Using the fppf topology

First of all, the fppf-topology is used because it behaves well with respect to descent. For example, if there are schemes ${\displaystyle X,Y\in \operatorname {Ob} (\mathrm {Sch} /S)}$ and ${\displaystyle X\to Y}$can be refined to an fppf-cover of ${\displaystyle Y}$, if ${\displaystyle X}$ is flat, locally finite type, or locally of finite presentation, then ${\displaystyle Y}$ has this property.^[8] this kind of idea can be extended further by considering properties local either on the target or the source of a morphism ${\displaystyle f:X\to Y}$. For a cover ${\displaystyle \{X_{i}\to X\}_{i\in I}}$ we say a property ${\displaystyle {\mathcal {P}}}$ is local on the source if

${\displaystyle f:X\to Y}$ has ${\displaystyle {\mathcal {P}}}$ if and only if each ${\displaystyle X_{i}\to Y}$ has ${\displaystyle {\mathcal {P}}}$.

There is an analogous notion on the target called local on the target. This means given a cover ${\displaystyle \{Y_{i}\to Y\}_{i\in I}}$

${\displaystyle f:X\to Y}$ has ${\displaystyle {\mathcal {P}}}$ if and only if each ${\displaystyle X\times _{Y}Y_{i}\to Y_{i}}$ has ${\displaystyle {\mathcal {P}}}$.

For the fppf topology, having an immersion is local on the target.^[9] In addition to the previous properties local on the source for the fppf topology, ${\displaystyle f}$ being universally open is also local on the source.^[10] Also, being locally Noetherian and Jacobson are local on the source and target for the fppf topology.^[11] 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 ${\displaystyle {\mathcal {M}}_{fg}}$ is an fpqc-algebraic stack^{[12]pg 40}.

Representable diagonal

By definition, a 1-morphism ${\displaystyle f:{\mathcal {X}}\to {\mathcal {Y}}}$ of categories fibered in groupoids is representable by algebraic spaces^[13] if for any fppf morphism ${\displaystyle U\to S}$ of schemes and any 1-morphism ${\displaystyle y:(Sch/U)_{fppf}\to {\mathcal {Y}}}$, the associated category fibered in groupoids

${\displaystyle (Sch/U)_{fppf}\times _{\mathcal {Y}}{\mathcal {X}}}$

is representable as an algebraic space,^[14][15] meaning there exists an algebraic space

${\displaystyle F:(Sch/S)_{fppf}^{op}\to Sets}$

such that the associated fibered category ${\displaystyle {\mathcal {S}}_{F}\to (Sch/S)_{fppf}}$^[16] is equivalent to ${\displaystyle (Sch/U)_{fppf}\times _{\mathcal {Y}}{\mathcal {X}}}$. There are a number of equivalent conditions for representability of the diagonal^[17] which help give intuition for this technical condition, but one of main motivations is the following: for a scheme ${\displaystyle U}$ and objects ${\displaystyle x,y\in \operatorname {Ob} ({\mathcal {X}}_{U})}$ the sheaf ${\displaystyle \operatorname {Isom} (x,y)}$ is representable as an algebraic space. In particular, the stabilizer group for any point on the stack ${\displaystyle x:\operatorname {Spec} (k)\to {\mathcal {X}}_{\operatorname {Spec} (k)}}$ 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 products

${\displaystyle {\begin{matrix}Y\times _{\mathcal {X}}Z&\to &Y\\\downarrow &&\downarrow \\Z&\to &{\mathcal {X}}\end{matrix}}}$

the representability of the diagonal is equivalent to ${\displaystyle Y\to {\mathcal {X}}}$ being representable for an algebraic space ${\displaystyle Y}$. This is because given morphisms ${\displaystyle Y\to {\mathcal {X}},Z\to {\mathcal {X}}}$ from algebraic spaces, they extend to maps ${\displaystyle {\mathcal {X}}\times {\mathcal {X}}}$ from the diagonal map. There is an analogous statement for algebraic spaces which gives representability of a sheaf on ${\displaystyle (F/S)_{fppf}}$ as an algebraic space.^[18]

Note that an analogous condition of representability of the diagonal holds for some formulations of higher stacks^[19] where the fiber product is an ${\displaystyle (n-1)}$-stack for an ${\displaystyle n}$-stack ${\displaystyle {\mathcal {X}}}$.

2-Yoneda lemma

The existence of an ${\displaystyle fppf}$ scheme ${\displaystyle U\to S}$ and a 1-morphism of fibered categories ${\displaystyle {\mathcal {U}}\to {\mathcal {X}}}$ which is surjective and smooth depends on defining a smooth and surjective morphisms of fibered categories. Here ${\displaystyle {\mathcal {U}}}$ is the algebraic stack from the representable functor ${\displaystyle h_{U}}$ on ${\displaystyle h_{U}:(Sch/S)_{fppf}^{op}\to Sets}$ upgraded to a category fibered in groupoids where the categories only have trivial morphisms. This means the set

${\displaystyle h_{U}(T)={\text{Hom}}_{(Sch/S)_{fppf}}(T,U)}$

is considered as a category, denoted ${\displaystyle h_{\mathcal {U}}(T)}$, with objects in ${\displaystyle h_{U}(T)}$ as ${\displaystyle fppf}$ morphisms

${\displaystyle f:T\to U}$

and morphisms are the identity morphism. Hence

${\displaystyle h_{\mathcal {U}}:(Sch/S)_{fppf}^{op}\to Groupoids}$

is a 2-functor of groupoids. Showing this 2-functor is a sheaf is the content of the 2-Yoneda lemma. Using the Grothendieck construction, there is an associated category fibered in groupoids denoted ${\displaystyle {\mathcal {U}}\to {\mathcal {X}}}$.

Representable morphisms of categories fibered in groupoids

To say this morphism ${\displaystyle {\mathcal {U}}\to {\mathcal {X}}}$ is smooth or surjective, we have to introduce representable morphisms.^[20] A morphism ${\displaystyle p:{\mathcal {X}}\to {\mathcal {Y}}}$ of categories fibered in groupoids over ${\displaystyle (Sch/S)_{fppf}}$ is said to be representable if given an object ${\displaystyle T\to S}$ in ${\displaystyle (Sch/S)_{fppf}}$ and an object ${\displaystyle t\in {\text{Ob}}({\mathcal {Y}}_{T})}$ the 2-fibered product

${\displaystyle (Sch/T)_{fppf}\times _{t,{\mathcal {Y}}}{\mathcal {X}}_{T}}$

is representable by a scheme. Then, we can say the morphism of categories fibered in groupoids ${\displaystyle p}$ is smooth and surjective if the associated morphism

${\displaystyle (Sch/T)_{fppf}\times _{t,{\mathcal {Y}}}{\mathcal {X}}_{T}\to (Sch/T)_{fppf}}$

of schemes is smooth and surjective.