This is probably a basic question but I haven't found anything satisfying yet.
I'm trying to understand the difference between inverse and direct limits other than the formal definition. In my mind, an inverse limit is like $\mathbb{Z}_p$ and a direct limit is like the germ of functions at a point on a manifold. Perhaps these aren't the best ways to think about it, but it leads me to believe that inverse limits feel "big" and direct limits feel "small." But I've come across some confusion when seeing definitions like
$$H^i(G, M) = \lim_{\to} H^i(G/H, M^H)$$
when I would have thought it would have gone in the other direction. Is it just a matter of formality depending on the direction of arrows, like how one calls left derived functors homology and right derived functors cohomology? Or is there a deeper distinction between these kinds of objects? Thanks.
Direct limits don't have to be small: any set is the direct limit of its finite subsets under inclusion, for instance. An inverse limit of some sets or groups is always a subset (subgroup) of their product, and dually a direct limit is a quotient of their disjoint union (direct sum), if that helps with intuition. I would say that in the end, yes, the difference is purely formal: every direct limit could be described as an inverse limit of a different diagram, although this would usually be very artificial.