The following selection is from Serre's FAC (Chapter 1, §3, n°22, page 26).
The relation `$\mathfrak{U}$ is finer than $\mathfrak{V}$' (which we denote hencforth by $\mathfrak{U} \prec \mathfrak{V}$) is a relation of a preorder$^{3}$ between coverings of $X$; moreover, this relation is filtered$^{4}$, since if $\mathfrak{U} = {U_i}_{i \in I}$ and $\mathfrak{V} = {V_j}_{j \in J}$ are two coverings, $\mathfrak{W} = {U_i \cap V_j}_{(i,j) \in I \times J}$ is a covering finer than $\mathfrak{U}$ and than $\mathfrak{V}$.
We say that two coverings $\mathfrak{U}$ and $\mathfrak{V}$ are equivalent if we have $\mathfrak{U} \prec \mathfrak{V}$ and $\mathfrak{V} \prec \mathfrak{U}$. Any covering $\mathfrak{U}$ is equivalent to a covering $\mathfrak{U'}$ whose set of indices is a subset of $\mathfrak{P}(X)$; in fact, we can take for $\mathfrak{U'}$ the set of open subsets of $X$ belonging to the family $\mathfrak{U}$. We can thus speak of the set of classes of coverings with respect to this equivalence relation; this is an ordered filtered set.$^{5}$
- $^{3}$i.e. quasiorder
- $^{4}$i.e. directed
- $^{5}$i.e. To the contrary, we cannot speak of the `set' of coverings, because a covering is a family whose set of indices is arbitrary.
Please let me explain my interpretation of what is taking place here, and ask some related questions along the way.
At this point we have defined cohomology for an arbitrary open cover, but the information heavily dependend on the open cover chosen. We want a way to make the cohomology intrinsic to the space $X$ - so now in these two paragraphs Serre is setting the stage to take a direct limit of cohomology groups indexed by open covers ordered by refinement. Some questions about that:
According to the wikipedia page, to take the direct limit we need the index set to be a directed preorder, but when I learned directed limits in Atiyah Macdonald (page 32) they require the index set be a directed partial order (thus also satisfy antisymmetry). What is the deal with this difference? After the equivalence relation Serre defines in paragraph 2 antisymmetry would hold (by definition), but is it required to take the limit in general?
I don't understand footnote 5, why do open covers not form a set? He seems to say it is because the index set of the open cover is arbitrary, but by definition of open cover we have $\mathfrak{U} = \{U_i\}_{i \in I}$. Re-index this by some other index set $\mathfrak{U'} = \{U_j\}_{j \in J}$ but since each $U_i, U_j$ is a subset of the space $X$, if we considered the collection of open covers of $X$ wouldn't these two open covers be the same since they are equal as sets? They contain precisely the same elements, the same subsets of $X$. Am I being naive?
Even once I understand why we have a set-theoretic issue, I am not sure I understand the solution. What is the solution Serre is proposing with the $\mathfrak{U'}$? -'Taking $\mathfrak{U'}$ the set of open subsets of $X$ belonging to the family $\mathfrak{U}$? Serre seems to do two things in paragraph 2, one is introduce the equivalence relation, and one is tell us how to choose a canonical index set? Can someone elaborate on what problems were presented and solved in paragraph 2?
Is the equivalence relation he puts forth to help deal with the set-theoretic issue, or just to whittle down the number of open covers we are dealing with, since as shown two open covers that are equal by this equivalence relation will give the same cohomology.
1) The concept of the direct limit of a direct system is defined in any category via a universal property (see the wikipedia article). So we must understand the concept of a direct system. You can define it for index sets which are directed preordered sets, or even more generally for small filtered categories. However, many authors restrict to directed partially ordered sets which suffices for most practical applications.
Let us work with directed preordered sets $(I,\le)$. We define a subset $I'' \subset I$ to be cofinal if each $i \in I$ admits $i'' \in I''$ such that $i \le i''$. Giving $I''$ the induced preorder, it is easy to verify that the direct limit of a direct system indexed by $I$ agrees with the direct limit of the cofinal subsystem indexed by $I''$.
Given a directed preordered set $(I,\le)$, we define $i \sim j$ iff $i \le j$ and $j \le i$. This is an equivalence relation and the set of equivalence classes $I'$ is a directed partially ordered set with $[i] \le [j]$ defined by $i \le j$. We may choose a subset $I'' \subset I$ containing precisely one representative from each equivelence class in $I'$. It inherits a preorder from $I$ and it is easy to see that $I''$ and $I'$ are isomorphic as preordered sets. Obviously $I''$ is a cofinal subset of $I$ which is a directed partially ordered set. This explains why many authors only consider the more special case of directed partially ordered sets.
2) The problem is that a cover is defined as an indexed family of sets, and the index set can be any set. But there is no set of all sets (it is a class), so there is no set of all covers of $X$.
For example, take any set $I$ and let $U_i = X$ for all $i$. This is not very interesting, but it is a cover.
3) The solution is that we may identify the family $\mathcal{U} = (U_i)_{i \in I}$ with the subset $s(\mathcal{U}) = \{ U_i \mid i \in I \} \subset \mathfrak{P}(X)$ which can be regarded again as a cover of $X$ (it is indexed by itself).
Doing so, we see that $s(\mathcal{U}) \prec \mathcal{U}$ and $\mathcal{U} \prec s(\mathcal{U})$, i.e. $s(\mathcal{U})$ and $\mathcal{U}$ are equivalent. This concept of equivalence resembles the definition in 1), the difference being that in 1) we started with a set, and here we start with a class. The effect is that we get a canonical set of representatives in the class of all covers.
4) In fact set theoretic issues are settled. But recall 1) - intuitively we may regard the set of coverings $s(\mathcal{U})$ as a "cofinal subsystem" of the class of all coverings. Note, however, that this set is not partially ordered which gives evidence that a more general concept of direct system is useful.