# References for Learning Inverse Limits (for Profinite Groups)

I'm doing some independent study on Profinite Groups this summer and, as I understand it, it is important to be familiar with the notion of an inverse limit before doing so. The trouble for me is that there seem to be multiple approaches to teaching them, each depending on a different subject that sends me down a rabbit hole. At first, I was told it would be useful to approach it from the standpoint of Category Theory––looking at limits and colimits––but another, seemingly less general approach seems to be by looking at inverse systems of groups. (I think that an inverse system of groups exists within the category $$\textbf{Grp}$$, so this may be an equivalent definition... Apologies if this is wrong.)

How would you recommend learning about inverse limits for the purpose of studying Profinite Groups and, eventually, the Profinite Topology? Specifically, could anyone please recommend some sources on learning the subject? In case it is asked: of course I have googled around, but there seem to be so many approaches and variety of names (e.g. limits, colimits, direct limits, projective limits, inverse limits, etc...) that it's a bit overwhelming. If it's of any use, I've taken Calculus (including Multivariable), Analysis, Set Theory, Algebra I (Group Theory), and Algebra II (ring, field, and Galois theory). I also know some topology.

• $\varprojlim \Bbb{Z/nZ}$ is the set (also a topological group and ring with the termwise addition and multiplication) of limits of sequences of integers that converge $\bmod n$ for all $n$. This understanding works in 99% of applications. In representation theory of $GL_n(\Bbb{Z}_p)$ it gets harder and the profinite group is understood abstractly as a topological group with particular kind of open/closed/compact sets. – reuns May 25 '20 at 23:12
• @reuns Thanks for that! Unfortunately, I do need more background for the writing, though. Would you recommend any sources? – Luke Poeppel May 25 '20 at 23:50
• tbh I think Wikipedia is pretty on point on this one – Daniel Plácido May 28 '20 at 0:50
• After getting some extra context in the Rotman text, it actually seems like the wiki pages for direct and inverse limits are pretty good. Thanks for pointing me back to that! – Luke Poeppel May 28 '20 at 11:52

This may be more than you need, but you can find an introduction to categories and universal algebra, including the concepts of limit and colimit, in Chapters 1 and 2 of Jacobson's Basic Algebra II.

For an even briefer treatment, you could refer to the appendix on category theory in An Introduction to Homological Algebra by Weibel.

• Thanks for the recommendations! (I also found a pretty good treatment in Rotman's Introduction to Homological Algebra) I look forward to checking these out. – Luke Poeppel May 28 '20 at 11:50