# On the notion of arrows in a category of pro-objects

Given a category $$C$$, the pro-objects of $$C$$ are defined as cofiltered diagrams of objects in $$C$$. However, morphisms are not defined as natural transformations between diagrams but by some other procedure.

I guess the idea is that we are thinking of the diagram in terms of its limit. So I believed that if cofiltered limits exist in a naturally enlarged category, then the morphisms between pro-objects are merely morphisms between the limits of the diagrams. I am wrong and the n-lab page on profinite groups (Remark 1.2) indicates that

In most cases, the limit would not actually exist in the category of finite groups, and while it would exist in the category of all groups, it would be “wrong” category-theoretically: maps between profinite groups are not the same as maps between their honest limits in Grp.

Now, I am really confused about the maps in a pro-object category. How should I imagine them? What is the distinction between the maps between limits of objects and maps between pro-objects (say in Sets, Groups...)?

• Yes, there is a category in which you can embed $\mathcal{C}$ as a full subcategory and realise pro-objects as limits and have the correct morphisms... the category of pro-objects. Of course, this is not helpful if you want to construct the category of pro-objects. Here's another candidate: $[\mathcal{C}, \textbf{Set}]^\textrm{op}$. Write out the concrete description of the hom sets and you will see the same definition used in constructing the category of pro-objects. – Zhen Lin Oct 25 at 10:39

## 1 Answer

A pro-object is a formal cofiltered limit of objects in $$C$$, and the morphisms are accordingly formal morphisms between such cofiltered limits ("what the morphisms would be if the cofiltered limits existed, given only that the cofiltered limits exist and nothing else"). The completion of $$C$$ under formal limits is $$[C, \text{Set}]^{op}$$ (at least if $$C$$ is essentially small) so you can take formal cofiltered limits there and consider natural transformations between these and you'll get $$\text{Pro}(C)$$.

If $$C$$ has finite limits (which is true for $$\text{FinGrp}$$) the essential image of the embedding $$\text{Pro}(C) \to [C, \text{Set}]^{op}$$ consists of the functors $$C \to \text{Set}$$ which preserve finite limits. In general it consists of the flat functors.

Speaking concretely about profinite groups, the issue is that if you take the honest limit in $$\text{Grp}$$ then you get the right underlying group but profinite groups can be thought of as topological groups (there's a fully faithful embedding into topological groups) and maps between the limits in $$\text{Grp}$$ can be discontinuous wrt the profinite topology.