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\tableofcontents
\section{Idea}
Homotopy type theory is a framework of dependent type theories which additionally consists of
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identity types
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dependent product types
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dependent sum types
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univalent univalence universes
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inductive types, 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:
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Since objective type theory lacks definitional equality,
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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
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The results in objective type theory are more general than in models which use definitional equality
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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.
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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.
In a similar manner, for simplicity and ease of presentation, we shall also avoid follow the HoTT use book of in universes, not and including have a separate single judgment and rather stipulating that every type is an element additional of type a former Russell universe. for equality of types. This allows us to express univalence as a rule in the type theory, without having to define universes.
Universes are relegated to a discussion of cardinals and size issues in various branches of mathematics, such as in order theory, category theory, set theory, and model theory, and won’t be described here.
\subsection{Judgments and contexts}
We Objective introduce type two theory consists of three judgments: type judgments in the model: typing judgments, where we judge , to where be we an judge element of , to be a type, typing judgments, where we judge , and to context be judgments, an where element we of judge , to be a context, . , Contexts and are context lists judgments, of where typing we judgments 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
\subsection{Structural rules}
There are three structural rules in objective type theory, the variable rule?, the weakening rule?, and the substitution rule?.
The variable rule states that we may derive a typing judgment if the typing judgment is in the context already:
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 \subsection{Dependent universes} types and sections}
Universes A are dependent the types of types in type theory. is a type in the context of the variable judgment , , they are usually written as to indicate its dependence upon .
We A introduce section or dependent term is a hierarchy term of universes indexed in by the natural context numbers of the variable judgment , in the metatheory, with rules for universe formation and cumulativity.. Sections are likewise usually written as to indicate its dependence upon .
\subsection{Equality of elements}
\subsection{Sections Equality and of dependent elements types} of a type in objective type theory is represented by a type known as theequality type or the type of equalities. The elements of the equality type are called equalities.
A Equality section of is elements comes with a term formation rule, an introduction rule, an elimination rule, and a computation rule: in the context of the variable judgment , . Sections are usually written as to indicate its dependence upon .
A Formation dependent rule type for is equality a types section of a elements: universe in the context of the variable judgment , , and since they are sections, they are usually written as .
\subsection{Equality}
Equality Introduction in rule type for theory equality is types represented by the identity type, which is also called the path type or identification type. The terms of the elements: 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 Elimination rule for identity equality types: types of elements:
Introduction Computation rule rules for identity equality types: types of elements:
Elimination \subsection{Equality rule of for types} identity types:
Equality of types is also an identity type, represented by the bare equality symbol for types and . As types do not belong to any type, the rules for the equality types of types is slightly different from the rules for the equality types of terms:
Computation Type rules formation for identity equality of types:
The type formation rule for equality types states that given the hypothesis that and are types in context , one could form the conclusion that in the contexts of one could form the equality type .
Element introduction for equality of types:
The element introduction rule for equality types states that given the hypothesis that is a type in context , one could form the conclusion that in the contexts of one could introduce the element , called reflexivity.
Element elimination for equality of types:
This says that given
Conversion rules for equality of 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 equality . Since Similarly, types are elements of universes, one defines a type to be a type if the type comes with an identification equality . This is formalized with the initialization operator by the following rules for definitions of terms
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 and a similarly proof for assistant definitions or of some types other program,elaboration is needed to expand out all the implicit arguments to get the right type .
\subsection{Natural numbers}
Formation rules for the natural numbers:
Introduction rules for the natural numbers:
Elimination rules for the natural numbers:
Computation rules for the natural numbers:
Uniqueness rules for the natural numbers: …
\subsection{Function types}
Formation rules for function types:
Introduction rules for function types:
Elimination rules for function types:
Computation rules for function types:
Uniqueness rules for function types:
\subsection{Pi types}
Pi types are the dependent versions of functions or products, depending on how one looks at it.
Formation rules for Pi types:
Introduction rules for Pi types:
Elimination rules for Pi types:
Computation rules for Pi types:
Uniqueness rules for Pi types:
\subsection{Product types}
We use the negative presentation for product types.
Formation rules for product types:
Introduction rules for product types:
Elimination rules for product types:
Computation rules for product types:
Uniqueness rules for product types:
\subsection{Sigma types}
We use the negative presentation for sigma types.
Formation rules for Sigma types:
Introduction rules for Sigma types:
Elimination rules for Sigma types:
Computation rules for Sigma types:
Uniqueness rules for Sigma types:
\subsection{Sum types}
\subsection{Empty type}
\subsection{Unit type}
\subsection{Boolean type}
\subsection{Interval type}
\subsection{Function extensionality}
\subsection{Singletons}
A singleton is a type with only one element up to identifications.
We define the function which states that a type is a singleton if there is an identification
\subsection{Fibers}
Given types and let the type family
comes with identifications
for each function and element . The type is known as the fiber of at .
\subsection{Equivalences}
We In define set the theory, type a family set is apointed set? such if that it comes has with at identifications least one chosen element; it is asubsingleton? if it has at most one element up to equality: every two elements in the set are equal to each other, and it is a singleton? if it has exactly one element up to equality: it is a pointed set and a singleton.
These same definitions carry over to objective type theory: A type is a pointed type if it has a chosen element , a type is a subsingleton if for every two elements and , there is an identification? , or equivalently, there is a dependent function
for all types and . The type is known as the type of equivalences between and .
and a type is a singleton if it has both an element and a dependent function
or equivalently, an element
In set theory, the fiber of a function with domain? and codomain? at an element in is the indexed disjoint union? of all elements such that . This definition could also be translated into objective type theory: the fiber of a function at the element is the type
In set theory, a function is a bijection? if all its fibers at every element of the codomain are singletons. This definition can be translated over to objective type theory, but the name used for such a function in objective type theory is equivalence?: A function is an equivalence if there is an element
The type of equivalences is given by
\subsection{Univalence}
For types and there is a function
inducively defined by
Univalence is the statement that one could form the witness that is an equivalence given types and , and is given by the rule
Thus, , and by univalence again, .
\subsection{Function extensionality}
\subsection{Empty type}
Formation rules for the empty type:
Elimination rules for the empty type:
Uniqueness rules for the empty type:
\subsection{Unit type}
Formation rules for the unit type:
Introduction rules for the unit type:
Elimination rules for the unit type:
Computation rules for the unit type:
Uniqueness rules for the unit type:
\subsection{Booleans}
Formation rules for the booleans:
Introduction rules for the booleans:
Elimination rules for the booleans:
Computation rules for the booleans:
Uniqueness rules for the booleans:
\subsection{Natural numbers}
Formation rules for the natural numbers:
Introduction rules for the natural numbers:
Elimination rules for the natural numbers:
Computation rules for the natural numbers:
Uniqueness rules for the natural numbers:
\section{References}
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Homotopy Type Theory: Univalent Foundations of Mathematics, The Univalent Foundations Program, Institute for Advanced Study, 2013. (web, pdf)
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Egbert Rijke, Introduction to Homotopy Type Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press (pdf) (478 pages)
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Benno van den Berg, Martijn den Besten, Quadratic type checking for objective type theory (arXiv:2102.00905)
\section{External links}