A pro-object of a category $C$ is a “formal cofiltered limit” of objects of $C$.
The category of pro-objects of $C$ is written $pro$-$C$. Such a category is sometimes called a pro-category, but notice that that is not the same thing as a pro-object in Cat.
“Pro” is short for “projective” (projective limit is an older term for limit). It is in contrast to “ind” in the dual notion of ind-object, standing for “inductive”, (and corresponding to inductive limit, the old term for colimit). In some (often older) sources, the term ‘projective system’ is used more or less synonymously for pro-object.
The definition of arrows in the category of pro-objects in $\mathcal{C}$ is perhaps more intuitive in the dual case of ind-objects (pro-objects in $C^{op}$), where it can be seen as stipulating that the objects of $C$ are finitely presentable in $ind$-$C$.
Let $\mathcal{C}$ be a small category. We write
for the Yoneda embedding of its opposite category, which we will often regard through its opposite functor
This is still a full subcategory-inclusion, now of $\mathcal{C}$ itself.
(category of pro-objects)
The category of pro-objects of $\mathcal{C}$, according to Grothendieck 1960, Section 2, is the full subcategory of the functor category $Func(\mathcal{C}, Sets)^{op}$ (2)
on those functors which are cofiltered limits of representable functors under the opposite Yoneda embedding (2), hence of the form
for $\mathcal{I}$ a cofiltered category. These objects of $Pro(\mathcal{C})$ are also called the pro-objects of $\mathcal{C}$.
Since $Func(\mathcal{C}, Sets)^{op}$ (2) is the free completion of $\mathcal{C}$, we may think of $Pro(\mathcal{C})$ (Def. ) as the free completion of $\mathcal{C}$ under cofiltered limits. See also at pro-representable functor.
Under the identification of the objects of a category with those of its opposite category, the pro-objects (4) are equivalently filtered colimits of presheaves:
since the limit of a functor is the colimit of its opposite functor.
The category $Pro(\mathcal{C})$ is the opposite category of that of ind-objects in the opposite catgegory of $\mathcal{C}$ (e.g. Kashiwara-Schapira 06, p. 138): $Pro(\mathcal{C}) \simeq (Ind(\mathcal{C}^{op}))^{op} \,.$
The pro-objects regarded as functors (4) are objectwise on $c \in \mathcal{C}$ given by
This is the composite of the following sequence of natural bijections:
where
the second step uses that colimits of presheaves are computed objectwise;
the third step is the Yoneda lemma;
the fourth step is the Yoneda embedding (1);
the fifth step is the definition of the opposite category.
(hom-sets between pro-objects as limits of colimits of hom-sets in $\mathcal{C}$)
The hom-sets in $Pro(\mathcal{C})$ (Def. ) are in natural bijection with limits of colimits of hom-sets in $\mathcal{C}$, as follows:
This is the composite of the following sequence of natural bijections:
Here
the first step is the fully faithfulness of the defining full subcategory-inclusion (3);
the second step is the definition of the opposite category, using Remark ;
the third step uses that hom-functors preserve limits, which means that $\mathcal{K}^{op} \xrightarrow{\;\mathcal{K}(-,k)\;} Sets$ takes colimits in $\mathcal{K}$ to limits in Sets, here for $\mathcal{K} = Func(\mathcal{C},Sets)$;
the fourth step is the Yoneda lemma;
We can give an explicit description of the arrows of pro-$\mathcal{C}$ as follows. First, for pro-objects $F: \mathcal{D} \rightarrow \mathcal{C}$ and $G: \mathcal{E} \rightarrow \mathcal{C}$ and for any object $e$ of $\mathcal{E}$, we introduce a relation $\sim$ on arrows with target $G(e)$ which identifies an arrow $f: F(d) \rightarrow G(e)$ with an arrow $f': F(d') \rightarrow G(e)$ for objects $d$ and $d'$ of $\mathcal{D}$ and an object $e$ of $\mathcal{E}$, if there is an object $d''$ of $\mathcal{D}$, an arrow $g: d'' \rightarrow d$ of $\mathcal{D}$, and an arrow $g': d'' \rightarrow d'$ of $\mathcal{D}$, such that $f \circ F(g) = f' \circ F(g')$.
This relation $\sim$ is in fact an equivalence relation. Symmetry is obvious. Reflexivity is immediately demonstrated using the identity arrows of $\mathcal{D}$. Transitivity would not hold for an arbitrary category, but follows from the assumption that $\mathcal{D}$ is cofiltered. Indeed, suppose that we have a zig-zag in $\mathcal{D}$ as follows.
The fact that $\mathcal{D}$ is cofiltered ensures that there is an object $d''$ of $\mathcal{D}$ fitting into the following diagram.
Suppose that we have arrows $f_{0} : F(d_{0}) \rightarrow G(e)$, $f_{1}: F(d_{1}) \rightarrow G(e)$, and $f_{2}: F(d_{2}) \rightarrow G(e)$ such that $g_{0}$ and $g_{1}$ exhibit that $f_{0} \sim f_{1}$, and such that $g_{2}$ and $g_{3}$ exhibit that $f_{1} \sim f_{2}$. Then
This exhibits that $f_{0} \sim f_{2}$, as required.
With this equivalence relation $\sim$ to hand, we can give our explicit description of the arrows of pro-$\mathcal{C}$: an arrow of pro-$\mathcal{C}$ from a pro-object $F: \mathcal{D} \rightarrow \mathcal{C}$ to a pro-object $G: \mathcal{E} \rightarrow \mathcal{C}$ can be taken to be a set $\left\{ f_{e} : F\left(d_{e}\right) \rightarrow G(e) \right\}$ of arrows of $\mathcal{C}$, one for every object $e$ of $\mathcal{E}$, such that, for every arrow $g: e \rightarrow e'$ of $E$, $G(g) \circ f_{e} \sim f_{e'}$.
In other words: a set $\left\{ f_{e} : F\left(d_{e}\right) \rightarrow G(e) \right\}$ of arrows of $\mathcal{C}$, one for every object $e$ of $\mathcal{E}$, such that, for every arrow $g: e \rightarrow e'$ of $E$, there is an object $d$ of $\mathcal{D}$, an arrow $g_{e} : d \rightarrow d_{e}$ of $\mathcal{D}$, and an arrow $g_{e'}: d \rightarrow d_{e'}$ of $\mathcal{D}$ such that $G(g) \circ f_{e} \circ F(g_{e}) = \circ f_{e'} \circ F(g_{e'})$.
Two such sets $\left\{ f_{e} \right\}_{e \in Ob(\mathcal{E})}$ and $\left\{ f'_{e} \right\}_{e \in Ob(\mathcal{E})}$ are equal, i.e. define the same arrow from $F$ to $G$, if $f_{e} \sim f'_{e}$ for every object $e$ of $\mathcal{E}$.
In some cases, pro-objects in a category $\mathcal{C}$ can be viewed as actual limits in a certain category. We prove here some results of this kind.
Let $\mathcal{C}$ be a category, and let $\mathcal{A}$ be a category with cofiltered limits. Suppose that we have a fully faithful functor $i: \mathcal{C} \rightarrow \mathcal{A}$ which lands in cocompact objects. Then $lim_{\mathcal{A}} \circ (i \circ) : Pro(\mathcal{C}) \to \mathcal{A}$ is fully faithful, and hence defines an equivalence onto its image.
Let $F:\mathcal{D} \to \mathcal{C}$ and $G:\mathcal{E} \to \mathcal{C}$ be pro-objects. We then have a sequence of (natural) bijections:
$Hom_{Pro(\mathcal{C})}(F,G)$
$= \underset{e:\mathcal{E}}{lim} \, \underset{d:\mathcal{D}}{colim} \, Hom_{\mathcal{C}}(F d, G e)$
$\cong \underset{e:\mathcal{E}}{lim} \, \underset{d:\mathcal{D}}{colim} \, Hom_{\mathcal{A}}(i (F d), i (G e))$
$\cong \underset{e:\mathcal{E}}{lim} \, Hom_{\mathcal{A}}(\underset{d:\mathcal{D}}{lim} (i \circ F) d, i (G e))$
$\cong Hom_{\mathcal{A}}(\underset{d:\mathcal{D}}{lim} (i \circ F) d, \underset{e:\mathcal{E}}{lim} (i \circ G) e)$
$= Hom_{\mathcal{A}}(lim (i \circ F), lim (i \circ G))$
With these bijections being by definition of pro-object morphisms, fully faithfulness, cocompactness of $i (G e)$, definition of a limit, and definition respectively.
Let $\mathcal{C}$ be the category Grp of groups, and let $\mathcal{A}$ be the category $\mathsf{Top-Grp}$ of topological groups. The fully faithful functor $\mathsf{Set} \rightarrow \mathsf{Top}$ sending a set to the discrete topological space on this set gives rise to a fully faithful functor $\mathsf{Grp} \rightarrow \mathsf{Top-Grp}$. Then (as finite discrete spaces are cocompact) Proposition implies that the category pro-$\mathsf{FinGrp}$ of pro-objects in $\mathsf{FinGrp}$, that is to say of profinite groups, is equivalent to the full sub-category of topological groups whose objects are obtained as a cofiltered limit of finite groups (viewed as topological groups via the discrete topology).
Though it is less well-known, one can in Example evidently replace $\mathsf{Top}$ with any category $\mathcal{A}$ for which there is a fully faithful functor $\mathsf{Set} \rightarrow \mathcal{A}$ which preserves finite products and lands in cocompact objects. See discrete object for one general setting in which finite product preserving functors exist.
Both Example and Remark generalise from $\mathsf{Grp}$ to any finite product theory, that is to say to the category of models of a finite product sketch. They generalise further to any finite limit theory, that is to say to the category of models of a finite limit sketch, if the functor $\mathsf{Set} \rightarrow \mathcal{A}$ moreover preserves finite limits.
Since every (dg-)coalgebra is the filtered colimit of its finite-dimensional subalgebras (see at coalgebra – as filtered colimit), the linear dual of a (dg-)coalgebra is canonically a pro-object in finite dimensional (dg-)algebras. This plays a role for instance for constructing model structures for L-infinity algebras, see there.
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 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’.
The form of shape theory developed by Mardešić and Segal, at about the same time as the work in algebraic geometry, again used an approach equivalent to that of the pro-homotopy category. Under the name `Čech homotopy theory', the use more formally of the pro-homotopy category, was introduced by Porter. Strong shape, developed by Edwards and Hastings, also by <a class='existingWikiWord' href='/nlab/show/Tim+Porter'>Porter</a> and then in further work by <a class='existingWikiWord' href='/nlab/show/Sibe+Mardesic'>Mardešić</a> and <a class='existingWikiWord' href='/nlab/show/Jack+Segal'>Segal</a>, 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.
pro-object / pro-object in an (∞,1)-category
Alexander Grothendieck, Section A.2 of: FGA Technique de descente et théorèmes d’existence en géométrie algébriques. II. Le théorème d’existence en théorie formelle des modules, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195, 22 p. (numdam:SB_1958-1960__5__369_0, English translation: pdf, web version)
(SGA4-1) Alexander Grothendieck, Jean-Louis Verdier, Préfaisceaux, Exp. 1 (retyped pdf) in Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4). Dirigé 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 269, Springer 1972. pdf of SGA 4, Tome 1
Michael Artin, Barry Mazur, appendix of Étale homotopy theory, Lecture Notes in Maths. 100, Springer-Verlag, Berlin 1969.
Masaki Kashiwara, and Pierre Schapira, Section 6 of: Categories and Sheaves, Grundlehren der mathematischen Wissenschaften 332 (2006)
Peter Johnstone, Section VI.1 of: Stone Spaces
Dan Isaksen, Calculating limits and colimits in pro-categories, Fund. Math. 175 (2002), no. 2, 175–194.
Jacob Lurie, Section 6.1 in: Ultracategories (pdf)
Jean-Louis Verdier, Equivalence essentielle des systèmes projectifs, C. R.A.S. Paris261 (1965), 4950 - 4953.
John Duskin, Pro-objects (after Verdier), Sém. Heidelberg- Strasbourg1966 -67, Exposé 6, I.R.M.A.Strasbourg.
A. Deleanu, P. Hilton, Borsuk shape and Grothendieck categories of pro-objects, Math. Proc. Camb. Phil. Soc.79-3 (1976), 473-482 MR400220
Walter Tholen, Pro-categories and multiadjoint functors, Canadian J. Math. 36:1 (1984) 144-155 doi
See also discussion in shape theory:
Jean-Marc Cordier, and Tim Porter, Shape Theory, categorical methods of approximation, Dover (2008) (This is a reprint of the 1989 edition without amendments.)
S. Mardešić, J. Segal, Shape theory, North Holland 1982