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Let $\mathcal{C}$ be a category. The pro-category pro-$\mathcal{C}$ is defined as (see this nLab page) follows: its objects are diagrams $F: D\to \mathcal{C}$ where $D$ is a small cofiltered category. The Hom set between $F: D\to \mathcal{C}$ and $G: E\to \mathcal{C}$ is given by $$ \text{pro-}\mathcal{C}(F,G):=lim_{e\in E}colim_{d\in D}(Fd,Ge). $$

My question is:

Why we need the cofiltered condition on the index category $D$? If we do not require $D$ to be cofitered, what will we lose?

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    $\begingroup$ I dunno if "lost" is the right word. You've just defined a different category, which you already know by another name: it's $Fun(C, Sets)^{op}$. Maybe one thing you lose is control. The inclusion $C \to Pro(C)$ preserves finite limits, which is not true of the inclusion $C \to Fun(C, Sets)^{op}$ in general. $\endgroup$
    – Dylan Wilson
    Commented May 1, 2017 at 17:40
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    $\begingroup$ Another relevant comment is that $Pro(C)$ views all the elements of $C$ as cocompact (maps into them from a cofiltered limit factor through some finite stage), whereas $Fun(C, Sets)^{op}$ requires the objects of $C$ to have a much stronger requirement: maps into them commutes with arbitrary limits. Such objects rarely appear in nature. $\endgroup$
    – Dylan Wilson
    Commented May 1, 2017 at 17:47
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    $\begingroup$ It should be mentioned that since $Pro(C) = Ind(C^{op})^{op}$, the question can be viewed as equivalent to asking why the Ind-construction is important, and here there are lots of familiar examples to illustrate this. For example, a $k$-algebra is the same thing as an ind-object in finitely-presentable $k$-algebras. Dually, an affine scheme over $Spec(k)$ is the same as a pro-object in affine schemes of finite type over over $Spec(k)$. On the ubiquity of the Ind-construction, see e.g. here. $\endgroup$
    – Tim Campion
    Commented May 4, 2017 at 18:37
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    $\begingroup$ To amplify @TimCampion's comment, the forgetful functor from the category of models of a finitary (essentially) algebraic theory (eg categories) creates filtered colimits. So they work smoothly, without additing new algebraic structure. Why this should apply to cofiltered limits, I don't understand. $\endgroup$
    – Paul Taylor
    Commented Feb 21, 2022 at 12:37

1 Answer 1

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The introduction of $Pro(C)$ is in Grothendieck's seminar Bourbaki 195, (1960). There you can see the properties that using pro-objects gives you. This is related to Dylan's two comments above. In the classical case of handling inverse systems (pro-objects) of fin. dim. vector spaces or finite groups, then the limit of the pro-object is compact in a nice way and you can rebuild the inverse system from it. Without the cofiltering condition you do not get that. Likewise, the classical theory of Cech homology naturally gives rise to inverse systems and not just general functors. We thus have a situation in which several important examples / applications lead to such pro-representable functors AND such functors are better behaved.... It therefore looks to be a GOOD THING to restrict attention to pro-representable functors in these contexts. (N.B. I am missing many other situations where a naturally constructed functor is pro-representable.)

Another N.B. is that there are other forms of pro-representability involving more general properties than the ones involved in pro-representability per se, but going to absolutely general functors / presheaves, (with no conditions) usually involves other methods.

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