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\begin{document}

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\section*{free cocompletion}

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

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

[[!include category theory - contents]]

\hypertarget{yoneda_lemma}{}\paragraph*{{Yoneda lemma}}\label{yoneda_lemma}

[[!include Yoneda lemma - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{technical_details}{Technical details}\dotfill \pageref*{technical_details} \linebreak
\noindent\hyperlink{proofs}{Proofs}\dotfill \pageref*{proofs} \linebreak
\noindent\hyperlink{free_cocompletion_of_large_categories}{Free cocompletion of large categories}\dotfill \pageref*{free_cocompletion_of_large_categories} \linebreak
\noindent\hyperlink{free_cocompletion_as_a_pseudomonad}{Free cocompletion as a pseudomonad}\dotfill \pageref*{free_cocompletion_as_a_pseudomonad} \linebreak
\noindent\hyperlink{in_higher_category_theory}{In higher category theory}\dotfill \pageref*{in_higher_category_theory} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Passing from a [[category]] $C$ to its [[presheaf]] category $PSh(C) := [C^{op},Set]$ may be regarded as the operation of ``freely adjoining [[colimit]]s to $C$''.

A slightly more precise version of this statement is that the [[Yoneda embedding]]

\begin{displaymath}
Y : C \hookrightarrow PSh(C)
\end{displaymath}
is the \textbf{free cocompletion} of $C$.

The [[universal property]] of the [[Yoneda embedding]] is expressed in terms of the [[Yoneda extension]] of any [[functor]] $F : C \to D$ to a category $D$ with colimits.

\hypertarget{technical_details}{}\subsection*{{Technical details}}\label{technical_details}

The rough statement is that the [[Yoneda embedding]]

\begin{displaymath}
y_S: S \to Set^{S^{op}}
\end{displaymath}
of a [[small category]] $S$ into the category $Set^{S^{op}}$ of [[presheaf|presheaves]] on $S$ is universal among functors from $S$ into [[cocomplete category|cocomplete categories]]. Technically, this should be understood in an appropriate [[2-category|2-categorical]] sense: given a functor $F: S \to D$ where $D$ is (small-)cocomplete, there exists a unique (up to isomorphism) [[cocontinuous functor|cocontinuous]] extension

\begin{displaymath}
\hat{F}: Set^{S^{op}} \to D,
\end{displaymath}
called the [[Yoneda extension]], meaning that $\hat{F} y_S \cong F$ and $\hat{F}$ preserves small colimits.

Put slightly differently: let $Cocomp$ denote the 2-category of cocomplete categories, cocontinuous functors, and natural transformations between them. Then for cocomplete $D$, the Yoneda embedding $y S: S \to Set^{S^{op}}$ induces by restriction a functor

\begin{displaymath}
res_{y S} D: Cocomp(Set^{S^{op}}, D) \to Cat(S, D)
\end{displaymath}
that is an [[equivalence of categories|equivalence]] for each $D$, one that is [[2-natural transformation|2-natural]] in $D$. In fact we show that the Yoneda extension $F \mapsto \hat{F}$ gives a functor $ext_{y S}D: Cat(S, D) \to Cocomp(Set^{S^{op}}, D)$ that is a left adjoint to $res_{y S}D$, giving a 2-natural adjoint equivalence.

As a first step, we construct the desired extension $\hat{F}$ as a [[adjoint functor|left adjoint]] to the functor

\begin{displaymath}
D \to Set^{S^{op}}: d \mapsto \hom_D(F-, d).
\end{displaymath}
Being a left adjoint, $\hat{F}$ is cocontinuous.

An explicit formula for the left adjoint is given by the [[weighted colimit]], [[end|coend]], or [[tensor product of functors]] formula

\begin{displaymath}
\hat{F}(X) = X \otimes_S F = \int^{s: S} X(s) \cdot F(s)
\end{displaymath}
where $S \cdot d$ is notation for [[power|copowering]] (or tensoring) an object $d$ of $D$ by a set $S$ (in this case, a coproduct of an $S$-indexed set of copies of $D$). This formula recurs frequently throughout this wiki; see also [[nerve]], [[Day convolution]].

This ``free cocompletion'' property generalizes to [[enriched category]] theory. If $V$ is complete, cocomplete, symmetric monoidal closed, and $S$ is a small $V$-enriched category, then the enriched presheaf category $V^{S^{op}}$ is a free $V$-cocompletion of $S$. The explicit meaning is analogous to the case where $V = Set$, where all ordinary category concepts are replaced by their $V$-enriched analogues; in particular, the notion of ``$V$-cocontinuous functor'' referes to preservation of enriched weighted colimits (not just ordinary conical colimits).

If $C$ is not small, then its free cocompletion still exists, but it is not the category of all presheaves on $C$. Rather, it is the category of [[small presheaves]] on $C$, i.e. presheaves that are small colimits of representables.

\hypertarget{proofs}{}\subsubsection*{{Proofs}}\label{proofs}

\begin{prop}
\label{FreeCocompletion}\hypertarget{FreeCocompletion}{}
For $\mathcal{C}$ a [[small category]], its [[Yoneda embedding]] $\mathcal{C} \overset{y}{\hookrightarrow} [\mathcal{C}^{op}, Set]$ exhibits the [[category of presheaves]] $[\mathcal{C}^{op}, Set]$ as the \emph{free co-completion} of $\mathcal{C}$, in that it is a [[universal morphism]] (as in \href{adjoint+functor#UniversalArrow}{this Def.} but ``up to natural isomorphism'') into a [[cocomplete category]], in that:

\begin{enumerate}%
\item for $\mathcal{D}$ any [[cocomplete category]] [[category]];


\item for $F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ any [[functor]];



\end{enumerate}
there is a [[functor]] $\widetilde F \;\colon\; [\mathcal{C}^{op}, Set] \longrightarrow \mathcal{D}$, unique up to [[natural isomorphism]], such that

\begin{enumerate}%
\item $\widetilde F$ [[preserved limit|preserves]] all [[colimits]],


\item $\widetilde F$ [[extension|extends]] $F$ through the [[Yoneda embedding]], in that the following [[commuting diagram|diagram commutes]], up to [[natural isomorphism]]:



\end{enumerate}
\begin{displaymath}
\itexarray{
    && \mathcal{C}
    \\
    & {}^{y}\swarrow &\swArrow& \searrow^{\mathrlap{F}}
    \\
    \mathrlap{ \!\!\!\!\!\!\!\!\!\!\!\!\! [\mathcal{C}^{op}, Set] }
      && \underset{ \widetilde F }{\longrightarrow} &&
    \mathcal{D}
  }
\end{displaymath}
\end{prop}
\begin{proof}
The last condition says that $\widetilde F$ is fixed on [[representable presheaves]], up to isomorphism, by

\begin{equation}
\widetilde F( y(c) ) \simeq F(c)
\label{YonedaRestriction}\end{equation}
and in fact [[natural isomorphism|naturally]] so:

\begin{equation}
\itexarray{
    c_1&&
    \widetilde F( y(c_1) ) &\simeq& F(c_1)
    \\
    {}^{\mathllap{f}}\big\downarrow
     && {}^{\mathllap{ F(y(f)) }}\big\downarrow && \big\downarrow^{\mathrlap{ F(f) }}
    \\
    c_2 && \widetilde F (y(c_2)) &\simeq& F(c_2)
  }
\label{FunctorialYonedaRestriction}\end{equation}
But the [[co-Yoneda lemma]] expresses every [[presheaf]] $\mathbf{X} \in [\mathcal{C}^{op}, Set]$ as a [[colimit]] of [[representable presheaves]]

\begin{displaymath}
\mathbf{X} \;\simeq\; \int^{c \in \mathcal{C}} y(c) \cdot \mathbf{X}(c)
  \,.
\end{displaymath}
Since $\tilde F$ is required to preserve any colimit and hence these particular colimits, \eqref{YonedaRestriction} implies that $\widetilde F$ is fixed to act, up to isomorphism, as

\begin{displaymath}
\begin{aligned}
    \widetilde F(\mathbf{X})
    & =
    \widetilde F
    \left(
      \int^{c \in \mathcal{C}} y(c) \cdot \mathbf{X}(c)
    \right)
    & \coloneqq
    \int^{c \in \mathcal{C}} F(c) \cdot \mathbf{X}(c)
    \;\;\;\;\in \mathcal{D}
  \end{aligned}
\end{displaymath}
(where the colimit (a [[coend]]) on the right is computed in $\mathcal{D}$!).

\end{proof}
\hypertarget{free_cocompletion_of_large_categories}{}\subsection*{{Free cocompletion of large categories}}\label{free_cocompletion_of_large_categories}

In general, given a large category $\mathcal{A}$, we can define a locally small category of \emph{small presheaves} on $\mathcal{A}$ as the full subcategory of the functor category $Fun(\mathcal{A}^{\mathrm{op}},\operatorname{Set})$ spanned by those functors $F:\mathcal{A}^{\mathrm{op}} \to \operatorname{Set}$ such that there exists a functor $\gamma:\mathcal{C} \to \mathcal{A}$ with $\mathcal{C}$ small and a presheaf $F': \mathcal{C}^{\mathrm{op}}\to \operatorname{Set}$ such that

\begin{displaymath}
F=\operatorname{Lan}_{\gamma} F'.
\end{displaymath}
In the case where $\mathcal{A}^{\mathrm{op}}$ is [[locally presentable]], the category of small presheaves on $\mathcal{A}$ is equivalent to the category of [[accessible functors]] $\mathcal{A}^{\mathrm{op}}\to \operatorname{Set}$. In this situation, the category of small functors has many nice properties and is often said to be \emph{almost a topos}. In particular, under this assumption, the category of small presheaves is complete, cocomplete, and Cartesian-closed. See \hyperlink{DayLack}{Day-Lack} for more details.

\hypertarget{free_cocompletion_as_a_pseudomonad}{}\subsection*{{Free cocompletion as a pseudomonad}}\label{free_cocompletion_as_a_pseudomonad}

[[David Corfield]]: So is this `free cocompletion' part of an adjunction between the category of categories and the category of cocomplete categories (modulo size worries?). Or should we think of it as part of a [[pseudoadjunction]] between \emph{2-categories}? (I would start a page on that, but how are naming conventions going in this area?)

[[John Baez]]: Equations between functors tends to hold only up to natural isomorphism. So, your first guess should not be that there's an \emph{adjunction} between the \emph{categories} $Cat$ and $CocompleteCat$, but rather, a \emph{pseudoadjunction} between the \emph{2-categories} $Cat$ and $CocompleteCat$.

If this were true, what would it mean? It would mean that there's a forgetful 2-functor:

\begin{displaymath}
U: CocompleteCat \to Cat
\end{displaymath}
together with a `free cocompletion' 2-functor, which right now we've been calling `hat':

\begin{displaymath}
F: Cat \to CocompleteCat
\end{displaymath}
\begin{displaymath}
F: C \mapsto \widehat{C} = Set^{C^{op}}
\end{displaymath}
And, it would mean there's an equivalence of categories

\begin{displaymath}
hom_{CocompleteCat} (F C, D) \simeq
   hom_{Cat} (C, U D)
\end{displaymath}
for every $C \in Cat$, $D \in CocompleteCat$. And finally, it would also be saying that this equivalence is [[pseudonatural transformation|pseudonatural]] as a function of $C$ and $D$.

If we have a pseudonatural equivalence of categories

\begin{displaymath}
hom_{CocompleteCat} (F C, D) \simeq
   hom_{Cat} (C, U D)
\end{displaymath}
instead of a natural isomorphism of sets, then we say we have a `pseudoadjunction' instead of an adjunction. A pseudoadjunction is the right generalization of adjunction when we go to 2-categories; if we were feeling in a modern mood we might just say `adjunction' and expect people to know we meant `pseudo'.

Naively, it seems we \emph{do} have such a pseudoadjunction, at least \emph{modulo size issues}---which unfortunately is sort of like saying ``modulo truth''! The problem is that if Cat is the 2-category of [[small category|small]] categories then to define the free cocompletion functor

\begin{displaymath}
F : Cat \to CocompleteCat
\end{displaymath}
we need $CocompleteCat$ to be the 2-category of [[large category|large]] categories. But if $CocompleteCat$ is the 2-category of [[large category|large]] categories then to define

\begin{displaymath}
U: CocompleteCat \to Cat
\end{displaymath}
we need $Cat$ to be the 2-category of large categories! So, instead of an honest pseudoadjunction that bounces us back and forth between two 2-categories, the size keeps ratcheting up each time we make a round trip!

In particular, if we try to define a [[pseudomonad]]

\begin{displaymath}
U F : Cat \to Cat
\end{displaymath}
we're stuck: the `$Cat$' at right contains larger categories than the one at left.

In \href{http://www.lacim.uqam.ca/~gambino/species.pdf}{their work on species}, Fiore, Gambino, Hyland and Winskel had to confront this issue. In one draft of this paper they had a very artful and sophisticated device for dealing with this size problem. In the latest draft they seem to have sidestepped it entirely: you'll see they discuss the `free symmetric monoidal category on a category' pseudomonad, but never the `free cocomplete category on a category' pseudomonad, even though they \emph{do} use the $\widehat{C}$ construction all over the place. Somehow they've managed to avoid the need to consider this construction as a pseudomonad!

\hypertarget{in_higher_category_theory}{}\subsection*{{In higher category theory}}\label{in_higher_category_theory}

One can ask for the notion of free cocompletion in the wider context of [[higher category theory]].

\begin{itemize}%
\item for [[(∞,1)-category]] theory there is [[free (∞,1)-cocompletion]] .

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

\begin{itemize}%
\item [[Brian Day]], [[Steve Lack]], \emph{Limits of small functors} (\href{https://arxiv.org/abs/math/0610439}{web})

\end{itemize}
This reference might also give helpful clues:

\begin{itemize}%
\item [[Daniel Dugger]], \emph{Sheaves and Homotopy Theory} (\href{http://www.uoregon.edu/~ddugger/cech.html}{web}, \href{http://ncatlab.org/nlab/files/cech.pdf}{pdf})

\end{itemize}
A pedagogical explanation of the universal property of the [[Yoneda embedding]] is given starting on page 7. On page 8 there's an explanation with lots of pictures how a presheaf is an ``instruction for how to build a colimit''. Then on p. 9 the universal morphism that we are looking for here is identified as the one that ``takes the instructions for building a colimit and actually \emph{builds} it''.

(This text, by the way, contains various other gems. A pity that it is left unfinished.)

[[!redirects free cocompletions]]

[[!redirects free colimit completion]] [[!redirects free colimit completions]]

[[!redirects free co-completion]] [[!redirects free co-completions]]



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