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\begin{document}

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\section*{reflexive object}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_a_closed_symmetric_monoidal_category}{In a closed (symmetric) monoidal category}\dotfill \pageref*{in_a_closed_symmetric_monoidal_category} \linebreak
\noindent\hyperlink{in_a_closed_monoidal_2category}{In a closed monoidal 2-category}\dotfill \pageref*{in_a_closed_monoidal_2category} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{terminal_object}{Terminal object}\dotfill \pageref*{terminal_object} \linebreak
\noindent\hyperlink{scotts__construction}{Scott's $D_\infty$ construction}\dotfill \pageref*{scotts__construction} \linebreak
\noindent\hyperlink{enumeration_operator_model}{Enumeration operator model}\dotfill \pageref*{enumeration_operator_model} \linebreak
\noindent\hyperlink{scotts_representation_theorem}{Scott's representation theorem}\dotfill \pageref*{scotts_representation_theorem} \linebreak
\noindent\hyperlink{higherorder_abstract_syntax}{Higher-order abstract syntax}\dotfill \pageref*{higherorder_abstract_syntax} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

A \emph{reflexive object} is a model of the pure (untyped) [[lambda calculus]] validating the $\beta$ law, and sometimes also the $\eta$ law.

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

A \textbf{reflexive object} in a [[cartesian closed category]] is an object $U$ equipped with a pair of maps

\begin{displaymath}
\itexarray{U & \overset{app}{\underset{lam}{\rightleftarrows}} & U^U}
\end{displaymath}
such that $app \circ lam = 1$. In other words, a reflexive object is an object $U$ together with data $U^U \lhd U$ exhibiting $U^U$ as a [[retract]] of $U$. A reflexive object is said to be \textbf{extensional} (or ``strict'') when also $lam \circ app = 1$, so that there is an isomorphism $U^U \cong U$.

Viewed as a model of lambda calculus, the equation $app \circ lam = 1$ of a reflexive object represents $\beta$-equality $(\lambda x.t)(u) = t[u/x]$, while the equation $lam \circ app = 1$ of an extensional reflexive object represents $\eta$-equality $\lambda x.t(x) = t$.

\hypertarget{in_a_closed_symmetric_monoidal_category}{}\subsubsection*{{In a closed (symmetric) monoidal category}}\label{in_a_closed_symmetric_monoidal_category}

The definition has a straightforward generalization to any [[symmetric monoidal closed category]] (i.e., not necessarily [[cartesian monoidal category|cartesian monoidal]]), where we just replace the exponential object $U^U$ by the [[internal hom]] $[U,U]$. Indeed, the definition also makes sense in any left-closed or right-closed monoidal category.

A reflexive object in a symmetric monoidal closed category provides a model of [[linear lambda calculus]].

\hypertarget{in_a_closed_monoidal_2category}{}\subsubsection*{{In a closed monoidal 2-category}}\label{in_a_closed_monoidal_2category}

The definition of reflexive object also has a natural generalization to any closed monoidal [[2-category]] (or cartesian closed 2-category), where the retraction should be replaced by an [[adjunction]] $app \dashv lam : [U,U] \to U$. Then the [[counit]] of the adjunction $app \circ lam \Rightarrow 1_{[U,U]}$ models [[beta-reduction]] $(\lambda x.t)(u) \to t[u/x]$, while the [[unit]] $1_U \Rightarrow lam \circ app$ models [[eta-expansion]] $t \to \lambda x.t(x)$.

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

\hypertarget{terminal_object}{}\subsubsection*{{Terminal object}}\label{terminal_object}

The [[terminal object]] of a ccc provides a degenerate example of a (extensional) reflexive object, and for cardinality reasons, this is the only reflexive object in [[Set]].

\hypertarget{scotts__construction}{}\subsubsection*{{Scott's $D_\infty$ construction}}\label{scotts__construction}

The first non-degenerate model of untyped lambda calculus was described by [[Dana Scott]] (see \hyperlink{Scott70}{Scott (1970)} and \hyperlink{Scott72}{Scott (1972)}), who solved the cardinality issue by replacing sets with [[algebraic lattices]], and arbitrary set-theoretic functions by [[Scott topology|Scott-continuous]] functions. The so-called ``$D_\infty$'' model is built by a limit construction, starting from $D_0 = D$ an arbitrary algebraic lattice, and taking $D_{n+1} = D_n \to D_n$.

\hypertarget{enumeration_operator_model}{}\subsubsection*{{Enumeration operator model}}\label{enumeration_operator_model}

Scott later gave a more concrete model of lambda calculus in \hyperlink{Scott76}{Scott (1976)}, defining a reflexive object with carrier the lattice $P\omega$ of all [[subsets]] of the non-negative integers, and with the maps $app$ and $lam$ (there called ``fun'' and ``graph'') defined as follows:

\begin{displaymath}
app(u)(x) = \{m \mid \exists e_n \subseteq x. (n,m) \in u\}
\end{displaymath}
\begin{displaymath}
lam(f) = \{(n,m) \mid m \in f(e_n)\}
\end{displaymath}
Here $e_n$ stands for the set whose elements are the exponents in the binary expansion of $n$ (thus $e_n$ is the $n$th subset in the standard enumeration of finite subsets of $\omega$), while ``$(n,m)$'' stands for the standard enumeration of pairs of integers

\begin{displaymath}
(n,m) = \frac{1}{2}(n+m)(n+m+1)+m
\end{displaymath}
Note that the enumeration operator model is not ``extensional'' in the sense that it only validates $\eta$-conversion as an inclusion $u \subseteq \ell(a(u))$, which is not an equality in general.

\hypertarget{scotts_representation_theorem}{}\subsubsection*{{Scott's representation theorem}}\label{scotts_representation_theorem}

\hyperlink{Scott80}{Scott (1980)} proved a sort of representation theorem for pure lambda calculus, showing that \emph{any} ``type-free theory'' (a.k.a., [[lambda theory]]) can be realized as a reflexive object. He began by constructing a category whose objects are the closed lambda terms $A$ such that one can prove $A = \lambda x.A(A(x))$, and whose morphisms $f : A \to B$ are (equivalence classes of) closed terms $f$ such that one can prove $f = \lambda x.B(f(A(x)))$ (cf. [[Karoubi envelope]]), with identity and composition defined by

\begin{displaymath}
1_A = A
\qquad
f \circ g = \lambda x.f(g(x))
\end{displaymath}
This category is cartesian closed, where products and exponentials are defined (on objects) by

\begin{displaymath}
A \times B = \lambda u\lambda z.z(A(u(\lambda x\lambda y.x)))(B(u(\lambda x\lambda y.y))) 
\qquad
A \to B = \lambda f.B \circ f \circ A
\end{displaymath}
Now, the identity term $U = \lambda x.x$ lives inside this category since obviously $U = U\circ U$, and moreover, every object $A$ is a retract of $U$, since $A : A \to U$ and $A : U \to A$ and $A \circ A = A = 1_A$. Thus, $U$ equipped with its retraction from $U \to U$ is a reflexive object.

A more abstract analysis of Scott's representation theorem appears in \hyperlink{Hyland13}{Hyland (2013)}.

\hypertarget{higherorder_abstract_syntax}{}\subsubsection*{{Higher-order abstract syntax}}\label{higherorder_abstract_syntax}

Another way of looking at the \emph{free} cartesian closed category containing a reflexive object is as a representation of pure lambda terms (up to $\beta$-equality) in the style of so-called [[higher-order abstract syntax]]. The operation $app : U \to U^U$ is the [[currying|curried]] form of application, while the operation $lam : U^U \to U$ represents lambda abstraction. The [[global elements]] of $U$ can be interpreted as closed lambda terms, and more generally, morphisms $U^n \to U$ can be interpreted as terms with $n$ free variables.

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

Dana Scott's original work on lattice models of the lambda calculus can be found in:

\begin{itemize}%
\item Dana Scott. Outline of a Mathematical Theory of Computation. In \emph{4th Annual Princeton Conference on Information Sciences and Systems}, pages 169-176, 1970. (\href{https://ropas.snu.ac.kr/~kwang/520/readings/sco70.pdf}{pdf})


\item Dana Scott. Continuous Lattices. \emph{Toposes, Algebraic Geometry and Logic} (ed.: E. Lawvere), pages 97-136, 1972. (\href{https://www.cs.ox.ac.uk/files/3229/PRG07.pdf}{pdf})


\item Dana Scott. Data types as lattices. \emph{SIAM Journal of Computing}, 5(3):522--587, September 1976. (\href{https://www.cs.ox.ac.uk/files/3287/PRG05.pdf}{pdf})



\end{itemize}
Scott made the underlying categorical structure explicit in a later paper, where he introduced the definition of a reflexive object in a ccc, and proved a representation theorem for theories of pure lambda calculus:

\begin{itemize}%
\item Dana Scott. Relating theories of the $\lambda$-calculus. In \emph{To H.B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism} (eds. Hindley and Seldin), Academic Press, 403--450, 1980.

\end{itemize}
For a modern analysis of Scott's representation theorem, see:

\begin{itemize}%
\item [[Martin Hyland]]. Classical lambda calculus in modern dress. To appear in \emph{Mathematical Structures in Computer Science}, 2013. \href{http://arxiv.org/abs/1211.5762}{arxiv}


\item [[Martin Hyland]], \emph{Towards a Notion of Lambda Monoid} , Electronic Notes in Theoretical Computer Science \textbf{303} (2014) pp.59-77.



\end{itemize}
Bicategorical and monoidal generalizations of the notion of reflexive object are discussed in:

\begin{itemize}%
\item [[R. A. G. Seely]]. Modelling Computations : a 2-categorical Framework. LICS 1987. (\href{http://www.math.mcgill.ca/rags/WkAdj/LICS.pdf}{pdf})


\item [[Bart Jacobs]]. Semantics of lambda-I and of other substructure lambda calculi. M. Bezem and J.F. Groote (eds.) \emph{Typed Lambda Calculi and Applications}, Springer LNCS 664, 1993, p. 195-208.


\item [[Noam Zeilberger]]. Linear lambda terms as invariants of rooted trivalent maps. 31 Jan 2016. (\href{http://arxiv.org/abs/1512.06751}{arXiv})



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

\begin{itemize}%
\item [[lambda calculus]]
\item [[domain theory]]
\item [[lambda theory]]

\end{itemize}


\end{document}