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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.

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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.

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  • $\begingroup$ Can we define etale fundamental group of an algebraic space or of an element of $\widehat{{\rm Sch}}$ ? $\endgroup$ – user48900 Sep 11 '13 at 13:23
  • $\begingroup$ I don't know, but probably. Isn't this a completely different question? $\endgroup$ – Martin Brandenburg Sep 11 '13 at 14:53

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