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\begin{document}

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\section*{dense functor}

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

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

[[!include category theory - 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{terminology_and_history}{Terminology and History}\dotfill \pageref*{terminology_and_history} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{warning}{Warning}\dotfill \pageref*{warning} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In [[topology]], a not necessarily continuous function $f:X\to Y$ between [[Hausdorff space|Hausdorff spaces]] is \emph{dominant}, or \emph{dense}, in the sense that the [[image]] of $f$ is [[dense subspace|dense]] in $Y$, precisely if every continuous map $g:Y\to Z$ to some other Hausdorff space $Z$ is uniquely determined by $g\circ f$.

The concept of a \textbf{dense functor} is a generalization of this concept to functors.

An important special case that was also historically the source of the concept, is the case of a [[dense subcategory]] inclusion: a [[subcategory]] $S$ of category $C$ is \emph{dense} if every object $c$ of $C$ is a [[colimit]] of a diagram of objects in $S$, in a canonical way.

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

A functor $i:S\to C$ is \textbf{dense} if it satisfies the following equivalent conditions.

\begin{enumerate}%
\item every object $c$ of $C$ is the vertex of the following colimit over the [[comma category]] $(i/c)$:

\begin{displaymath}
\mathrm{colim}((i/c)\stackrel{\mathrm{pr}_S}{\longrightarrow} S \stackrel{i}{\to} C)
\end{displaymath}

\item every object $c$ of $C$ is the $C(i-,c)$-weighted colimit of $i$. This version generalizes more readily to the [[enriched category|enriched]] context.


\item the corresponding [[restricted Yoneda embedding]] $C \to [S^{op},Set]$ is [[full and faithful functor|fully faithful]].



\end{enumerate}
\hypertarget{terminology_and_history}{}\subsection*{{Terminology and History}}\label{terminology_and_history}

[[John Isbell]] introduced [[dense subcategory|dense subcategories]] in a seminal paper \hyperlink{MR0175954}{(Isbell 1960)} under the name \emph{left adequate}. The dual notion of \emph{right adequate} was also introduced and subcategories satisfying both were called \emph{adequate}. It was also shown that while the relation of being left (or right) adequate is not transitive, being adequate is transitive. He also brought out interesting connections with [[set theory]] and [[measurable cardinals]].

Later in the mid 60s, [[Friedrich Ulmer]] considered the concept for more general functors $F:C\to D$, not only inclusions $I:C\hookrightarrow D$, and introduced the name \emph{dense} for them.

Independently, [[Pierre Gabriel]] worked on this concept and their work coalesced to what was to become the concept of a [[locally presentable category]] of their \hyperlink{GabrielUlmer71}{1971 monograph}. It is also good to keep in mind the `\emph{Abelian}' subcontext in the background, in particular the developments in module theory e.g. Lazard's (1964) characterization of flat modules as filtered colimits of finitely generated free modules.

More recently, [[Jacob Lurie]] has referred to the analogue notion for [[(∞,1)-categories]] as \emph{strongly generating} in a version (arXiv v4) of his [[Higher Topos Theory|HTT]], but that term normally means [[strong generator|something different]].

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

\begin{enumerate}%
\item Let $V$ be a category of [[algebras]] and $n \in \mathbb{N}$ such that $V$ has a presentation with operations of at most arity $n$. Let $v$ be the free $V$-algebra on $n$ generators. Then the full subcategory with object $v$ is dense in $V$.


\item In $\Set$, a singleton space is dense.



\end{enumerate}
\hypertarget{warning}{}\subsection*{{Warning}}\label{warning}

There is a different notion of a dense subcategory, often used in [[shape theory]], which has a bit of the same spirit. A [[full subcategory]] $D\subset C$ is \textbf{dense} in this second sense, if every object in $C$ admits a $D$-expansion.

A \emph{$D$-expansion} of an object $X$ in $C$ is a morphism $X\to \mathbf{X}$ in $\mathrm{pro}C$ such that $\mathbf{X}$ is in $\mathrm{pro}D$ and $X$ is the rudimentary system (constant inverse system) corresponding to $X$; moreover one asks that the morphism is universal among all such morphisms $X\to\mathbf{Y}$.

Given a dense subcategory $D\subset C$ one defines an abstract shape category $\mathrm{Sh}(C,D)$ which has the same objects as $C$, but the morphisms are the equivalence classes of morphisms in $\mathrm{pro}D$ of $D$-expansions.

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[dense subcategory]]


\item [[codensity monad]]


\item [[space and quantity]]


\item [[dominant geometric morphism]]



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

\begin{itemize}%
\item [[William Lawvere]], \emph{John Isbell's Adequate Subcategories}, TopCom \textbf{11} no.1 2006. (\href{http://at.yorku.ca/t/o/p/d/65.htm}{link})


\item [[Pierre Gabriel|Peter Gabriel]], [[Friedrich Ulmer]], \emph{Lokal pr\"a{}sentierbare Kategorien} , LNM \textbf{221} Springer Heidelberg 1971. (\S{} 3, pp.38-44)


\item [[John Isbell]], \emph{Adequate subcategories} , Illinois J. Math. \textbf{4} (1960) pp.541-552. \href{http://www.ams.org/mathscinet-getitem?mr=0175954}{MR0175954}. (\href{https://projecteuclid.org/euclid.ijm/1255456274}{euclid})


\item [[John Isbell]], \emph{Subobjects, adequacy, completeness and categories of algebras} , Rozprawy Mat. \textbf{36} (1964) pp.1-32. (\href{http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.desklight-0dbcb276-0b92-49eb-b504-a9963119ea3e}{toc}, \href{http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.desklight-0dbcb276-0b92-49eb-b504-a9963119ea3e/c/rm36_01.pdf}{pdf})


\item [[Max Kelly]], \emph{Basic Concepts of Enriched Category Theory} , Cambridge UP 1982. (Reprinted as \href{http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html}{TAC reprint no.10} (2005); chapter 5, pp.85-112)


\item [[Saunders Mac Lane]], \emph{Categories for the Working Mathematician} , Springer Heidelberg 1998$^2$. (section X.6, pp.245ff, 250)


\item Horst Schubert, \emph{Kategorien II} , Springer Heidelberg 1970. (section 17.2, pp.29ff)


\item [[Friedrich Ulmer]], \emph{Properties of dense and relative adjoint functors} , J. of Algebra \textbf{8} (1968) pp.77-95.



\end{itemize}
[[!redirects adequate subcategory]] [[!redirects left adequate subcategory]]



\end{document}