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\section*{type of types}

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

\hypertarget{universes}{}\paragraph*{{Universes}}\label{universes}

[[!include universe - contents]]

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

[[!include type theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{formalizations}{Formalizations}\dotfill \pageref*{formalizations} \linebreak
\noindent\hyperlink{RussellStyle}{Type Universe \`a{} la Russell}\dotfill \pageref*{RussellStyle} \linebreak
\noindent\hyperlink{TarskiStyle}{Type Universe \`a{} la Tarski}\dotfill \pageref*{TarskiStyle} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{UniverseEnlargement}{Universe enlargement}\dotfill \pageref*{UniverseEnlargement} \linebreak
\noindent\hyperlink{CategoricalSemantics}{Categorical semantics}\dotfill \pageref*{CategoricalSemantics} \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 [[type theory]], a \emph{type of (small) types} -- usually written $Type$ -- is a [[type]] whose [[terms]] are themselves [[type|types]]. Thus, it is a [[universe]] of (small) [[type|types]], a \emph{universe in type theory}.

 One also speaks of $Type$ as being a \emph{reflection} of the type system in itself (e.g. \hyperlink{MartinLoef74}{MartinL\"o{}f 74, p. 6}, \hyperlink{Palmgren}{Palmgren, pp. 2-3}, \hyperlink{Rathjen}{Rathjen, p. 1}, \hyperlink{Luo11}{Luo 11, section 2.5}, \hyperlink{Luo12}{Luo 12, p. 2}, \hyperlink{SEP}{Stanf. Enc. Phil.}), following the \emph{[[reflection principle]]} in [[set theory]].

In [[homotopy type theory]] a type of (small) types is what in [[(∞,1)-category theory|higher]] [[categorical semantics]] is interpreted as a (small) \emph{[[object classifier]]}. Thus, the type of types is a refinement of the [[type of propositions]] which only contains the [[(-1)-truncated]]/[[h-level]]-1 types (and is semantically a [[subobject classifier]]).

In the presence of a type of types a [[judgement]] of the form

\begin{displaymath}
\vdash A : Type
\end{displaymath}
says that $A$ is a [[term]] of [[type]] $Type$, hence is a (small) [[type]] itself. More generally, a [[hypothetical judgement]] of the form

\begin{displaymath}
x : X \vdash A(x) : Type
\end{displaymath}
says that $A$ is an $X$-[[dependent type]].

In [[homotopy type theory]] the type of types $Type$ is often assumed to satisfy the [[univalence]] [[axiom]]. This is a reflection of the fact that in its [[categorical semantics]] as an [[object classifier]] is part of an [[internal (∞,1)-category]] in the ambient [[(∞,1)-topos]]: the one that as an [[indexed category]] is the small [[codomain fibration]].

[[Per Martin-Lof]]`s original type theory contained a type of \emph{all} types, which therefore in particular contained itself, i.e. one had $Type : Type$. But it was pointed out by [[Jean-Yves Girard]] that this was inconsistent; see [[Girard's paradox]]. Thus, modern type theories generally contain a hierarchy of types of types, with $Type_0 : Type_1$ and $Type_1 : Type_2$, etc.

\hypertarget{formalizations}{}\subsection*{{Formalizations}}\label{formalizations}

\hypertarget{RussellStyle}{}\subsubsection*{{Type Universe \`a{} la Russell}}\label{RussellStyle}

A \textbf{universe \`a{} la Russell} is a [[type]] whose terms \emph{are} types. In the presence of a separate [[judgment]] ``$A \;type$'', this can be formulated as a [[natural deduction|deduction rule]] of the form

\begin{displaymath}
\frac{A:U}{A \;type}
\end{displaymath}
Thus, the \href{natural+deduction#IntroductionAndElimination}{type formers} have rules saying which universe they belong to, such as:

\begin{displaymath}
\frac{A:U\quad B:A\to U}{\Pi\, A\, B : U}
\end{displaymath}
With universes \`a{} la Russell, we can also omit the judgment ``$A\; type$'' and replace it everywhere by a judgment that A is a term of some universe. This is the approach taken by the [[Homotopy Type Theory -- Univalent Foundations of Mathematics|HoTT textbook]] and by [[Coq]].

\hypertarget{TarskiStyle}{}\subsubsection*{{Type Universe \`a{} la Tarski}}\label{TarskiStyle}

A \textbf{universe \`a{} la Tarski} (\hyperlink{Hofmann}{Hofmann, section 2.1.6}, \hyperlink{Luo12}{Luo 12}, \hyperlink{Gallozzi14}{Gallozzi 14, p. 40}) is a type $U$ together with an ``interpretation'' operation allowing us to regard its [[terms]] as \emph{codes} or \emph{names} for actual types. Thus we have a rule such as

\begin{displaymath}
\frac{A:U}{El(A)\;type}
\end{displaymath}
saying that for each term $A$ of the type universe $U$ there is an actual type $El(A)$. (Conversely, with notation as used at \emph{[[object classifier in an (infinity,1)-topos]]}, one might write $A = 'El(A)'$ to indicate that $A$ is the \emph{name} of the type $El(A)$ in the type universe.)

We usually also have operations on the universe corresponding to (but not identical to) type formers, such as

\begin{displaymath}
\frac{A:U\quad B:A\to U}{\pi(A, B) : U}
\end{displaymath}
with an [[equality]] $El(\pi(A,B))=\Pi \, El(A)\, El(B)$. Usually this latter equality (and those for other type formers) is a [[judgmental equality]]. If it is only an [[equivalence in homotopy type theory|equivalence]] (i.e. we have a rule which gives us a canonical term of the equivalence type), we may speak of a \textbf{weakly \`a{} la Tarski universe} (\hyperlink{Gallozzi14}{Gallozzi 14, p. 49-50}).

We can give a slightly different definition of weakly \`a{} la Tarski universe using [[propositional equality]] and a larger universe. More precisely, we can consider two (or many) universes $U$ and $U'$ with the usual rules for the relative reflection $el(a):U'$ for any $a:U$, a choice of weakly or strongly a la Tarski [[computation rules]] for the reflections $El$ and $El'$, and a computation rule for the relative reflection el of $U$ inside $U'$ based on propositional equality, which gives us canonical elements of the identity types $Id_{U'}(\pi'(el(a),el(b)),el(\pi(a,b)))$ and similarly for the other type formers.

If the containing universe is univalent the two definitions turn out to coincide.

Universes defined internally via [[induction-recursion]] are (strongly) \`a{} la Tarski. Weakly \`a{} la Tarski universes are easier to obtain in [[semantics]] (see \hyperlink{CategoricalSemantics}{below}): they are somewhat more annoying to use, but probably suffice for most purposes.

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

\hypertarget{UniverseEnlargement}{}\subsubsection*{{Universe enlargement}}\label{UniverseEnlargement}

Both [[Coq]] and [[Agda]] support [[universe polymorphism]] to deal with the issue of universe enlargement. Moreover, Coq supports [[typical ambiguity]].

\hypertarget{CategoricalSemantics}{}\subsubsection*{{Categorical semantics}}\label{CategoricalSemantics}

[[univalence|Univalent]] type universes \hyperlink{RussellStyle}{\`a{} la Russell} have been shown to be interpreted in [[type-theoretic model categories]] presenting the base [[(∞,1)-topos]] [[∞Grpd]] (\hyperlink{KapulkinLumsdaineVoevodsky12}{Kapulkin-Lumsdaine-Voevodsky 12}) and more generally presenting [[(∞,1)-toposes]] of [[(∞,1)-presheaves]] over [[elegant Reedy categories]] (\hyperlink{Shulman13}{Shulman 13}).

Discussion for general [[(∞,1)-toposes]] (of [[(∞,1)-sheaves]]) that should have implementation \hyperlink{TarskiStyle}{weakly \`a{} la Tarski} (\hyperlink{Gallozzi14}{Gallozzi 14, p. 49-50}) is in (\hyperlink{GepnerKock12}{Gepner-Kock 12}).

For more on this see the respective sections at \emph{[[relation between type theory and category theory]]}.

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

\begin{itemize}%
\item [[resizing axiom]]


\item [[universe polymorphism]]


\item [[univalence]]


\item [[Prop]], the [[type of propositions]],

\begin{itemize}%
\item [[subobject classifier]]

\end{itemize}

\item [[object classifier]]


\item [[parametric polymorphism]]


\item [[Girard's paradox]]


\item [[Awodey's conjecture]]



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

Some of the text above is adapted from the entry \emph{[[homotopytypetheory:universe]]} at the [[homotopytypetheory:HomePage|homotopy type theory web]].

Type universes in [[Martin-Löf type theory]] originate around

\begin{itemize}%
\item [[Per Martin-Löf]], \emph{An intuitionistic theory of types: predicative part}, In Logic Colloquium (1973), ed. H. E. Rose and J. C. Shepherdson (North-Holland, 1974), 73-118. (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.926}{web})

\end{itemize}
Basic discussion of the syntax of type universes is in

\begin{itemize}%
\item [[Erik Palmgren]], \emph{On Universes in Type Theory} (\href{http://www2.math.uu.se/~palmgren/universe.pdf}{pdf})

\end{itemize}
Definition of type universes weakly \`a{} la Tarski is in

\begin{itemize}%
\item [[Martin Hofmann]], section 2.1.6 of \emph{Syntax and semantics of dependent types} (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.8985}{web})


\item [[Zhaohui Luo]], \emph{Notes on Universes in Type Theory}, 2012 (\href{http://www.cs.rhul.ac.uk/home/zhaohui/universes.pdf}{pdf})


\item [[Cesare Gallozzi]], \emph{Constructive Set Theory from a Weak Tarski Universe}, MSc thesis (2014) (\href{http://xxx.tau.ac.il/abs/1411.5591}{arXiv:1411.5591})



\end{itemize}
Detailed discussion of the type of types in [[Coq]] is in

\begin{itemize}%
\item [[Adam Chlipala]], \emph{\href{http://adam.chlipala.net/cpdt/}{Certified programming with dependent types}} -- \emph{\href{http://adam.chlipala.net/cpdt/html/Universes.html}{Library Universes}}

\end{itemize}
See also around slide 8 of the survey

\begin{itemize}%
\item Frade, \emph{Calculus of inductive constructions} (2008/2009) (\href{http://www3.di.uminho.pt/~mjf/pub/SFV-CIC-2up.pdf}{pdf})

\end{itemize}
A formal proof in [[homotopy type theory]] that the type of [[homotopy n-types]] is not itself a homotopy $n$-type (it is an $(n+1)$-type) is in

\begin{itemize}%
\item Nicolai Kraus, C. Sattler, \emph{The universe $\mathcal{U}_n$ is not an $n$-type} May 2013 (\href{http://red.cs.nott.ac.uk/~ngk/universes.pdf}{pdf})

\end{itemize}
[[(∞,1)-category theory|(∞,1)-Categorical]] semantics for [[univalence|univalent]] type universes is discussed in

\begin{itemize}%
\item [[Chris Kapulkin]], [[Peter LeFanu Lumsdaine]], [[Vladimir Voevodsky]], \emph{The Simplicial Model of Univalent Foundations} (\href{http://arxiv.org/abs/1211.2851}{arXiv:1211.2851})


\item [[Michael Shulman]], \emph{The univalence axiom for elegant Reedy presheaves} (\href{http://arxiv.org/abs/1307.6248}{arXiv:1307.6248})


\item [[David Gepner]], [[Joachim Kock]], \emph{Univalence in locally cartesian closed ∞-categories} (\href{http://arxiv.org/abs/1208.1749}{arXiv:1208.1749})



\end{itemize}
Relation to [[injective types]]:

\begin{itemize}%
\item [[Martín Escardó]], \emph{Injectives types in univalent mathematics} (\href{https://arxiv.org/abs/1903.01211}{arXiv:1903.01211})

\end{itemize}
See also

\begin{itemize}%
\item [[Michael Rathjen]], \emph{The strength of Martin-L\"o{}f type theory with superuniverse. Part I} \href{https://www1.maths.leeds.ac.uk/~rathjen/Super.pdf}{pdf}


\item Stanford Encyclopedia of Philosophy, \emph{\href{http://plato.stanford.edu/entries/type-theory/#6}{Type theory -- Extensions of type systems, Polymorphism, Paradoxes}}


\item [[Zhaohui Luo]], \emph{Contextual analysis of word meanings in type-theoretical semantics}, in Pogodalla, Prost (eds.) \emph{Logical Aspects of Computational Linguistics}, 2011 (\href{http://www.cs.rhul.ac.uk/home/zhaohui/LACL11.pdf}{pdf})



\end{itemize}
[[!redirects types of types]]

[[!redirects universe of types]] [[!redirects universes of types]]

[[!redirects Type]]

[[!redirects type universe]] [[!redirects type universes]]

[[!redirects universe in type theory]] [[!redirects universes in type theory]]



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