I’ve been reading an interesting paper of Adams, Haeberly, Jackowski, and May on the Atiyah-Segal completion theorem. One of the surprising features of this paper is the heavy use of pro-abelian groups to deal with the inconvenient fact that inverse limits are generally not exact in abelian groups. I’d like to blog about the proof in this paper, but first I’d like to go through some of the background on pro-objects. In this post, I’ll describe the entirely dual picture of ${\mathrm{Ind}}$-objects, which is (at least for me) easier to understand.

1. Definition

Let ${\mathcal{A}}$ be a small abelian category. Then there is an imbedding

$\displaystyle \mathcal{A} \hookrightarrow \mathrm{Ind}(\mathcal{A}),$

of ${\mathcal{A}}$ into the larger category of ind-objects of ${\mathcal{A}}$. One benefit of doing this is that ${\mathrm{Ind}(\mathcal{A})}$ is a larger abelian category containing ${\mathcal{A}}$, in which there are enough injectives.

I always found the traditional definition of these confusing, so let me describe another definition (which happens to generalize nicely to the ${\infty}$-categorical case, and which is where I learned it from).

Let ${\mathcal{C}}$ be any category. Then we know that the category ${P(\mathcal{C}) = \mathrm{Fun}(\mathcal{C}^{op}, \mathbf{Sets})}$ is the “free cocompletion” of ${\mathcal{C}}$: that is, given any cocomplete category ${\mathcal{D}}$, we have an equivalence

$\displaystyle \mathrm{Fun}(\mathcal{C}, \mathcal{D}) \simeq \mathrm{Fun}^{L}( P(\mathcal{C}), \mathcal{D})$

between functors ${\mathcal{C} \rightarrow \mathcal{D}}$ and colimit-preserving functors ${P(\mathcal{C}) \rightarrow \mathcal{D}}$. The ${\mathrm{Ind}}$-category is defined to have an analogous universal property, except that one just takes filtered colimits. (more…)