What mathematical objects permit "taking of limits"?

Question

Background

I have been reading a lot of abstract algebra recently (at the level of Artin/Dummit & Foote/Herstein Topics in Algebra for those of you familiar with these books). I have noticed that many of the abstract objects like groups and rings lack a certain property in general. Namely, we cannot "take limits" in these settings generally (at least, I don't know how to define such a thing).

Question

What is the property of fields like the reals that allows us to "take a limit" that fields like $\mathbb{Z}/7\mathbb{Z}$ lack?

Answer 1

Many categories are complete or cocomplete, which means that you can take limits or colimits of diagrams in your category. Every category of algebraic structures of a given type is complete and cocomplete. For example:

1. The ring of $p$-adic numbers $\mathbb{Z}_p$ is the limit of the rings $\mathbb{Z}/p^n$, where $n \geq 0$
2. The group $\mathbb{Q}/\mathbb{Z}$ is the colimit of the finite cyclic groups $\mathbb{Z}/n$, where $n>0$ w.r.t. to divisibility.
3. If $E/K$ is a Galois extension, then it is the colimit of the finite Galois extensions $E'/K$ where $E' \subseteq E$, and for the corresponding Galois groups this implies that $\mathrm{Gal}(E/K)$ is the limit of the finite groups $\mathrm{Gal}(E'/K)$, thus it is a profinite group.
4. If $R$ is a ring, the colimit of the groups $\mathrm{GL}_n(R)$, where the transition maps are $A \mapsto \mathrm{diag}(A,1)$, equals $\mathrm{GL}(R)$, the group of infinite matrices which are the identity up to finitely many entries. This group is important in K-theory.
5. One can show that limits in topological spaces are special cases of limits in the sense of category theory. See MO/9951.

But often one also wants to take limits of sequences (or nets, or filters) in your favorite algebraic object. This is possible for topological algebraic structures. The most important examples are topological groups, topological rings, Banach algebras, and C*-algebras. For example, we have $p^n \to 0$ in $\mathbb{Z}_p$, and $x^n \to 0$ in $\mathbb{Z}[[x]]$. For more information, see topological algebra and the references given there.

Every set can be equipped with the discrete topology, which means that a sequence converges iff it becomes eventually constant. This is the usual topology one puts on finite groups.

Answer 2

First of all we need a concept of arbitrarily near points, in terms of neighborhood of a point. So we need to define a topology on it.

If you want the uniqueness of the limit (If exists (just think about $\lbrace \frac{1}{n} \rbrace_n$ in $(0,1)$ you need to verify that the Hausdorff property of separation holds for your space.

Answer 3

It is worthy to note that we actually can take limits in $\mathbb Z / 7 \mathbb Z$ (using the discrete topology).

When viewed this way, for a sequence of numbers $\{ s_n \}$, $\lim_{n \to \infty} s_n = s$ if $s_n = s$ for all but finitely many $n$.

This satisfies a notion of limit, but the standard notion of "limit" on $\mathbb R$ has much nicer and much more interesting properties, of course.