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\section*{pro-object}

\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{via_formal_cofiltered_limits}{Via formal co-filtered limits}\dotfill \pageref*{via_formal_cofiltered_limits} \linebreak
\noindent\hyperlink{via_filtered_limits_of_presheaves}{Via filtered limits of presheaves}\dotfill \pageref*{via_filtered_limits_of_presheaves} \linebreak
\noindent\hyperlink{as_formal_duals_of_indobjects}{As formal duals of ind-objects}\dotfill \pageref*{as_formal_duals_of_indobjects} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{tale_homotopy_theory}{\'E{}tale homotopy theory.}\dotfill \pageref*{tale_homotopy_theory} \linebreak
\noindent\hyperlink{shape_theory}{Shape theory.}\dotfill \pageref*{shape_theory} \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}

A \textbf{pro-object} of a [[category]] $C$ is a ``formal [[filtered category|cofiltered]] [[limit]]'' of [[objects]] of $C$.

The category of pro-objects of $C$ is written $pro$-$C$. Such a category is sometimes called a \textbf{pro-category}, but notice that that is \emph{not} the same thing as a pro-object in [[Cat]].

``Pro'' is short for ``projective''. ( \emph{[[projective limit|Projective limit]]} is an older term for \emph{[[limit]]}.) It is in contrast to ``ind'' in the dual notion of [[ind-object]], standing for ``inductive'', (and corresponding to \emph{[[inductive limit]]}, the old term for \emph{[[colimit]]}). In some (often older) sources, the term `projective system' is used more or less synonymously for pro-object.

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

\hypertarget{via_formal_cofiltered_limits}{}\subsubsection*{{Via formal co-filtered limits}}\label{via_formal_cofiltered_limits}

The [[objects]] of the [[category]] $pro$-$C$ are [[diagrams]] $F:D\to C$ where $D$ is a [[small category|small]] [[filtered category|cofiltered]] category. The [[hom set]] of [[morphisms]] between $F:D\to C$ and $G:E\to C$ is

\begin{displaymath}
pro\text{-}C(F,G) = \underset{e\in E}{lim}\, \underset{d\in D}{colim} C(F d, G e)
\end{displaymath}
Notice that here the [[limit]] and [[colimit]] is taken in the category [[Set]] of sets.

Cofiltered limits in [[Set]] are given by sets of [[thread]]s and filtered colimits by [[germs]] (classes of equivalences), thus a representative of $s\in\mathrm{pro}C(F,G)$ is a thread whose each component is a germ:\newline $s = (germ_e(s))_{e\in E}$ which can be more concretely written as $([s_{d_e,e}])_e$; thus $[s_{d_e,e}]\in colim_{d\in D} C(F d, G e)$ where $s_{d_e,e}\in C(F d_e, G e)$ is some representative of the class; there is at least one $d_e$ for each $e$; if the domain $E$ is infinite, we seem to need an axiom of choice in general to find a function $e\mapsto d_e$ which will choose one representative in each class $germ_e(s)$. Thus $s$ is given by the (equivalence class) of the following data

\begin{itemize}%
\item function $e\mapsto d_e$


\item correspondence $e\mapsto s_{d_e,e}\in C(F d_e, G e)$



\end{itemize}
such that $([s_{d_e,e}])_e$ is a thread, i.e. for any $\gamma: e\to e'$ we have an equality of classes (germs) $[G(\gamma)\circ s_{d_e,e}] = [s_{d_{e'},e'}]$. This equality holds if there is a $d'$ and morphisms $\delta_e: d'\to d_e$, $\delta_{e'}: d'\to d_{e'}$ such that $G(\gamma)\circ s_{d_e,e}\circ F\delta_e = s_{d_{e'},e'}\circ F\delta_{e'}$. (Usually in fact people consider the dual of $D$ and the dual of $C$ as filtered domains). Now if we chose a different function $e\mapsto\tilde{d}_e$ instead then, $([s_{d_e,e}])_e = ([s_{\tilde{d}_e,e}])_e$, hence by the definition od classes, for every $e$ there is a $d''\in D$ and morphisms $\sigma_e : d''\to d_e$, $\tilde\sigma_e:d''\to \tilde{d}_e$ such that $s_{\tilde{d}_e,e}\circ F(\tilde\sigma_e) = s_{d_e,e}\circ F(\sigma_e)$.

This definition is perhaps more intuitive in the dual case of [[ind-object|ind-objects]] (pro-objects in $C^{op}$), where it can be seen as stipulating that the objects of $C$ are [[finitely presentable object|finitely presentable]] in $ind$-$C$.

\hypertarget{via_filtered_limits_of_presheaves}{}\subsubsection*{{Via filtered limits of presheaves}}\label{via_filtered_limits_of_presheaves}

Another, equivalent, definition is to let $pro$-$C$ be the [[full subcategory]] of the [[opposite category|opposite]] [[functor category]]/[[category of presheaves|presheaf category]] $[C,Set]^{op}$ determined by those functors which are cofiltered limits of [[representable functor|representables]]. This is reasonable since the [[presheaf category|copresheaf category]] $[C,Set]^{op}$ is the [[free completion]] of $C$, so $pro$-$C$ is the ``free completion of $C$ under cofiltered limits.'' See also at [[pro-representable functor]].

The equivalence with the previous definition is seen as follows. To a functor $F: I \to C$, compose with the [[Yoneda lemma|co-Yoneda embedding]] $C \to [C,Set]^{op}$ to obtain a functor $\tilde F: I \to [C, Set]^{op}$, and then take $|F| = lim \tilde F \in [C,Set]^\mathrm{op}$. Explicitly, $|F|(c) = colim \tilde F^{op}$. This yields a functor $Pro(C) \to [C,Set]^{op}$, and its essential image manifestly consists of the functors which are cofiltered limits of the duals of representables. To see that this functor is fully faithful, we compute, for $F: I \to C$ and $G: J \to C$:

$Hom(|F|,|G|) = Nat(colim \tilde G^\mathrm{op}, colim \tilde F^\mathrm{op})$

$= lim_{J^{op}} Nat(\tilde G^\mathrm{op}, colim \tilde F^\mathrm{op})$

$= lim_{J^{op}} colim_{I^\mathrm{op}} Nat(\tilde G^\mathrm{op}, \tilde F^\mathrm{op})$

$= lim_{J^{op}}colim_{I^\mathrm{op}} Hom_\mathcal{C}(F,G)$

as in $Pro(C)$. Here we have used the definition of a colimit, the fact that representables are [[compact objects]] (this follows from the fact that colimits are computed ``levelwise'' in a functor category), and the Yoneda lemma.

\hypertarget{as_formal_duals_of_indobjects}{}\subsubsection*{{As formal duals of ind-objects}}\label{as_formal_duals_of_indobjects}

\begin{remarl}
\label{ProObjectAsFormalDualOfIndObject}\hypertarget{ProObjectAsFormalDualOfIndObject}{}
The category of pro-objects in $\mathcal{C}$ is the [[opposite category]] of that of [[ind-objects]] in the opposite catgegory of $\mathcal{C}$:

\begin{displaymath}
Pro(\mathcal{C})
    \simeq
  (Ind(\mathcal{C}^{op}))^{op}
  \,.
\end{displaymath}
\end{remarl}
(e.g. \hyperlink{KashiwaraSchapira06}{Kashiwara-Schapira 06, p. 138})

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

\begin{itemize}%
\item [[profinite set]], [[profinite space]]


\item [[profinite group]], [[progroup]]


\item [[pro-étale morphism of schemes]], [[pro-étale site]]


\item Since every ([[dg-coalgebra|dg-]])[[coalgebra]] is the [[filtered colimit]] of its finite-dimensional subalgebras (see at \emph{\href{coalgebra#AsFilteredColimits}{coalgebra -- as filtered colimit}}), the [[linear dual]] of a (dg-)coalgebra is canonically a pro-object in finite dimensional ([[dg-algebra|dg-]])[[associative algebra|algebras]]. This plays a role for instance for constructing [[model structures for L-infinity algebras]], see \href{model+structure+for+L-infinity+algebras#OnProAlg}{there}.


\item [[pro-category of towers]]



\end{itemize}
\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{tale_homotopy_theory}{}\subsubsection*{{\'E{}tale homotopy theory.}}\label{tale_homotopy_theory}

Procategories were used by Artin and Mazur in their work on [[étale homotopy]] theory. They associated to a scheme a `pro-homotopy type'. (This is discussed briefly at [[étale homotopy]].) The important thing to note is that this was a pro-object in the \emph{homotopy category} of simplicial sets, i.e., in the pro-homotopy category. Friedlander rigidified their construction to get an object in the pro-category of simplicial sets, and this opened the door to use of `homotopy pro-categories'.

\hypertarget{shape_theory}{}\subsubsection*{{Shape theory.}}\label{shape_theory}

The form of [[shape theory]] developed by Marde\v{s}i and Segal, at about the same time as the work in algebraic geometry, again used the pro-homotopy category. Strong shape, developed by Edwards and Hastings, Porter and also in further work by Marde\v{s}i and Segal, used various forms of rigidification to get to the pro-category of spaces, or of simplicial sets. There methods of model category theory could be used.

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

\begin{itemize}%
\item [[ind-object]] / [[ind-object in an (∞,1)-category]]


\item \textbf{pro-object} / [[pro-object in an (∞,1)-category]]


\item [[pro-representable functor]]


\item [[ind-pro-object]]


\item [[pro-left adjoint]]


\item [[pro-homotopy theory]], [[profinite completion of a group]]



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

\begin{itemize}%
\item [[Alexander Grothendieck]], \emph{Techniques de d\'e{}scente et th\'e{}or\`e{}mes d'existence en g\'e{}om\'e{}trie alg\'e{}brique, II: le th\'e{}or\`e{}me d'existence en th\'e{}orie formelle des modules}, Seminaire Bourbaki \textbf{195}, 1960, \href{http://archive.numdam.org/article/SB_1958-1960__5__369_0.pdf}{(pdf)}.


\item (SGA4-1) [[Alexander Grothendieck]], [[Jean-Louis Verdier]], \emph{Pr\'e{}faisceaux}, Exp. 1 (\href{http://www.math.polytechnique.fr/~orgogozo/SGA4/01/01.pdf}{retyped pdf}) in \emph{Th\'e{}orie des topos et cohomologie \'e{}tale des sch\'e{}mas. Tome 1: Th\'e{}orie des topos}, S\'e{}minaire de G\'e{}om\'e{}trie Alg\'e{}brique du Bois-Marie 1963--1964 ([[SGA 4]]). Dirig\'e{} par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. Lecture Notes in Mathematics \textbf{269}, Springer 1972. \href{http://www.iecn.u-nancy.fr/~gaillapy/SGA/grothendieck_sga_4.1.pdf}{pdf of SGA 4, Tome 1}


\item [[Michael Artin]], [[Barry Mazur]], appendix of \emph{\'E{}tale homotopy theory}, Lecture Notes in Maths. 100, Springer-Verlag, Berlin 1969.


\item [[Jean-Marc Cordier]], and [[Tim Porter]], \emph{Shape Theory}, categorical methods of approximation, Dover (2008) (This is a reprint of the 1989 edition without amendments.)


\item [[Masaki Kashiwara]], and [[Pierre Schapira]], section 6 of \emph{[[Categories and Sheaves]]}, Grundlehren der mathematischen Wissenschaften 332 (2006)


\item [[Peter Johnstone]], section VI.1 of \emph{[[Stone Spaces]]}


\item [[Dan Isaksen]], \emph{Calculating limits and colimits in pro-categories}, Fund. Math. 175 (2002),


\item [[Sibe Mardesic|S. Marde\v{s}i]], J. Segal, \emph{Shape theory}, North Holland 1982


\item [[Jean-Louis Verdier]], \emph{Equivalence essentielle des syst\`e{}mes projectifs}, C. R.A.S. Paris261 (1965), 4950 - 4953.


\item [[John Duskin]], \emph{Pro-objects (after Verdier)}, S\'e{}m. Heidelberg- Strasbourg1966 -67, Expos\'e{} 6, I.R.M.A.Strasbourg.


\item A. Deleanu, P. Hilton, Borsuk shape and Grothendieck categories of pro-objects, Math. Proc. Camb. Phil. Soc.79-3 (1976), 473-482 \href{http://www.ams.org/mathscinet-getitem?mr=400220}{MR400220}



\end{itemize}
no. 2, 175--194.

\begin{itemize}%
\item Tholen

\end{itemize}
[[!redirects pro-objects]] [[!redirects pro object]] [[!redirects pro-category]]



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