Contents

category theory

Contents

Idea

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.

Definition

Via formal co-filtered limits

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

$pro\text{-}C(F,G) = \underset{e\in E}{lim}\, \underset{d\in D}{colim} C(F d, G e)$

Notice that here the limit and colimit is taken in the category Set of sets.

Cofiltered limits in Set are given by sets of threads 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:
$s = (germ_e(s))_{e\in E}$ 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

• function $e\mapsto d_e$

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

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-objects (pro-objects in $C^{op}$), where it can be seen as stipulating that the objects of $C$ are finitely presentable in $ind$-$C$.

Via filtered limits of presheaves

Another, equivalent, definition is to let $pro$-$C$ be the full subcategory of the opposite functor category/presheaf category $[C,Set]^{op}$ determined by those functors which are cofiltered limits of representables. This is reasonable since the 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 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.

As formal duals of ind-objects

Remark

The category of pro-objects in $\mathcal{C}$ is the opposite category of that of ind-objects in the opposite catgegory of $\mathcal{C}$:

$Pro(\mathcal{C}) \simeq (Ind(\mathcal{C}^{op}))^{op} \,.$

Applications

Étale 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 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’.

Shape theory.

The form of shape theory developed by Mardešić 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 Mardešić 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.

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• Tholen