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\section*{complete small category}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits}

[[!include infinity-limits - contents]]

\hypertarget{complete_small_categories}{}\section*{{Complete small categories}}\label{complete_small_categories}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_classical_logic}{In classical logic}\dotfill \pageref*{in_classical_logic} \linebreak
\noindent\hyperlink{in_grothendieck_toposes}{In Grothendieck toposes}\dotfill \pageref*{in_grothendieck_toposes} \linebreak
\noindent\hyperlink{in_more_general_toposes_and_constructive_mathematics}{In more general toposes and constructive mathematics}\dotfill \pageref*{in_more_general_toposes_and_constructive_mathematics} \linebreak
\noindent\hyperlink{modelling_impredicative_type_theory}{Modelling impredicative type theory}\dotfill \pageref*{modelling_impredicative_type_theory} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \textbf{complete small category} (or \textbf{small complete category}) is a [[category]] which is both [[small category|small]] (has only a [[set]] of [[objects]] and [[morphisms]]) and [[complete category|complete]] (has a [[limit]] of every small [[diagram]]).

(Note that the phrase ``[[small-complete category]]'' is sometimes used to mean merely a complete category, i.e. one which is ``complete relative to small diagrams''. Therefore, ``complete small category'' is a safer term for our concept.)

\hypertarget{in_classical_logic}{}\subsection*{{In classical logic}}\label{in_classical_logic}

In the presence of [[classical logic]], complete small categories reduce to [[complete lattices]] by the following theorem of Freyd.

\begin{utheorem}
If (in some [[universe]] $U$) a small [[category]] $D$ has [[product]]s of families of objects whose size is at least that of its set of morphisms, then $D$ is a [[preorder]]. In particular, any complete small category is a [[preorder]].

\end{utheorem}
\begin{proof}
Let $x$, $y$ be any two objects, and suppose (contrary to $D$ being a preorder) that there are at least two different morphisms $x \,\underoverset{s}{r}{\rightrightarrows}\, y$. Then the set of morphisms

\begin{displaymath}
x \to \prod_{f \in Mor(D)} y
\end{displaymath}
has [[cardinality]] at least $2^{|Mor(D)|} \gt {|Mor(D)|}$, which is a contradiction.

\end{proof}
\hypertarget{in_grothendieck_toposes}{}\subsection*{{In Grothendieck toposes}}\label{in_grothendieck_toposes}

The [[internal logic]] of a [[topos]] is not, in general, classical, so the above proof does not apply when considering [[internal categories]] in a topos. However, in a [[Grothendieck topos]], one can show that the conclusion still holds by essentially reducing the question to the analogous one in $Set$. A brief description of the argument can be found in the answer to \href{http://mathoverflow.net/questions/43433/small-complete-categories-in-a-grothendieck-topos}{this question}.

\hypertarget{in_more_general_toposes_and_constructive_mathematics}{}\subsection*{{In more general toposes and constructive mathematics}}\label{in_more_general_toposes_and_constructive_mathematics}

However, it is possible to have non-preorder complete small categories in non-Grothendieck topoi. In particular, the [[effective topos]], along with most other [[realizability topos|realizability toposes]], does contain such internal categories, arising for example from the [[modest sets]] or [[partial equivalence relations]]. See \hyperlink{Hyland88}{Hyland88} and \hyperlink{HRR90}{HRR90} for details.

It follows that, in [[constructive mathematics]], it is impossible to prove even that every complete small category in $Set$ or in a Grothendieck topos must be a preorder.

There is a subtlety, however, because in the absence of the [[axiom of choice]], there is a difference between a category being ``strongly complete'' in the sense of being equipped with limit-assigning [[functors]], and being ``weakly complete'' in the sense that there merely \emph{exists} a limit for every diagram. The complete small categories in realizability toposes are only weakly complete, which categorically means that only the [[stack]] completions of their corresponding [[indexed categories]] are complete in the relevant indexed-category sense.

It seems to be unknown whether there exists a (necessarily non-Grothendieck) topos containing a \emph{strongly} complete small category. However, one of the complete small categories in the effective topos does become strongly complete when the ambient context is restricted to the subcategory of [[assemblies]] rather than the entire effective topos.

\hypertarget{modelling_impredicative_type_theory}{}\subsection*{{Modelling impredicative type theory}}\label{modelling_impredicative_type_theory}

Complete small categories have applications to the modeling of impredicative [[polymorphism]]. Suppose that there is a full subcategory $\mathbf C$ of $\mathbf{Set}$ that is small and complete. Then we can interpret an impredicatively-quantified type as

\begin{displaymath}
\llbracket \forall X.T \rrbracket = \prod_{A\in \mathbf{C}} \llbracket T\rrbracket_{A/X}
\end{displaymath}
although there are more elaborate formulations that use a subset of this product.

At least at first glance, this requires $\mathbf{C}$ to be \emph{strongly} complete. Hence it works in the category of assemblies, but not the entire realizability topos.

An alternative approach is to suppose that there is a full [[replete subcategory]] $\mathcal{C}$ of $\mathbf{Set}$ which is complete in the sense of being closed under set-indexed products and equalizers, yet [[essentially small]], i.e. with a small skeleton $\mathbf C$. (Being replete, $\mathcal{C}$ will not itself be small.) Such a category can be obtained as the repletion of any \emph{weakly} complete \emph{actually} small subcategory, such as exist in realizability toposes. See \hyperlink{RS04}{RS04} and \hyperlink{MS09}{MS09}. The interpretation of the impredicative $\forall$ is then achieved by a product over the sets in the small skeleton $\mathbf C$. However, to eliminate terms of type $\forall$, one must pick an isomorphism to a set in the skeleton, and be confident that the choice does not matter, and for this reason a relational parametricity is built in to the interpretation of $\forall$. In more detail:

\begin{displaymath}
\llbracket \forall X.T \rrbracket = \{\pi\in \prod_{A\in \mathbf{C}} \llbracket T\rrbracket_{A/X} | \forall R\subseteq A\times B. \mathcal{R}_R(
\pi(A),\pi(B))\}
\end{displaymath}
where $\mathcal{R}_R$ is a binary logical relation, suitably defined, and

\begin{displaymath}
\llbracket (t:\forall X.T)(U)\rrbracket = 
gpd(i)(\llbracket t\rrbracket_D)
\end{displaymath}
where $i\colon D \to \llbracket U\rrbracket$ is a bijection with a set in the skeleton $\mathbf C$, and we are using the fact that for any bijection $j\colon C\to D$, there is a functorial action $gpd(j)\llbracket T\rrbracket_{C/X}\to \llbracket T\rrbracket_{D/X}.$ For discussion about whether this works, see the nForum thread for this page.

Note that to model polymorphic simple type theory such as [[System F]], it is not necessary for this purpose that $\mathbf{C}$ be \emph{complete}; all it needs is to have products indexed by its own set of \emph{objects}. Freyd's theorem does not apply in this case, since there might be fewer objects than morphisms; and indeed there are plenty of categories even classically that have products indexed by their set of objects, e.g. any one-object category. It is true, however, that (assuming classical logic) no \emph{full subcategory of $Set$} can have products indexed by its own set of objects; this (and a bit more) was shown in \hyperlink{Reynolds84}{Reynolds 84}, again by reducing to Cantor's diagonal argument. On the other hand, to break this argument we do not have to move to realizability; Grothendieck toposes suffice, as shown in \hyperlink{Pitts87}{Pitts 87} via the [[Yoneda embedding]] of a syntactic model into its topos of presheaves.

In contrast, something more like a true complete small category may be necessary to model an impredicative dependent type theory such as the [[calculus of constructions]] with a proof-relevant impredicative sort (traditionally called ``Set''), where the impredicative universe admits [[Pi-types]] with \emph{arbitrary} domain.

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

Freyd's theorem may not have been published by him.

The original reference for the complete small categories in the effective topos is:

\begin{itemize}%
\item A small complete category, [[Martin Hyland]], Annals of Pure and Applied Logic 40 (1988), \href{https://webdpmms.maths.cam.ac.uk/~martin/Research/Oldpapers/smallcomplete88.pdf}{pdf}

\end{itemize}
It was expanded further in the following paper, which discusses the strong/weak completeness issue and the relation to stacks in more detail:

\begin{itemize}%
\item Hyland, J. M., Robinson, E. P. and Rosolini, G. (1990), The Discrete Objects in the Effective Topos. Proceedings of the London Mathematical Society, s3-60: 1-36. \href{https://doi.org/10.1112/plms/s3-60.1.1}{doi:10.1112/plms/s3-60.1.1}

\end{itemize}
There was much related activity around that time, by [[Peter Freyd]], [[Eugenio Moggi]], [[Andrew Pitts]], [[John Reynolds]], Guiseppe Rosolini, and others, including:

\begin{itemize}%
\item [[John Reynolds]], Polymorphism is not set-theoretic. In: Kahn G., MacQueen D.B., Plotkin G. (eds) Semantics of Data Types. SDT 1984. Lecture Notes in Computer Science, vol 173. Springer, Berlin, Heidelberg


\item [[Andy Pitts]], Polymorphism is Set Theoretic, Constructively. In Category Theory and Computer Science, Proceedings, Edinburgh 1987, Lecture Notes in Computer Science Vol. 283 (Springer-Verlag, Berlin, 1987), pp 12-39, \href{https://www.cl.cam.ac.uk/~amp12/papers/polist/polist.pdf}{PDF}



\end{itemize}
A more recent survey of this type of model for polymorphic type theory is in

\begin{itemize}%
\item Andrea Asperti and Simone Martini, Categorical models of polymorphism, Information and Computation, Volume 99, Issue 1, 1992, Pages 1-79, \href{https://doi.org/10.1016/0890-5401(92}{DOI}90024-A.)

\end{itemize}
The idea of using a replete complete subcategory which is essentially small in [[constructive set theory|IZF]] is discussed in

\begin{itemize}%
\item Using Synthetic Domain Theory to Prove Operational Properties of a Polymorphic Programming Language Based on Strictness``, Guiseppe Rosolini and [[Alex Simpson]], unpublished, 2004. \href{https://www.researchgate.net/profile/Giuseppe_Rosolini/publication/237887060_Using_Synthetic_Domain_Theory_to_Prove_Operational_Properties_of_a_Polymorphic_Programming_Language_Based_on_Strictness/links/0deec53356d020dc85000000/Using-Synthetic-Domain-Theory-to-Prove-Operational-Properties-of-a-Polymorphic-Programming-Language-Based-on-Strictness.pdf}{ResearchGate}


\item Relational parametricity for computational effects, Rasmus Møgelberg and [[Alex Simpson]], Logical Methods in Computer Science, Vol 5 (3:7), 2009. \href{https://arxiv.org/abs/0906.5488}{arxiv}.



\end{itemize}
[[!redirects complete small category]] [[!redirects complete small categories]] [[!redirects small complete category]] [[!redirects small complete categories]]



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