# 3.3. 下降

The idea of descents should be that, they are like sheaf axiom for fibered categories.

In this case, the naive sheaf axioms fit into the picture, i.e. we getand this diagram is not exact.

We are missing the “cocycle” condition (to make the above diagram exact/equalizer). This means that, on , we get diagram

In other word, the right “exact” diagram we need will be something like

Therefore, we want to formalize this triple arrow thing in fibered categories.

Let be a fibered category. Given in , let be the category defined as follows (this is called category of descent data). The object should be with andwhere the triple arrows are and the double arrows are , and is an isomorphism in such that we get the following commutative diagramwhere means canonical isomorphism. This is called the cocycle condition.

First, recall that are defined as the pullback of the following diagramSimilarly, are defined as the pullback of the following diagram

Second, we note are projections come from the universal property of (fibered) products. In other word, note we would define as the unique object satisfies the following diagram

Finally, a word on the isomorphisms . Continue with the above diagram (where now we let ), we get the followingwhere the two pullbacks along and must be the same object living over , hence the canonical isomorphisms between . The others are similar.

The idea is that, if we have , then stack would be fibered in groupoids where is an equivalence1.

Last time, we let be a fibered category (not necessarily fibered in groupoids). Given in , we defined a category be the category of descent data.

The objects of this category is where and is an isomorhism between . Here we have a diagramand hence the two pullbacks of should be isomorphic, and we just choose a particular . However, this cannot be arbitrary, as we need one additional condition called the cocycle condition.

The cocycle condition says that, all the ways to pullback should all be commuting (we have three arrows from to , and two arrows from to , then we want ).

The point is that, we then get byThis is because, and are pullbacks of along . For example, we get by the following diagram (where is equal both and at the same time)

Therefore we get a canonical map .

In this case, we say is an effective descent morphism if is equivalence of categories. In this case we also say satisfies descent for .

Before we say explain what this means, we recall the morphisms of is the following. From to , a morphism is in such that we get the following commuting diagram

More generally, we can define as such that and is an isomorhism, whereWe also need the cocycle condition .

Normally, we do not need to think about this more general case because of the following lemma.

Note this is coproduct, not co-fiber product. Co-fiber product may not exists in schemes. On the other hand, coproducts exists in schemes, as they are just disjoint union of schemes.

Thus, the point is that, we can always think as , which means we back to the first case.