Limits and colimits in the category of schemes

Question

What is the smallest category enlarging the category of schemes over a field $k$ which is:

1. Complete? Cocomplete?

2. Admits a cogenerator? generator?

I admit there is some overlap with my previous questions. Sorry about that.

Answer 1

If $C$ is an arbitrary category, then the "smallest cocomplete category containing $C$" is usually called the free cocompletion $\widehat{C}$ of $C$ and is defined to satisfy the following universal property: If $D$ is a cocomplete category, then the category of functors $C \to D$ is naturally equivalent to the category of cocontinuous functors $\widehat{C} \to D$. It can be constructed as follows: First consider the cocomplete category of presheaves on $C$, i.e. functors $C^{op} \to \mathsf{Set}$. By Yoneda the full subcategory of representable functors is equivalent to $C$. Now, take $\widehat{C}$ to be the closure of this full subcategory under small colimits.

Of course this can be applied to the category of schemes $\mathsf{Sch}$, but I don't know if $\widehat{\mathsf{Sch}}$ is of much use in algebraic geometry. Instead, one looks at full subcategories of $\widehat{\mathsf{Sch}}$, for example of sheaves with respect to some Grothendieck topology on $\mathsf{Sch}$ (e.g. Zariski, étale, fpqc, Nisnevich). An algebraic space is an étale sheaf on $\mathsf{Sch}$ which has a surjective étale morphism from a (functor represented by) scheme (and some condition on the diagonal). In contrast to the category of schemes, the category of algebraic spaces is closed under quotients by étale equivalence relations.