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I can't understand the notion of pro-category and ind-category.

According to [Kashiwara-Schapira, Categories and Sheaves], an ind-object of category $C$ is a contravariant functor $F\in Set^{C^{op}}$ which is isomorphic to a filtered colimit of representable functors, and $Ind(C)$ denotes their full subcategory of $Set^{C^{op}}$. A pro-object and $Pro(C)$ is also defined similarly. However, I can't understand what this category means.

I thought this is a certain completion or cocompletion of categories. Because the category of finite sets does not have arbitrary colimits and its indization $Ind(Set_f)$ is isomorphic to the category of sets. However, according to nlab it seems that the category of groups $Grp$ is cocomplete but its indization $Ind(Grp)$ is not isomorphic to $Grp$.

Furthermore, in [Artin-Mazur, Etale homotopy], homology and cohomology functor is extended to pro-categories. In detail, let $H_{CW}$ be a homotopy category of CW complexes and denote its homology and cohomology functor by $H_*:H_{CW}\to Ab, H^*:H_{CW}^{op}\to Ab$. Then, these are extended to pro-category $H_*:Pro(H_{CW})\to Pro(Ab), H^*:Pro(H_{CW}^{op})\to Pro(Ab)$. However, the image of latter functor is actually an abelian group, while the former is just a pro-abelian group. I don't understand this difference.

To sum up, what I want to ask is

  • What is the motivation of this concept? Although I know every presheaf is canonically isomorphic to a colimit of representable functors, to restrict its index category to be filtered is seemed to be unnatural.
  • Are there any criterion for when the category $C$ and its indization $Ind(C)$ are isomorphic?
  • What is the difference of homology and cohomology in the above case? And when the image of the pro-extension $F:Pro(C)\to Pro(D)$ of a functor $F:C\to D$ is always in $D$?
  • Why this concept is important? I want to know some applications.
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There are excellent reasons to look at ind-completions: for one thing, we have a complete axiomatisation of all categories equivalent to $\mathbf{Ind}(\mathcal{C})$ for a locally small category $\mathcal{C}$ – they are precisely the class $\aleph_0$-accessible categories. A special case are the $\aleph_0$-accessible categories, and very many of the categories of algebraic structures are examples of this: sets, groups, rings, modules, fields, chain complexes of abelian groups, presheaves, (small) categories etc. On the other hand $\mathbf{Pro}(\mathcal{C})$ is simply $\mathbf{Ind}(\mathcal{C}^\mathrm{op})^\mathrm{op}$. There are not as many examples of such categories, but one example is the category of Stone spaces (a.k.a. profinite sets).

Another thing I should point out is that $\mathbf{Ind}(\mathcal{C})$ is almost never equivalent to $\mathcal{C}$, let alone isomorphic to $\mathcal{C}$, because it is a free completion – emphasis on free! This is not unlike the fact that $\mathbb{C} \otimes_\mathbb{R} M$ is rarely isomorphic to $M$ even when $M$ is a $\mathbb{C}$-module.

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  • $\begingroup$ You mean $Set, Grp, Mod$ etc are the indization of $FinSet, FinGrp, FinMod$? Even if it is true, I can't still understand why such a "finite" description is so useful? $\endgroup$ – Fujita Tomomi Nov 21 '13 at 21:53
  • $\begingroup$ Not quite. They are the ind-completion of the finitely presentable objects. Such categories have some of the properties familiar from algebra, e.g. filtered colimits preserve finite limits. $\endgroup$ – Zhen Lin Nov 21 '13 at 21:57
  • $\begingroup$ I just saw it at nlab but there are few examples... In the above examples, what are their finitely presentable objects? $\endgroup$ – Fujita Tomomi Nov 21 '13 at 22:06
  • $\begingroup$ Surely you know what a finitely presentable group/module/etc. is? The abstract definition generalises those examples. $\endgroup$ – Zhen Lin Nov 21 '13 at 22:08
  • $\begingroup$ I see. Is this concept defined over any algebraic theories? $\endgroup$ – Fujita Tomomi Nov 21 '13 at 22:24
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Indization provides a $2$-functor which is left adjoint to the forgetful $2$-functor from the $2$-category of categories with directed colimits with functors preserving directed colimits to the $2$-category of all categories with functors. Thus, the idea is to adjoin directed colimits to a given category freely. We consider the given category only as a category, any additional random properties such as the existence of certain colimits are ignored. Thus, even if it already has directed colimits, the indization will change the category, since it simply doesn't know that directed colimits are already there. At least this is my intuition. One might ask if it is possible to remember the colimits which already exist, but I doubt that this has a universal solution. Perhaps the confusion is also connected to the following: When one thinks of other adjunctions, for example that of torsion-free abelian groups and abelian groups, the left adjoint which mods out the torsion subgroup of course doesn't change torsion-free groups. More generally, if $F$ is left adjoint to $G$ and $G$ is fully faithful, then the counit of the adjunction $FG \to 1$ is an isomorphism. But this is not the case when $G$ is only faithful, i.e. corresponds to the inclusion of a category which is not assumed to be full and therefore may have less morphisms. The same remark applies to $2$-adjunctions, and in particular to the one mentioned above: Between categories with directed colimits, we consider only those functors preserving directed colimits.

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