Gluing, descent, sheaves and more

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

Gluing, descent, sheaves and more

Gluing functions

Gluing vector bundles

Gluing categories of vector bundles

Sheaves are functors that satisfy descent

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