§ Direct and Inverse limits
§ Direct limit: definition
A direct limit consists of injections .
It leads to a limit object , which as a set is equal to the union of all
the . 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 has an equivalence class representative in .
§ Direct limit: prototypical example
We can inject the symmetric groups .
However, we cannot project back some permutation of (say) to :
if I have (swap 2 and 1), then I can't project this back into .
This is prototypical; in general, we will only have injections into the limit,
not projections out of the limit.
§ Prufer group
Given a topological space and functions to the reals
on open sets , we define the restricted
function spaces .
Given two open sets , we can restrict functions on
(a larger set) to functions on (a smaller set). So we get maps
So given a function on a larger set , we can inject into a smaller set .
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 /
Given an alphabet set , we can construct a finite limit of strings of length
, strings of length , and so on for strings of any given length .
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 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 .
It leads to a limit object , which as a set is equal to a subset of the
product of all the , where we only allow elements that "agree downwards"
.Formally, we write this as:
So from each element in , we get the projection maps that give us the component .
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
xxxc*, and so on? There's no canonical choice!
Hence, we only have projections, but no injections.
Consider the 7-adics written as infinite strings of digits in .
Formally, we start by:
The point is that we define the -adics by projecting back solutions
from . It's impossible to correctly embed
into : The naive map
that sends the "digit i" to the "digit i" fails, because:
- Having solutions to some equation in
- Finding a solution in that restricts to the same solution in
- Keep going.
So . Hece, we
don't have injections, we only have projections.
- in we have that .
- in .
Let be some infinite set. Let be a sequence of partitions
such that is finer than . That is, every element of
is the union of some elements of . 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
The inverse limit is:
But we only care about "adjacent consistency", since that generates the other
consistency conditions; So we are left with:
But unravelling the definition of , we get:
So the inverse limit is the "path" in the "tree of partitions".
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).