
\subsection{Empty type}
\subsection{Unit type}
\subsection{Booleans}
\subsection{Interval}
\subsection{Function extensionality}
\subsection{Equivalences}
\subsection{Univalence}
\subsection{Natural numbers}
\section{References}
Homotopy Type Theory: Univalent Foundations of Mathematics, The Univalent Foundations Program, Institute for Advanced Study, 2013. (web, pdf)
Egbert RijkeEgbert Rijke, , Introduction to Homotopy Type Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press (pdf) (478 pages)
Benno van den BergBenno van den Berg, Martijn den Besten, , Martijn den Besten?, Quadratic type checking for objective type theory (arXiv:2102.00905)
\section{External links}