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\begin{document}

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\section*{universe enlargement}

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

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

[[!include universe - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{notation}{Notation}\dotfill \pageref*{notation} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{models_of_a_theory}{Models of a theory}\dotfill \pageref*{models_of_a_theory} \linebreak
\noindent\hyperlink{LocallyPresentableEnlargement}{Locally presentable enlargement}\dotfill \pageref*{LocallyPresentableEnlargement} \linebreak
\noindent\hyperlink{accessible_enlargement}{Accessible enlargement}\dotfill \pageref*{accessible_enlargement} \linebreak
\noindent\hyperlink{monoidal_enlargements}{Monoidal enlargements}\dotfill \pageref*{monoidal_enlargements} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

Let $\mathbf{U}$ and $\mathbf{V}$ be set-theoretic [[universes]] (such as [[Grothendieck universes]] or [[universe in a topos|universes in an ambient topos]]) with $\mathbf{V}$ ``bigger than'' $\mathbf{U}$, and let $C$ be a [[category]] ``in the world of $\mathbf{U}$,'' i.e. whose objects are $\mathbf{U}$-[[small category|small]] and which itself is $\mathbf{U}$-[[large category|large]]. Its \textbf{universe enlargement} is supposed to be a category ``in the world of $\mathbf{V}$'' which is ``the same'' or at least similar to $C$, and perhaps better behaved.

\hypertarget{notation}{}\subsection*{{Notation}}\label{notation}

An enlargement of a specific category, such as [[Set]], [[Cat]], [[Grp]], or [[Top]], is often denoted by writing its name in all capitals: SET, CAT, GRP, TOP.

As a notation for the operation on a general category $C$, the notation $\Uparrow C$ has been proposed.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Since the enlargement is not well-specified by the above intuitive description, there are several different definitions, which are often equivalent but not always.

Henceforth we fix two universes $\mathbf{U}$ and $\mathbf{V}$ with $\mathbf{U}\in \mathbf{V}$. We will refer to sets in $\mathbf{U}$ as \emph{small}, sets in $\mathbf{V}$ as \emph{large}, and sets not necessarily in $\mathbf{V}$ (or in the next larger universe $\mathbf{W}$) as \emph{very large}.

\hypertarget{models_of_a_theory}{}\subsubsection*{{Models of a theory}}\label{models_of_a_theory}

If $C$ is the category of [[small set|small]] [[models]] of some [[theory]], then we can take $\Uparrow C$ to be the category of large [[models]] of the same theory. For instance:

\begin{itemize}%
\item the [[large category]] [[Set]] of [[small sets]] is the category of small models of the theory with one sort and no operations or relations. The category of large models of this theory is just the very large category $SET$ of [[large set|large sets]].


\item Similarly, from categories such as [[Grp]], [[Cat]], and [[Top]] of small [[groups]], [[categories]], and [[topological spaces]], we obtain the categories $GRP$, $CAT$, and $TOP$ of large groups, categories, and topological spaces.



\end{itemize}
Note that the theory could be [[algebraic theory|algebraic]], as for $Grp$, or [[essentially algebraic theory|essentially algebraic]], as for $Cat$, or even [[higher-order logic|higher-order]], as for $Top$. We will refer to this as the \textbf{logical enlargement}. In practice, this enlargement usually suffices, but for theoretical reasons it would be nice to have a construction which works on \emph{any} large category.

\hypertarget{LocallyPresentableEnlargement}{}\subsubsection*{{Locally presentable enlargement}}\label{LocallyPresentableEnlargement}

For any large category $C$, the very large [[presheaf category]] $[C^{op},SET]$ contains $C$ as a [[full subcategory]]. Furthermore, $[C^{op},SET]$ is $SET$-bicomplete, i.e. it has all large [[limits]] and [[colimits]], whether or not $C$ was $Set$-complete or cocomplete, and the [[Yoneda embedding]] $C \hookrightarrow [C^{op},SET]$ [[preserved limit|preserves]] all limits that exist in $C$.

However, it hardly preserves any \emph{colimits} (since it is a [[free cocompletion]], after all). If we want an enlargement of $C$ which preserves some class $\Phi$ of colimits in $C$, then we can restrict to the full subcategory of $[C^{op},SET]$ consisting of the presheaves $C^{op}\to SET$ which preserved limit the colimits $\Phi$ (i.e. take them to limits in $SET$). Let's denote this category by $\Uparrow_\Phi C$. Since [[representable functors]] preserve all limits, the Yoneda embedding of $C$ factors through $\Uparrow_\Phi C$, and all the colimits in $\Phi$ are preserved by this restricted embedding.

Moreover, $\Uparrow_\Phi C$ is closed under limits in $[C^{op},SET]$, so it is itself $SET$-complete. And as long as the (diagrams sizes of the) colimits in $\Phi$ are at most \emph{large}, then (by theorems of Day etc.) $\Uparrow_\Phi C$ is reflective in $[C^{op},SET]$, and hence also $SET$-cocomplete. Since usually, the only colimits we might care about in a large category $C$ are \emph{small}, a natural choice for $\Phi$ is the class of all small colimits existing in $C$.

We will refer to this $\Uparrow_\Phi C$ as the \textbf{locally presentable enlargement}, since it is a [[locally presentable category]] relative to $\mathbf{V}$. (In fact, it is locally $\kappa$-presentable, where $\kappa$ is the [[cardinality]] of the first universe $\mathbf{U}$.) Moreover, we have the following (where $\lambda Cts(A,B)$ denotes the category of $\lambda$-small-limit preserving functors).

\begin{utheorem}
Suppose $C$ is locally presentable relative to $\mathbf{U}$, so that there is a small cardinal $\lambda$ and a small category $A$ with $\lambda$-small colimits such that $C\simeq \lambda Cts(A^{op},Set)$. If $\Phi$ denotes the class of all small colimits in $C$, then $\Uparrow_\Phi C \simeq \lambda Cts(A^{op},SET)$.

\end{utheorem}
Note that in this case, $\lambda Cts(A^{op},SET)$ is the category of large models of the theory (namely $A^{op}$) of which $C$ is the category of small ones. Thus, the theorem says that when $C$ is locally presentable, the locally presentable enlargement is the same as the logical one.

\begin{proof}
Write $C' \coloneqq \lambda Cts(A^{op},SET)$. Then we have a full embedding $C\hookrightarrow C'$, which preserves limits and colimits since they are calculated in the same way in both categories.

Now every object of $C$ is a small colimit of objects of $A$, and the objects of $A$ are all $\lambda$-presentable in $C$ and in $C'$. Thus, every object of $C$ is $\kappa$-presentable in $C'$, where $\kappa$ is the size of the universe $\mathbf{U}$. Conversely, since $C'$ is locally $\lambda$-presentable (relative to $\mathbf{V}$), any $\kappa$-presentable object in it is a $\kappa$-small colimit of $\lambda$-presentable objects---so since $C$ is closed under small colimits in $C'$, it must consist exactly of the $\kappa$-presentable objects. However, since $C'$ is locally $\lambda$-presentable, it is also locally $\kappa$-presentable, so this implies that $C' \simeq \kappa Cts(C^{op},SET)$. But $\kappa Cts(C^{op},SET)$ is precisely $\Uparrow_\Phi C$.

\end{proof}
However, if $C$ is not locally presentable, then its logical and locally-presentable enlargements can disagree. In fact, they will \emph{usually} disagree in this case, since if $C$ is not locally presentable relative to $\mathbf{U}$, its logical enlargement will usually not be locally presentable relative to $\mathbf{V}$ either, whereas its locally-presentable enlargement is, by definition, locally presentable relative to $\mathbf{V}$. For example, the logical and locally presentable enlargements of $Top$ are quite different.

\hypertarget{accessible_enlargement}{}\subsubsection*{{Accessible enlargement}}\label{accessible_enlargement}

There is a simple modification of the locally-presentable enlargement which makes it agree with the logical enlargement for all [[accessible categories]], not just locally presentable ones. Namely, instead of $\kappa Cts(C^{op},SET)$, we consider $\kappa Flat(C^{op},SET)$, the category of $\kappa$-[[flat functors]] (functors that ``would preserve all $\kappa$-small limits if they existed''). We call this the \textbf{accessible enlargement}.

Essentially the same proof as for the theorem above shows that if $C$ is accessible relative to $\mathbf{U}$, hence $C\simeq \lambda Flat(A^{op},Set)$, then the accessible enlargement is equivalent to $\lambda Flat(A^{op},SET)$, which is of course the logical enlargement.

Note that if $C$ has all small limits, then its accessible and locally-presentable enlargements are the same, since a functor defined on a category with $\kappa$-small limits is $\kappa$-continuous iff it is $\kappa$-flat. However, if $C$ does not have small limits, then its locally-presentable enlargement \emph{does} have all large limits, whereas its accessible enlargement does not.

Thus, when $C$ is ``poorly behaved,'' its accessible enlargement is ``closer to it'' (in particular, identical to its logical enlargement if $C$ is accessible) and thus equally poorly behaved, whereas its locally-presentable enlargement is further from it, but better behaved (complete and cocomplete). For some purposes one may want one of these behaviors; for other purposes one may want the other.

\hypertarget{monoidal_enlargements}{}\subsubsection*{{Monoidal enlargements}}\label{monoidal_enlargements}

One may further ask for an enlargement which inherits whatever additional structure $C$ has, such as [[monoidal category|monoidal structure]]. The logical enlargement will usually inherit such structure trivially, while the [[Day convolution]] theorem implies that when $C$ is monoidal, so is its locally-presentable enlargement---and moreover, the latter is [[closed monoidal category|closed]] even if $C$ is not!

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

\begin{itemize}%
\item [[inter-universal Teichmüller theory]]

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

\begin{itemize}%
\item [[universe polymorphism]]

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

\begin{itemize}%
\item Sections 2.6 and 3.11--3.12 of \href{http://tac.mta.ca/tac/reprints/articles/10/tr10abs.html}{Basic Concepts of Enriched Category}.


\item \href{http://golem.ph.utexas.edu/category/2010/11/universe_enlargement.html}{Blog post}


\item [[Zhen Lin Low]], \href{http://mathoverflow.net/a/130639/381}{MO comment on universe enlargement}



\end{itemize}
[[!redirects universe enlargements]] [[!redirects change of universe]] [[!redirects changes of universe]]

[[!redirects universe extension]]



\end{document}