What is the inverse limit?

Question

In his book of Differential Topology, Hirsch starts a little detour in his theory in order to present a way to see things in a general perspective. The precise fragment I'm referring to is the following:

I have never encountered a inverse limit before, and I don't have any intuition about what it is. I've searched the internet and it didn't help me either (for instance, the Wikipedia Page on Inverse Limit). He then proceeds to give an example:

It is not clear at all to me why it is continuous. My question is three-fold:

1. Is there a (less general) definition of inverse limit adequate to this setting, or is the general definition adequate and acessible enough?
2. Is there an intuition?
3. Why is the structure functor above obviously continuous?

Answer 1

The first cases of inverse limits I saw were of the form:

$$X_1\mathop{\leftarrow}_{f_1} X_2\mathop{\leftarrow}_{f_2} X_3\mathop{\leftarrow}_{f_3}\cdots$$

Then the inverse limit of this sequence of sets is the set of tuples:

$$\left\{(x_1,x_2,\cdots,x_n,\cdots)\in X_1\times X_2\cdots\mid \forall i( f_i(x_{i+1})=x_i)\right\}$$

An example of this, from the categories of rings, is the sequence:

$$\mathbb Z_p\leftarrow \mathbb Z_{p^2}\leftarrow \mathbb Z_{p^3}\cdots$$

Where $$f_i: \mathbb Z_{p^{i+1}}\rightarrow \mathbb Z_{p^i}$$ is the natural map.

The inverse limit here is the $p$-adic integers.

An even more basic example is if $X_{i+1}\subseteq X_i$ with $f_i$ the usual inclusion. Then the inverse limit is the same as $\bigcap X_i$.

Answer 2

Perhaps this will help.

Basically an inverse limit is like this.

Fix a category (for example the category of sets, groups, rings, topological spaces, vector spaces). It's easy to define the inverse limit if your objects have an underlying "set", upon which possibly additional structure is imposed. (ie, groups are a set together with a group operation, topological spaces are a set together with a topology...etc)

Suppose you have a bunch of objects $X_i$ (possibly infinitely many) of that category. Suppose you have a bunch of morphisms $\phi_{ij} : X_i\rightarrow X_j$.

Then the inverse limit $\lim X_i$ over the objects $X_i$ and morphisms $\phi_{ij}$ is by definition the subset of the product $\prod_i X_i$ consisting of those tuples $(\ldots,x_i,\ldots,x_j,\ldots)$ such that for each morphism $\phi_{ij}$, $\phi_{ij}(x_i) = x_j$.

As a subset of the product, it comes with natural projection maps $pr_i : \lim X_i\rightarrow X_i$.

Then on this set there is usually a canonical "minimal" structure (group structure, topology,...etc) which makes the inverse limit an object in your category such that all the projection maps are morphisms in your category (ie, group/ring homomorphisms, or continuous maps...etc).

In other words, an inverse limit is a subset of the product consisting of tuples where the coordinates are all "compatible" with the morphisms $\phi_{ij}$. Note that if you take the inverse limit over a system $(\{X_i\}_i, \{\phi_{ij}\}_{ij})$, where the set of morphisms $\phi_{ij}$ is empty, then the inverse limit is precisely the product $\prod_i X_i$.

The $p$-adic numbers that Thomas Andrews describes is a good example. In that case, you're taking an inverse limit of infinitely many finite rings of the form $\mathbb{Z}/p^n\mathbb{Z}$ (each having positive characteristic), and ending up with an infinite ring of characteristic 0.

Another example is the classical cantor set. This can be seen as the 3-adic integers, where each $\mathbb{Z}/3^n\mathbb{Z}$ is seen as a finite set equipped with the discrete topology. Then the topology on the inverse limit is defined to be the smallest/weakest topology such that all the projection maps are continuous (it's a good exercise to work through the definitions to prove to yourself that this is homeomorphic to the usual definition of the cantor set)