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\section*{two-level type theory}

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

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

[[!include type theory - contents]]

\hypertarget{deduction_and_induction}{}\paragraph*{{Deduction and Induction}}\label{deduction_and_induction}

[[!include deduction and induction - contents]]

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

[[!include constructivism - contents]]

\hypertarget{foundations}{}\paragraph*{{Foundations}}\label{foundations}

[[!include foundations - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{type_theories}{Type theories}\dotfill \pageref*{type_theories} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{semisimplicial_types}{Semisimplicial types}\dotfill \pageref*{semisimplicial_types} \linebreak
\noindent\hyperlink{variations}{Variations}\dotfill \pageref*{variations} \linebreak
\noindent\hyperlink{natural_numbers}{Natural numbers}\dotfill \pageref*{natural_numbers} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\textbf{Two-level type theory} (2LTT) refers to versions of [[Martin-Lof type theory]] that combine two type theories: one level as a [[homotopy type theory]], which may include [[univalence axiom|univalent]] universes and [[higher inductive types]], and the second level as a traditional form of type theory validating [[uniqueness of identity proofs]]. The second layer may be understood as the internalised meta-theory of the first.

The rules are much inspired by the homotopical semantics in [[fibration categories]] and [[model categories]], according to which the types of the homotopy layer are sometimes called \textbf{[[fibrant object|fibrant]] types} and the others \textbf{non-fibrant types}. An alternative terminology is simply \textbf{types} (since these are the objects of real interest) and \textbf{pretypes} (an auxiliary structure used to study the types).

In other words, where default [[homotopy type theory]] has [[categorical semantics]] (see at \emph{[[relation between category theory and type theory]]}) in suitable [[type-theoretic model categories]] but in such a way that only the [[(infinity,1)-category]] [[presentable (infinity,1)-category|presented]] by that really matters, in two-level type theory one adds explicit control over the presenting model category (or other kind of [[fibration category]]), thus apparently breaking the $(\infty,1)$-categorical ``[[principle of equivalence]]'' but providing more tools for handling the presentation. It is an open question to what extent the principle of equivalence is actually broken, i.e. whether results proven in two-level type theory can be transferred to any model categorical presentation.

\hypertarget{type_theories}{}\subsection*{{Type theories}}\label{type_theories}

\begin{itemize}%
\item The first proposal for two-level type theory was [[Vladimir Voevodsky]]`s [[Homotopy Type System]]. This system had a reflection rule collapsing the non-fibrant ``exact equality'' to [[judgmental equality]], making some things easier but making type-checking undecidable.


\item More recently the proposals of \hyperlink{ACK17}{ACK} simply assumes [[uniqueness of identity proofs]] for the exact equality; this seems to suffice for most if not all purposes.


\item [[computational type theory|Computational]] [[cubical type theory]] can also include a non-fibrant layer: its syntax is interpreted as cubical sets, where ``fibrancy'' is a \emph{defined} condition on pretypes (existence of Kan operations).



\end{itemize}
\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{semisimplicial_types}{}\subsubsection*{{Semisimplicial types}}\label{semisimplicial_types}

Two-level type theories (including HTS and ACK) were motivated to a large extent by the technical difficulties encountered in formalizing [[semisimplicial types]] in default [[homotopy type theory]] (HoTT). This problem comes precisely from the fact that HoTT is really the [[internal language]] of some [[(infinity,1)-topos]] and hence defining [[simplicial objects]] or similar here means to speak of [[simplicial objects in an (infinity,1)-category]], which means that the [[simplicial identities]] hold only up to [[coherence|coherent]] higher [[homotopy]]. The problem of syntactically encoding the infinite amount of this [[coherence]] data remains unsolved to date. For [[semisimplicial types]] the difficulties greatly reduce (since the evident iterated [[dependent type]]-definition of a semisimplicial type is automatically interpreted by a [[Reedy model structure|Reedy]]-[[fibrant object]] in the given [[type-theoretic model category]], which takes care of all the homotopy coherence by the power of [[model category]] theory, see at \emph{[[internal (infinity,1)-category]]} for more on this) but technical problems remain even in formulating the plain 1-categorical [[simplicial identities]] to all degrees.

The ``exact equality'' of two-level type theories solves this problem because such equalities can be defined by induction.

\hypertarget{variations}{}\subsection*{{Variations}}\label{variations}

\hypertarget{natural_numbers}{}\subsubsection*{{Natural numbers}}\label{natural_numbers}

One important choice to be made in writing down a two-level type theory is whether there is one natural numbers type or two. There must certainly be a fibrant natural numbers type; the question is whether the elimination rule for this ``fibrant nat'' allows elimination also into non-fibrant types. If it does not, then there should also be a ``non-fibrant nat'' which is not fibrant, but whose elimination principle can eliminate into other non-fibrant types.

Semantically, this distinction corresponds to whether the natural numbers object of the model category presentation is already fibrant, or whether it needs to be fibrantly replaced to represent a (fibrant) type. In some models, such as [[simplicial sets]], it is already fibrant; but in others, such as [[local model structure on simplicial presheaves]], it is not. It is an open question whether any [[(infinity,1)-topos]] can be presented by a model category in which the natural numbers object is fibrant, and thus serve as semantics for a two-level type theory with only one natural numbers type.

Since the definition of semisimplicial types requires an inductive definition of a (non-fibrant) exact equality, if there are two nats then it must use the non-fibrant one. This means that, without additional rules, a two-level type theory with two nats cannot define a \emph{fibrant} type of \emph{untruncated} semisimplicial types: for each non-fibrant $n$ it can define a fibrant type of $n$-truncated semisimplicial types, but the limit of these types over the \emph{non-fibrant} nat is no longer fibrant. Assuming only one nat is sufficient to solve this problem, but as mentioned above this may exclude desirable models. Another weaker axiom, which \emph{is} satisfied in all sufficiently nice model categories, is that fibrant types are closed under limits of towers of fibrations indexed by the non-fibrant nat. (In two-level type theory terminology, this is a strengthening of the assumption that the non-fibrant nat is ``cofibrant''.)

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

\begin{itemize}%
\item [[homotopytypetheory:HomePage|Homotopy Type Theory Wiki]], \emph{[[homotopytypetheory:Homotopy Type System]]}


\item Thorsten Altenkirch, Paolo Capriotti, Nicolai Kraus, \emph{Extending Homotopy Type Theory with Strict Equality}, (\href{https://arxiv.org/abs/1604.03799}{arXiv:1604.03799})


\item Danil Annenkov, Paolo Capriotti, Nicolai Kraus, \emph{Two-Level Type Theory and Applications}, (\href{https://arxiv.org/abs/1705.03307}{arXiv:1705.03307})



\end{itemize}
[[!redirects 2-level type theory]] [[!redirects 2-level type theories]] [[!redirects 2LTT]] [[!redirects HTS]] [[!redirects Homotopy Type System]] [[!redirects fibrant type]] [[!redirects fibrant types]] [[!redirects non-fibrant type]] [[!redirects non-fibrant types]]



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