July | 2020 | Arithmetic variety

Gluing functions

Suppose $\{U_i\}$ is a cover of a topological space $X$ . What does it take to glue continuous functions $f_i$ on $U_i$ into a function on all of $X$ ?

Well, all we need is that they need to agree on intersections. If $f_i|U_i \cap U_j = f_j|U_i\cap U_j$ for all pairs $i, j$ , then they glue uniquely to a function $g$ on $X$ such that $g|U_i = f_i$ .

Let’s reformulate this slightly. Let $U$ be the disjoint union of the $U_i$ s. Then we have a surjective “cover” $U \to X$ . Combine the $f_i$ s into a function on $U$ called $f$ . Then what we are trying to do is see if the function $f$ on $U$ can be “descended” to $X$ i.e. we are trying to find functions $g$ on $X$ , if any, such that when such a function is pulled back to $U$ it is $f$ . Note here that by “is” we mean the functions are literally identical. We can also reformulate what is means for the $f_i$ s to agree on intersections. The fiber product of $U$ with itself over $X$ has two maps, left and right projection, down to $U$ :

Call them $p_1$ and $p_2$ . Then for the $f_i$ to “agree on intersections” is equivalent to the statement $f \circ p_1 = f \circ p_2$ as functions on $U \times_{X} U$ (think about this!). We’ll return to this later. Now, let’s step up one level, from functions to vector bundles.

Gluing vector bundles

Suppose $\{U_i\}$ is a cover of a topological space $X$ . What does it take to glue vector bundles (or sheaves) $V_i \to U_i$ to a vector bundle $V \to X$ ? We also know the answer to this: transition maps satisfying the cocyle condition. More precisely, we need for each pair $i, j$ an isomorphism of vector bundles $\phi_{i,j}:V_i \to V_j$ restricted to $U_i \cap U_j$ , such that for every triple $i, j, k$ , $\phi_{j,k} \circ \phi_{i,j} = \phi_{i,k}$ on the triple intersection $U_i \cap U_j \cap U_k$ .

Instead of just needing the “local objects” to agree on intersections, we need data specifying how they agree; furthermore, we need the data to be mutually compatible

Let’s try to reformulate this in terms of the disjoint union of the $U_i$ s again. Let’s call $U$ the disjoint union of the $U_i$ and combine the $V_i$ together as a single vector bundle $V \to U$ . What we’re trying to do is “descend” this vector bundle over $U$ to $X$ .

Since we have triple intersections, we’ll need to consider the fiber product $U \times_{X} U \times_{X} U$ . It has three projections down to $U \times_{X} U$ (by omitting any of the three factors).

Let’s call these projections $p_{12}, p_{23}, p_{31}$ .

Now we are ready to state the answer to the question of when does the vector bundle descend. The isomorphism on intersections is given by an isomorphism $\phi: p_1^*V \to p_2^*V$ of vector bundles, and the cocycle condition translates to $p_{12}^*(\phi) \circ p_{23}^*(\phi) = p_{31}^*(\phi)$

Should we go up one more level?

Gluing categories of vector bundles

No, I’m not ready yet.

Sheaves are functors that satisfy descent

Let’s fix a base category. Consider the functor $F$ which assigns to every base object the set of (insert adjective) functions on it. Then such a functor (presheaf) is a sheaf if, for every function on $U$ which, along the two projections $p_1, p_2: U \times U \to U$ , is pulled to identical functions, is the unique pullback of a function on $X$ . In other words $F(X)$ is the equalizer $F(U) \rightrightarrows F(U \times U)$ .

Each element of $F(X)$ is: an element of $F(U)$ , and a truth value on $F(U \times U)$ (the truth value is whether the “functions” agree).

Let’s move one level up. Consider the functor $F$ which assigns to every base object the category of sheaves over it. Then such a functor is a 2-sheaf if for every sheaf on $U$ , there is an isomorphism between the pullbacks along the two projections $U \times U \to U$ , and these isomorphisms, pulled back to $U \times U \times U$ (along the three projections $p_{12}, p_{23}, p_{13}: U \times U \times U \to U \times U$ ), satisfy a cocyle condition. If such a thing made sense, we might say $F(X)$ is equivalent, as a category, to the following “2-limit of categories”:

Each object of $F(X)$ is: an object of $F(U)$ , isomorphism of $F(U\times U)$ , and truth value on $F(U\times U\times U)$ . Each morphism of $F(X)$ is: morphism of $F(U)$ , truth value on $F(U\times U)$ . (morphism of descent datum)

Consider the functor $F$ which assigns to every base object the 2-category of sheaves of 1-categories over it.

Each object of $F(X)$ is: an object of $F(U)$ , isomorphism of $F(U\times U)$ , 2-isomorphism of $F(U\times U\times U)$ , and a truth value on $F(U\times U\times U\times U)$ . A morphism of $F(X)$ is a morphism of $F(U)$ , a 2-morphism of $F(U\times U)$ , and a truth value in $F(U\times U\times U)$ . A 2-morphism of $F(X)$ is a 2-morphism in $F(U\times U)$ and a truth value on $F(U \times U \times U)$ .

And so on…

Acknowledgements: Thanks to David Corwin for explaining these ideas to me.

Arithmetic variety

Monthly Archives: July 2020

Gluing, descent, sheaves and more

Gluing functions

Gluing vector bundles

Gluing categories of vector bundles

Sheaves are functors that satisfy descent