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\begin{document}

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\section*{total category}

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \textbf{total category} is a category with a well-behaved [[Yoneda embedding]] endowing the category with very good completeness and cocompleteness properties but still admitting most types of categories occurring ``in practice''.

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

\begin{defn}
\label{}\hypertarget{}{}
A [[locally small category]] $C$ is \textbf{total} if its [[Yoneda embedding]] $Y \;\colon \;C\longrightarrow [C^{op},Set]$ has a [[left adjoint]] $L$.

If the [[opposite category]] $C^{op}$ is total, $C$ is called \textbf{cototal}.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The definition above requires some [[set theory|set-theoretic]] assumption to ensure that the [[functor category]] $[C^{op},Set]$ exists, but it can be rephrased to say that the [[colimit]] of $Id_C:C\to C$ [[weighted limit|weighted]] by $W$ exists, for any $W:C^{op}\to Set$. (This still involves [[quantification]] over large objects, however, so some foundational care is needed.) This version has an evident generalization to [[enriched category|enriched]] categories.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Since the [[Yoneda embedding]] is a [[full and faithful functor]], a total category $C$ induces an [[idempotent monad]] $Y \circ L$ on its [[category of presheaves]], hence a [[modality]]. One says that $C$ is a [[totally distributive category]] if this modality is itself the [[right adjoint]] of an [[adjoint modality]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The $(L \dashv Y)$-[[adjunction]] of a total category is closely related to the $(\mathcal{O} \dashv Spec)$-[[adjunction]] discussed at \emph{[[Isbell duality]]} and at \emph{[[function algebras on ∞-stacks]]}. In that context the $L Y$-[[modality]] deserves to be called the \emph{[[affine modality]]}.

\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{itemize}%
\item Total categories satisfy a very satisfactory [[adjoint functor theorem]]: any colimit-preserving functor from a total category to a locally small category has a [[right adjoint]].


\item Although the definition refers explicitly only to colimits, every total category is also [[complete category|complete]], i.e. has all small limits. It also has some large limits. In fact, it has ``all possible'' large limits that a locally small category can have: if $F\colon D\to C$ is a functor such that $lim_d Hom_C(X,F d)$ is a small set for all $X\in C$, then $F$ has a limit.


\item A total category $\mathcal{C}$ is [[cartesian closed category|cartesian closed]] iff $L$ preserves binary products (cf. \hyperlink{Wood82}{Wood 1982}, thm. 9).



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

\begin{prop}
\label{Day}\hypertarget{Day}{}
Any [[cocomplete category|cocomplete]] and [[epi-cocomplete category|epi-cocomplete]] category with a [[generator]] is total. More generally, any cocomplete and $E$-complete category with an $E$-generator is total, for a suitable class $E$.

\end{prop}
See (\hyperlink{Day}{Day}), theorem 1, for a proof. This includes:

\begin{itemize}%
\item [[locally presentable category|locally presentable categories]], hence in particular [[Grothendieck toposes]], or the category of [[abelian sheaves]] on a small [[site]].

\end{itemize}
Also, totality lifts along [[solid functors]]; that is, if the [[codomain]] of a solid functor is total, then so is its domain. See (\hyperlink{Tholen}{Tholen}) for a proof. This implies that the following types of categories are total:

\begin{itemize}%
\item any [[reflective subcategory]] of a total category

\end{itemize}
For example

\begin{itemize}%
\item any category which is [[monadic]] over [[Set]]


\item any category admitting a [[topological functor]] to [[Set]]


\item The category [[Grp]] of groups as a category monadic over $Set$ is total, but it is not cototal; see below.


\item The category of [[topological groups]] is total as well since it is topological over the total category [[Grp]].


\item If $C$ is total and $J$ is small, then $C^J$ is total, morally because it is a reflective subcategory of $Set^{C^{op} \times J}$; see section 6 of \hyperlink{Kelly}{Kelly}.



\end{itemize}
Thus, ``most naturally-occurring'' cocomplete categories are in fact total.

In practice, i.e., in naturally occurring concrete cases, cototality is more rare. For example, it is frequently \emph{not} the case that categories that are monadic over $Set$ are cototal. This is well-illustrated by the following two examples:

\begin{itemize}%
\item The category of groups [[Grp]] is not cototal; if it were, then any continuous functor $Grp \to Set$ would be representable. To see this is not the case, it suffices to produce a class of [[simple groups]] $G_\alpha$ of unbounded [[cardinality]] (for example, for any [[infinite set]] $X$, the [[alternating group]] $Alt(X)$, consisting of permutations of finite support that are even, is simple and of cardinality equal to that of $X$). For any group $G$, the hom-set $\hom(G_\alpha, G)$ consists of a single element (the trivial homomorphism) as soon as the cardinality of $G_\alpha$ exceeds that of $G$. Thus the class-indexed product $\prod_\alpha \hom(G_\alpha, G)$ is bounded in size, and defines a continuous functor $F = \prod_\alpha \hom(G_\alpha, -): Grp \to Set$. But it is clear this functor is not representable; e.g., for any group $G$, one can find $G_\alpha$ such that $F(G_\alpha)$ is much larger in size than $\hom(G, G_\alpha)$. This example is given in \hyperlink{Wood82}{Wood 1982}.


\item By a similar construction, the category of commutative rings is not cototal. For each infinite cardinal $\alpha$, choose a field $F_\alpha$ of size $\alpha$, e.g., an algebraically closed field over $\mathbb{Q}$ of transcendence degree $\alpha$. Put $A_\alpha = \mathbb{Z} \times F_\alpha$. Then, for any commutative ring $R$, there is exactly one homomorphism $A_\alpha \to R$ as soon as $\alpha$ exceeds the cardinality of $R$. Then one argues that $\prod_\alpha \hom(A_\alpha, -): CRing \to Set$ is continuous but not representable.



\end{itemize}
But cototal categories do occur:

\begin{itemize}%
\item [[Set]] is cototal (as well as total).


\item By dualizing Proposition \ref{Day}, [[Ab]] is cototal (as well as total), because it is complete, well-powered, and has a cogenerator (e.g., $\mathbb{Q}/\mathbb{Z}$). Similarly, the category of [[modules]] $R Mod$ is cototal (and total) for any [[ring]] $R$. For that matter, any [[well-powered category|well-powered]] [[Grothendieck category]], such as the category of [[abelian sheaves]] on a small [[site]], is cototal.


\item Similarly, the category $CH$ of [[compact Hausdorff spaces]] is cototal (as well as total, being monadic over $Set$), because like $Ab$ it is complete, well-powered, and has a cogenerator $I = [0, 1]$ (cf. [[Urysohn's lemma]]).


\item If $C$ is cototal and $J$ is small, then $C^J$ is cototal.


\item Any [[presheaf category]] of a [[small category]] is cototal (as well as total). Indeed, any [[Grothendieck topos]] is both cototal and total.


\item Any category admitting a [[topological functor]] to [[Set]] is cototal (as well as total).


\item Any [[totally distributive category]] is cototal (as well as total).


\item Any [[coreflective subcategory]] of a cototal category is cototal, e.g., the category of [[compactly generated spaces]] is cototal.



\end{itemize}
\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item [[totally distributive category]]


\item [[lex total category]]



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

\begin{itemize}%
\item [[Ross Street]], [[Bob Walters]], \emph{Yoneda structures on 2-category}, (contains the original definition of total categories)


\item [[Max Kelly]], \emph{A survey of totality for enriched and ordinary categories}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques, 27 no. 2 (1986), p. 109-132, \href{http://www.numdam.org/item?id=CTGDC_1986__27_2_109_0}{numdam}



\end{itemize}
\begin{itemize}%
\item [[Walter Tholen]], \emph{Note on total categories}, Bulletin of the Australian Mathematical Society \href{http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=4759056}{cambridge journals}

\end{itemize}
\begin{itemize}%
\item [[Brian Day]], \emph{Further criteria for totality}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques, 28 no. 1 (1987), p. 77-78, \href{http://www.numdam.org/item?id=CTGDC_1987__28_1_77_0}{numdam}

\end{itemize}
\begin{itemize}%
\item [[Richard Garner]], \emph{Topological=Total} (\href{https://arxiv.org/abs/1310.0903}{arXiv:1310.0903})


\item [[Ross Street]], \emph{The family approach to total cocompleteness and toposes} , Trans. A. M. S. \textbf{284} (1984) pp.355-369, ()


\item [[Richard J. Wood]], \emph{Some remarks on total categories}, J. Algebra \textbf{75\_:2, 1982, 538--545 }



\end{itemize}
[[!redirects total categories]] [[!redirects cototal category]] [[!redirects cototal categories]]



\end{document}