can be extended further by considering properties local either on the target or the source of a morphism . For a cover we say a property is local on the source if has if and only if each has .There is an analogous notion on the target called local on the target. This means given a cover has if and only if each has .For the fppf topology, having an immersion is local on the target. In addition to the previous properties local on the source for the fppf topology, being universally open is also local on the source. Also, being locally Noetherian and Jacobson are local on the source and target for the fppf topology. This does not hold in the fpqc topology, making it not as "nice" in terms of technical properties. Even though this is true, using algebraic stacks over the fpqc topology still has its use, such as in chromatic homotopy theory. This is because the Moduli stack of formal group laws is an fpqc-algebraic stackpg 40. Representable diagonal By definition, a 1-morphism of categories fibered in groupoids is representable by algebraic spaces if for any fppf morphism of schemes and any 1-morphism , the associated category fibered in groupoidsis representable as an algebraic space, meaning there exists an algebraic spacesuch that the associated fibered category is equivalent to . There are a number of equivalent conditions for representability of the diagonal which help give intuition for this technical condition, but one of main motivations is the following: for a scheme and objects the sheaf is representable as an algebraic space. In particular, the stabilizer group for any point on the stack is representable as an algebraic space. Another important equivalence of having a representable diagonal is the technical condition that the intersection of any two algebraic spaces in an algebraic stack is an algebraic space. Reformulated using fiber productsthe representability of the diagonal is equivalent to being representable for an algebraic space . This is because given morphisms from algebraic spaces, they extend to maps from the diagonal map. There is an analogous statement for algebraic spaces which gives representability of a sheaf on as an algebraic space. Note that an analogous condition of representability of the diagonal holds for some formulations of higher stacks where the fiber product is an -stack for an -stack . Surjective and smooth atlas 2-Yoneda lemma The existence of