§ Direct and Inverse limits

§ Direct limit: definition

A direct limit consists of injections A1A2A_1 \rightarrow A_2 \rightarrow \dots. It leads to a limit object LL, which as a set is equal to the union of all the AiA_i. It is equipped with an equivalence relation. We can push data "towards" the limit object, hence it's a "direct" limit. So each element in AiA_i has an equivalence class representative in LL.

§ Direct limit: prototypical example

§ SnS_n

We can inject the symmetric groups S1S2S_1 \rightarrow S_2 \rightarrow \dots. However, we cannot project back some permutation of S2S_2 (say) to S1S_1: if I have (2,1)(2, 1) (swap 2 and 1), then I can't project this back into S1S_1. This is prototypical; in general, we will only have injections into the limit, not projections out of the limit.

§ Prufer group

§ Stalks

Given a topological space (X,T)(X, T) and functions to the reals on open sets F{UR}F \equiv \{ U \rightarrow \R \}, we define the restricted function spaces FU{FU:UR:fF}F|_U \equiv \{ F_U : U \rightarrow \mathbb R : f \in F \}. Given two open sets UWU \subseteq W, we can restrict functions on WW (a larger set) to functions on UU (a smaller set). So we get maps FWFUF|_W \rightarrow F|_U. So given a function on a larger set WW, we can inject into a smaller set UU. But given a function on a smaller set, it's impossible to uniquely extend the function back into a larger set. These maps really are "one way". The reason it's a union of all functions is because we want to "identify" equivalent functions. We don't want to "take the product" of all germs of functions; We want to "take the union under equivalence".

§ Finite strings / A*

Given an alphabet set AA, we can construct a finite limit of strings of length 00, strings of length 11, and so on for strings of any given length nNn \in \mathbb N. Here, the "problem" is that we can also find projection maps that allow us to "chop off" a given string, which makes this example not-so-great.

§ Categorically

Categorically speaking, this is like some sort of union / sum (coproduct). This, cateogrically speaking, a direct limit is a colimit.

§ Inverse limit: definition

An inverse limit consists of projections A1A2A_1 \leftarrow A_2 \leftarrow \dots. It leads to a limit object LL, which as a set is equal to a subset of the product of all the AiA_i, where we only allow elements that "agree downwards" .Formally, we write this as:
L{a[:]iAi:proj(αω)(a[ω])=a[α] αω} L \equiv \{ a[:] \in \prod_i A_i : \texttt{proj}(\alpha \leftarrow \omega)(a[\omega]) = a[\alpha] ~ \forall \alpha \leq \omega \}
So from each element in LL, we get the projection maps that give us the component a[α]a[\alpha].
These 'feel like' cauchy sequences, where we are refining information at each step to get to the final object.

§ Inverse limit: prototypical example

§ infinite strings

We can consider the set of infinite strings. Given an infinite string, we can always find a finite prefix as a projection. However, it is impossible to canonically inject a finite prefix of a string into an infinite string! Given the finite string xxx, how do we make it into an infinite string? do we choose xxxa*, xxxb*, xxxc*, and so on? There's no canonical choice! Hence, we only have projections, but no injections.

§ P-adics

Consider the 7-adics written as infinite strings of digits in {0,1,,6}\{0, 1, \dots, 6\}. Formally, we start by:
  1. Having solutions to some equation in Z/7Z\mathbb{Z}/7\mathbb{Z}
  2. Finding a solution in Z/49Z\mathbb{Z}/49\mathbb{Z} that restricts to the same solution inZ/7Z\mathbb{Z}/7\mathbb{Z}
  3. Keep going.
The point is that we define the 77-adics by projecting back solutions from Z/49Z\mathbb{Z}/49\mathbb{Z}. It's impossible to correctly embed Z/7Z\mathbb{Z}/7\mathbb{Z} into Z/49Z\mathbb{Z}/49\mathbb{Z}: The naive map that sends the "digit i" to the "digit i" fails, because:
  • in Z/7Z\mathbb{Z}/7\mathbb{Z} we have that 2×412 \times 4 \equiv 1.
  • in Z/49Z\mathbb{Z}/49\mathbb{Z} 2×482 \times 4 \equiv 8.
So ϕ(2)×ϕ(7)ϕ(2×7)=ϕ(4)\phi(2) \times \phi(7) \neq \phi(2 \times 7) = \phi(4). Hece, we don't have injections, we only have projections.

§ Partitions

Let SS be some infinite set. Let {Πn}\{ \Pi_n \} be a sequence of partitions such that Πn+1\Pi_{n+1} is finer than Πn\Pi_n. That is, every element of Πn\Pi_n is the union of some elements of Πn+1\Pi_{n+1}. Now, given a finer partition, we can clearly "coarsen" it as desired, by mapping a cell in the "finer space" to the cell containing it in the "coarser space". The reverse has no canonical way of being performed; Once again, we only have projections, we have no injections. The inverse limit is:
{(P0,P1,P2,)i=0nΠn:Pa=projaz(Pz)az}. \{ (P_0, P_1, P_2, \dots) \in \prod_{i=0}^n \Pi_n : P_a = \texttt{proj}_{a \leftarrow z}(P_z) \forall a \leq z \}.
But we only care about "adjacent consistency", since that generates the other consistency conditions; So we are left with:
{(P0,P1,P2,)i=0nΠn:Pa=projab(Pb)a+1=b}. \{ (P_0, P_1, P_2, \dots) \in \prod_{i=0}^n \Pi_n : P_a = \texttt{proj}_{a \leftarrow b}(P_b) \forall a +1 = b \}.
But unravelling the definition of proj\texttt{proj}, we get:
{(P0,P1,P2,)i=0nΠn:Pa(Pb)a+1=b}. \{ (P_0, P_1, P_2, \dots) \in \prod_{i=0}^n \Pi_n : P_a \supseteq (P_b) \forall a +1 = b \}.
So the inverse limit is the "path" in the "tree of partitions".

§ Categorically

Categorically speaking, this is like some sort of product along with equating elements. This, cateogrically speaking, a inverse limit is a limit (recall that categorical limits exist iff products and equalizers exist).

§ Differences