The Formalism of l-adic Sheaves

The ℓ-adic product formula discussed in Chapter 4 will need to make use of analogous structures, which are simply not visible at the level of the triangulated category Dℓ(X). This chapter attempts to remedy the situation by introducing a mathematical object Shvℓ (X), which refines the triangulated category Dℓ (X). This object is not itself a category but instead is an example of an ∞-category, which is referred to as the ∞-category of ℓ-adic sheaves on X. The triangulated category Dℓ (X) can be identified with the homotopy category of Shvℓ (X); in particular, the objects of Dℓ (X) and Shvℓ (X) are the same. However, there is a large difference between commutative algebra objects of Dℓ (X) and commutative algebra objects of the ∞-category Shvℓ (X). We can achieve (b') by viewing the complex B as a commutative algebra of the latter sort.