The motivation for this post is to go through some details necessary in the formal definition of Cech Cohomology.  In that definition, we come across the notion of the direct limit of a directed system.  This post will serve to:

1. Explicitly write out some of the details needed to see why the direct limit of a directed system of groups is, in fact, a group.
2. State and prove an important proposition which shows why good covers are relevant to computing Cech Cohomology.
3. Offer some exercises for anyone reading which pertain to related facts we know are true but which have not been written out in this post.

Here we go!

Definition: A directed system of groups, $\mathcal{A}$, consists of the following data:

• A directed set, $I$, which is to say a set with a reflexive and transitive binary relation, $\le$, so that each pair of elements has a lower bound.
• A collection of groups, $\{ A_i \}_{i \in I}$, indexed over this set.
• A collection of group homomorphisms, $f_{ij}: A_j \to A_i$, which satisfy the cocycle conditions:
• $f_{ij} \circ f_{jk} = f_{ik}$ for all $i \le j \le k$.
• $f_{ii} = id_{A_i}$

Definition/Proposition: The direct limit of a directed system, $\lim(\mathcal{A}) = \lim\limits_{\substack{\rightarrow\\ i \in I}} A_i$, is defined to be the group whose set of elements is the disjoint union of all of the $A_i$‘s modulo a relation,

$\lim\limits_{\substack{\rightarrow\\ i \in I}} A_i = \coprod\limits_{i \in I} A_i \Big{/} \sim,$

where $(x_i \in A_i) \sim (y_j \in A_j)$ if there exists a lower bound $k$ for $\{ i,j\}$ so that $f_{ki}(x_i) = f_{kj}(y_j)$.

## Checking the Details: Direct Limit of Groups is a Group

The reason why this definition above is listed as both a definition and a proposition, is that it is not immediately obvious why this set is a group.  In general, the disjoint union of a collection of groups is not a group, but somehow the relation we place on the disjoint union makes everything work nicely.  I will now prove that (i) addition is well defined in this direct limit and (ii) that there is a well defined identity element.

### Claim (i): Addition is well defined

For $[x_i]$ and $[y_j]$ in $\lim\limits_{\rightarrow} A_i$, we define

$[x_i] + [y_j] := [ f_{ki}(x_i) + f_{ki}(y_j)]$

where $k \in I$ is some upper bound for $\{ i,j \}$.  Before we prove anything, let’s make sure we understand what is happening here.  Since $x_i \in A_i$ and $y_j \in A_j$, I can’t add these two elements since they live in two different groups, so I need to push them forward into a common group, add them, and then take the resulting equivalence class.  How do I know there is even a common group to push them both into?  This existence of a common group comes from the condition for our directed system which states that for $\{ i,j\}$ there exists a lower bound $k \in I$.

#### Part (a): Addition is independent of choice of lower bound

OK, but what if we chose a different lower bound?  Well suppose $l\in I$ was another lower bound and that we had $l \le k$.  We would like to show that

$[ f_{ki}(x_i) + f_{ki}(y_j)] = [ f_{li}(x_i) + f_{lj}(y_j)]$

Recall that two classes are equal if they are equal when we push them forward into a common group.  Since $( f_{ki}(x_i) + f_{ki}(y_j)) \in A_k$ and  $( f_{li}(x_i) + f_{li}(y_j)) \in A_l$, we choose a lower bound $p$ of $\{l, k\}$ and note that:

\begin{aligned} f_{pk}\left( f_{ki}(x_i) + f_{kj}(y_j)\right) &= (f_{pk} \circ f_{ki})(x_i) + (f_{pk} \circ f_{ki})(y_j) \\ &=f_{pi}(x_i) + f_{pj}(y_j) \\ &=(f_{pl} \circ f_{li})(x_i) + (f_{pl} \circ f_{lj})(y_j) \\ &= f_{pl} \left( f_{li}(x_i) + f_{lj}(y_j) \right) \end{aligned}

and so we have shown that the two classes we obtain from using a different lower bound in our definition of addition, are equal.`

#### Part (b): Addition is independent of choice of representative in each class

Now we consider the case where we want to add $[x_i] + [y_j]$, where $l$ is a lower bound for $\{i,j\}$, but would like to show that if $[x_i] = [z_k]$ then $[x_i] + [y_j] = [z_k] + [y_j]$.  As before, we use the fact that two equivalence classes are equal in the direct limit if they are equal in some group that they can be pushed forward into.  So, from $[x_i] = [z_k]$, there must exist $p$, a lower bound of $\{i, k\}$ where $f_{pi}(x_i) = f_{pk}(z_k)$.  Finally, in order to make the addition work, let us choose a lower bound $q$ for $\{p,j\}$, which consequently is a lower bound for $\{ i,j\}$, and then we can demonstrate the desired equality:

\begin{aligned} \ [ x_i ] + [y_j] &= [f_{qi}(x_i) + f_{qj}(y_j)] \\ &= [ (f_{qp} \circ f_{pi})(x_i) + f_{qj}(y_j)] \\ &= [ (f_{qp} \circ f_{pk})(z_k) + f_{qj}(y_j)] \\ &= [ (f_{qk})(z_k) + f_{qj}(y_j)] = [z_k] + [y_j] \end{aligned}

### Claim (ii): There is a well-defined identity element

Since each group, $A_i$ has it’s own identity, $0_i$, and we are initially taking the disjoint union of all of these groups, it isn’t immediately clear that there is a single identity.  However, we will now show that, after quotienting, there is one equivalence class which holds all of the identity elements and it serves as an identity for our addition as defined above.

#### Part (a): $[0_i] = [0_j]$ for any $i,j \in I$.

Let $k$ be a lower bound for $\{ i,j\}$ and note that

$f_{ki}(0_i) = 0_k = f_{kj}(0_j),$

and this is all that is needed to show the two classes $[0_i]$ and $[0_j]$ are equal.  Thus, all identity elements live in the same class.

#### Part (b): $[0_i] +[y_j] = [y_j]$ for any $i \in I$ and $y_j \in A_j$.

Let $k$ be a lower bound for $\{ i,j\}$ and observe

$[ 0_i ]+ [y_j] = [f_{ki}(0_i) + f_{kj}(y_j)] = [ 0_k + f_{qj}(y_j)] = [ f_{qj}(y_j)] = [ y_j ]$

While there are a few other details to check when you want to show that you have a group, I think that at this point it is clear how to you how all of those details would flow.  Let’s then consider ourselves satisfied with the fact that the direct limit of a group is a group.

## In Practice: Stable Elements in the Direct Limit

In real life, when someone actually wants to work with an object defined as a direct limit, $\lim\limits_{\rightarrow} A_i$, they usually use a stable representative, $G$, which is isomorphic to the limiting group: $G \simeq \lim\limits_{\rightarrow} A_i$.  We know state and prove a proposition which is useful for this purpose of representing the limit.

Proposition: If $B = A_{i_B}$ is a group in the direct system of groups, $\mathcal{A}$, such that for all $j \le i_B$, $f_{j i_B} : A_{i_B} \to A_j$ is an isomorphism, then $\lim(\mathcal{A}) \simeq B$.  We will refer to the group $B$ as a stable group in the limit.

To prove this theorem, we recall that $B$ must be equal to some $A_{i_B}$ with $i_B \in I$, and we use the canonical homomorphism, $A_{i_B} \xrightarrow{\pi} lim(\mathcal{A})$ which takes each element $b \in A_{i_B}$ and maps it to its equivalence class $[b] \in \lim(\mathcal{A})$.

Exercise: Show that this is in fact a well-defined homomorphism.

We will show here that it is injective and surjective.  For the sake of the proof, let us assume   To prove injectivity, suppose that

$\pi(b) = [b] = [b'] = \pi(b').$

Then there exists an index, $j \le i_B$, such that $f_{j i_B} (b) = f_{j i_B} (b')$.  However since we are assuming in this proposition that $f_{j i_B}$  is an isomorphism, then we have $b = b'$ and so $pi$ is injective.

To show that $pi$ is surjective, consider an element $[x_i] \in \lim(\mathcal{A})$.  Let $k$ be a lower bound for $\{i, i_B\}$ and consider the other representative $[f_{k i}(x_i)]$.  Since $f_{k i_B}: A_{i_B} \to A_k$ is surjective, and $f_{k i}(x_i) \in A_k$, then there exists an element $b \in A_{i_B}$ such that $f_{k i_B} (b) = f_{ki}(x_i)$.  But this is exactly what it means for $[b] = [x_i]$!  Thus, $\pi$ is surjective and the isomorphism has been proved.

## Who cares about Direct Limits?

To end this post, I will end with two exercises (in the form of stories with many details to check) that I hope some of my students will eventually make progress on.

### The Cech Cohomology of a Good Cover

Given a topological space, $X$, and an open cover, $\mathcal{V}$, we can compute the Cech Cohomology, $H^{\bullet}(\mathcal{V}, \mathbb{R})$, with values in the sheaf, $\mathbb{R}$, of locally constant $\mathbb{R}$-valued functions.  It turns out that if an open cover, $\mathcal{U}$, is good, then it satisfies that for all refinements $\mathcal{V}$, the associated cohomology groups are isomorphic.  Since these Cech Cohomology groups form a directed system over the directed set of open covers on a fixed topological space, it then follows from the above proposition that

$H^p(X, \mathbb{R}):= \lim\limits_{\rightarrow}\left( H^p(\mathcal{V}, \mathbb{R} ) \right) \simeq H^p(\mathcal{U}, \mathbb{R}) .$

Note: I currently have no idea how to prove, without changing the sheaf or cohomology theory we are working with, that good covers have this “stable” property in the direct limit.    I have been, and will continue, to think about this and provide an update when I make progress.

### The Direct Limit as a Categorical coLimit

A directed system, $\mathcal{A}$, can be thought of as a (covariant) functor, $F: I \to Gr$, to the category of Groups, where $I$ is thought of as a small functor.  In this setting, the direct limit is equal to the colimit of this functor, $F$.