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.

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 $ind$-$C$.

## Details

###### Definition

A pro-object in a category $\mathcal{C}$ is a functor $F: \mathcal{D} \rightarrow \mathcal{C}$ for some small cofiltered category $\mathcal{D}$.

Pro-objects in a category $\mathcal{C}$ assemble into a category as follows.

###### Definition

Let $\mathcal{C}$ be a category. The category of pro-objects in $\mathcal{C}$ is the category defined as follows.

1. The objects are pro-objects in $\mathcal{C}$.
2. The set of arrows from a pro-object $F: \mathcal{D} \rightarrow \mathcal{C}$ to a pro-object $G: \mathcal{E} \rightarrow \mathcal{C}$ is the limit of the functor $\mathcal{D}^{op} \times \mathcal{E} \rightarrow \mathsf{Set}$ given by $Hom_{\mathcal{C}}\left(F(-), G(-)\right)$.
3. Composition of arrows arises, given pro-objects $F: \mathcal{D}_{0} \rightarrow \mathcal{C}$, $G: \mathcal{D}_{1} \rightarrow \mathcal{C}$, and $H: \mathcal{D}_{2} \rightarrow \mathcal{C}$ of $\mathcal{C}$, by applying the limit functor for diagrams $\mathcal{D}^{op} \times \mathcal{E} \rightarrow \mathsf{Set}$ to the natural transformation of functors $Hom_{\mathcal{C}}\left(F(-), G(-)\right) \times Hom_{\mathcal{C}}\left(G(-), H(-)\right) \rightarrow Hom_{\mathcal{C}}\left( F(-), H(-) \right)$ given by composition in $\mathcal{C}$.
4. The identity arrow on a pro-object $F: \mathcal{D} \rightarrow \mathcal{C}$ arises, using the universal property of a limit, from the identity arrow $Hom_{\mathcal{C}}\left(F(c), F(c)\right)$ for every object $c$ of $C$.

That the associativity and identity axioms hold follows immediately from the fact that they hold in $\mathcal{C}$.

###### Notation

We denote the category of Definition by pro-$\mathcal{C}$.

###### Remark

For brevity, we sometimes write the hom set between $F: \mathcal{D} \to \mathcal{C}$ and $G: \mathcal{E} \to \mathcal{C}$ as

$\underset{e \in Ob(\mathcal{E})}{lim}\, \underset{d\in Ob(\mathcal{D})}{colim} \mathcal{C}(F d, G e),$

where the limit and colimit is taken in the category Set of sets.

###### Remark

We can give an explicit description of the arrows of pro-$\mathcal{C}$ as follows. First, 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

\begin{aligned} f_{0} \circ F(g_{0} \circ g'_{0}) &= f_{0} \circ F(g_{0}) \circ F(g'_{0}) \\ &= f_{1} \circ F(g_{1}) \circ F(g'_{0}) \\ &= f_{1} \circ F(g_{2}) \circ F(g'_{1}) \\ &= f_{2} \circ F(g_{3}) \circ F(g'_{1}) \\ &= f_{2} \circ F(g_{3} \circ g'_{1}). \end{aligned}

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 $\mathcal{E}$.

## Alternative points of view

### 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 $\mathcal{C}$:

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

## Characterisations

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.

###### Proposition

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.

###### Proof

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.

###### Example

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).

###### Remark

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.

###### Remark

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.

## 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, and also 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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