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\begin{document}

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\section*{h-set}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

\hypertarget{constructivism_realizability_computability}{}\paragraph*{{Constructivism, Realizability, Computability}}\label{constructivism_realizability_computability}

[[!include constructivism - contents]]

\hypertarget{hsets}{}\section*{{h-Sets}}\label{hsets}

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{equivalent_characterizations}{Equivalent characterizations}\dotfill \pageref*{equivalent_characterizations} \linebreak
\noindent\hyperlink{RelationToInternalSets}{Relation to internal sets}\dotfill \pageref*{RelationToInternalSets} \linebreak
\noindent\hyperlink{PretoposOfHsets}{Pretopos of hsets}\dotfill \pageref*{PretoposOfHsets} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea}

In [[homotopy type theory]] an \emph{h-set} is a [[type]] $X$ -- hence a [[homotopy type]] -- with the special property that any two of its [[terms]] $x,y : X$ are [[equality|equal]] ([[equivalence|equivalent]]) in an at most essentially unique way, hence that the [[identity type]] $(x = y) : Type$ is an [[h-proposition]].

The notion of h-set is an [[internalization]] of the notion of [[0-truncated]] object into homotopy type theory, essentially an internalization of the notion of \emph{[[set]]} (or possibly of \emph{[[preset]]}). See below in \emph{\hyperlink{RelationToInternalSets}{Relation to internal sets}} for more on this.

h-Sets can also be regarded as a way of embedding [[extensional type theory]] into [[intensional type theory]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $A$ be a [[type]] in [[intensional type theory|intensional]] [[type theory]] with [[dependent sums]], [[dependent products]], and [[identity types]]. We define a new type $isSet(A)$ as follows:

\begin{displaymath}
isSet(A) \coloneqq \prod_{x\colon A} \prod_{y\colon A} isProp(x=y)
\end{displaymath}
(using any equivalent definition of the [[predicate]] [[isProp]] for [[h-propositions]]; and where ``$\prod$'' denotes [[dependent product]] types and ``$=$'' denotes [[identity types]]).

In other words, the only relationship between two elements of an h-set is whether they are equal; there is no room for more than one path between them. By [[beta reduction|beta-reducing]] this definition, we can express it as

\begin{displaymath}
isSet(A) \coloneqq \prod_{x,y\colon A} \prod_{p,q\colon x=y} (p=q)
\end{displaymath}
In other words, any two parallel paths in $A$ are equal.

A provably [[equivalence in homotopy type theory|equivalent]] definition is

\begin{displaymath}
isSet(A) \coloneqq \prod_{x\colon A} \prod_{p\colon x=x} (p=id_x)
\end{displaymath}
This says that a version of Streicher's ``[[axiom K]]'' holds for h-sets. (See also at \emph{[[axiom UIP]]}.)

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item Most (non-[[higher inductive type|higher]]) [[inductive types]] are h-sets (assuming that all their parameters and indices are so). In particular, the type of [[natural numbers]] is an h-set. This can be proven from Theorem \ref{DecidableIsSet} below.

\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{equivalent_characterizations}{}\subsubsection*{{Equivalent characterizations}}\label{equivalent_characterizations}

One interesting consequence of this definition is the following, first proven in (\hyperlink{Hedberg}{Hedberg})

\begin{theorem}
\label{DecidableIsSet}\hypertarget{DecidableIsSet}{}
(\textbf{[[Hedberg's theorem]]})

Suppose that $A$ is a [[type]] which has [[decidable equality]] in the [[propositions as types]] [[logic]] (which is not the logic of h-propositions usually used in HoTT). In other words, the projection

\begin{displaymath}
\itexarray{Paths_A + (0\to A\times A)^{(Paths_A\to A\times A)}\\
\downarrow\\
A\times A}
\end{displaymath}
(where $Paths_A$ is the [[path object|path type]] of $A$, ``+'' forms the [[sum type]], and on the right we have the $A \times A$-[[dependent type|dependent]] [[function type]] into the [[empty type]]), has a section.

Then $A$ is a h-set.

\end{theorem}
\begin{proof}
Let $d$ be the given [[section]]. Thus, for any $x,y\colon A$, $d(x,y)$ is either a path from $x$ to $y$ or a function from $Paths(x,y)$ to the empty type (implying that $Paths(x,y)$ is also empty).

It suffices to exhibit an operation connecting any endo-path $p \in Paths(x,x)$ to the identity path $1_x$. Given such a path, define $q = d(x,x)$. If $d(x,x)$ lies in the second case, then $Paths(x,x)$ is empty, a contradiction since we know it contains $1_x$; hence we may assume $q\in Paths(x,x)$ as well.

Let $r$ be the image of $(1_x,p) \in Paths_{A\times A}((x,x),(x,x))$ under the section $d$. This is a path in the total space $Paths_A$ lying over the path $(1_x,p)$ in $A\times A$. Equivalently, it is a path in the fiber over $x$ from $(1_x,p)_*(d(x,x))$ to $d(x,x)$, where $(1_x,p)_*$ denotes transport in the fibration $Paths_A \to A\times A$ along the path $(1_x,p)$. However, we have defined $d(x,x) = q$, and transport in a path-space is just composition, so $r$ may be regarded as a path from $q p$ to $q$. Canceling $q$, we obtain a path from $p$ to $1_x$.

\end{proof}
Not every h-set has decidable equality (unless the law of [[excluded middle]] hold), but there are some other related equivalent characterizations.

\begin{itemize}%
\item A type $A$ is an h-set if and only if all its identity types $x=_A y$ have [[split support]], i.e. $\prod_{(x,y:A)} \Vert x=y\Vert \to (x=y)$. This is proven in \hyperlink{KECA}{(KECA)}.


\item More generally, $A$ is an h-set if and only if there is some $R:A\to A\to Prop$ which is reflexive (i.e. $\prod_{(x:A)} R(x,x)$) and such that $\prod_{(x,y:A)} R(x,y) \to (x=y)$. This is Theorem 7.2.2 in the [[HoTT Book]].



\end{itemize}
\hypertarget{RelationToInternalSets}{}\subsubsection*{{Relation to internal sets}}\label{RelationToInternalSets}

When using [[homotopy type theory]] as the ambient [[foundations]], h-sets generally play the role of the [[sets]]. When homotopy type theory is the [[internal logic]] of some [[(∞,1)-category]], then the h-sets are the ``internal sets'' in this internal logic. (Not to be confused with the other meaning of [[internal set]].)

Note, though, that this notion of ``internal set'' is of a different sort from the usual notions of [[internal category]] or [[internal groupoid]]. If an internal set is an h-set, then an ``internal groupoid'' should mean a 1-truncated type, whereas an internal groupoid usually means some kind of [[groupoid object in an (∞,1)-category]]. Conversely, the usual meaning of ``internal groupoid'' suggests that the meaning of ``internal set'' should be something more like a [[setoid]], with the h-sets being more like [[presets]]. This latter meaning is how ``sets'' are more often defined by constructive type theorists.

The point is that to be worthy of the name ``set'', a notion ought to come with ``[[quotients]] of [[equivalence relations]]''. If we start with a notion which does \emph{not} have quotients, such as the types in ordinary [[Martin-Löf dependent type theory]], then in order to get a good notion of ``set'' we need to ``freely add quotients'', which semantically means passing to the [[exact completion]] whose objects are [[setoids]]. But if we start with a notion that does have quotients, then this is unnecessary. In homotopy type theory, h-sets do have quotients, which can be constructed using [[higher inductive types]]; thus it makes sense to call them ``sets'' rather than ``presets''.

A good way to reconcile these seemingly clashing terminologies is to talk about [[exact completions]] of [[unary sites]] or [[(∞,1)-sites]]. The presence of a [[Grothendieck topology]] allows us to ``remember'' to what extent our given notion has well-behaved quotients: if we have no quotients, then we use the [[trivial topology]], whereas if we have quotients, we can use the [[regular topology]]. And the exact completion builds in quotients ``freely'' but preserving those which the topology asserts to already exist. In particular, if we start with quotients (an [[exact category]] or $(\infty,1)$-category), then the exact completion of the regular topology is idempotent, whereas if we start with a trivial topology, then the exact completion gives a category of setoids. Thus, in general, the good notion of ``internal set'' in a unary site is ``object of the exact completion''.

\hypertarget{PretoposOfHsets}{}\subsubsection*{{Pretopos of hsets}}\label{PretoposOfHsets}

The h-sets in [[HoTT]] form a [[ΠW-pretopos]] (\hyperlink{RijkeSpitters13}{Rijke-Spitters 13}). See also at \emph{[[structural set theory]]}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

[[!include homotopy n-types - table]]

[[!include types and logic - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Michael Hedberg, \emph{A coherence theorem for Martin-L\"o{}f's type theory}, J. Functional Programming, (1998)


\item Nicolai Kraus, \emph{A direct proof of Hedberg's theorem}, \href{http://homotopytypetheory.org/2012/03/30/a-direct-proof-of-hedbergs-theorem/}{blog post}


\item [[Nicolai Kraus]], [[Martin Escardo]], [[Thierry Coquand]] , [[Thorsten Altenkirch]], \emph{Generalizations of Hedberg's theorem}, in M. Hasegawa (Ed.): TLCA 2013, LNCS 7941, pp. 173-188. Springer, Heidelberg 2013. (\href{http://www.cs.bham.ac.uk/~mhe/papers/hedberg.pdf}{pdf})



\end{itemize}
Formalization of [[set theory]] via h-sets in [[homotopy type theory]] is discussed in

\begin{itemize}%
\item [[Egbert Rijke]], [[Bas Spitters]], \emph{Sets in homotopy type theory}, Mathematical Structures in Computer Science, Volume 25, Issue 5 (From type theory and homotopy theory to Univalent Foundations of Mathematics) (\href{http://arxiv.org/abs/1305.3835}{arXiv:1305.3835})

\end{itemize}
which became one chapter in

\begin{itemize}%
\item [[Univalent Foundations Project]], \emph{[[Homotopy Type Theory – Univalent Foundations of Mathematics]]}

\end{itemize}
[[!redirects h-set]] [[!redirects h-sets]] [[!redirects hSet]] [[!redirects hSets]] [[!redirects 0-truncated type]] [[!redirects 0-truncated types]] [[!redirects h-level 2]] [[!redirects h-level 2 type]] [[!redirects h-level 2 types]]

[[!redirects Hedberg's theorem]] [[!redirects Hedberg theorem]]



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