This might be obvious to experts, but I'm not sure where to look for the answer. On a reasonably nice, at least noetherian, scheme (or variety, algebraic space, stack), can the category of coherent sheaves be constructed categorically from the category of vector bundles? I am thinking of Coh being some kind of 'abelian envelope' of Vect.
(in the affine case this question can be rephrased as: can the category of finitely generated modules be defined via the category of projective modules of finite rank)?)
Best Answer
Here are a few comments that might be useful. I don't think there is a chance that this can work unless the scheme in question has the resolution property (meaning every coherent sheaf is a quotient of a locally free sheaf of finite rank). Otherwise the category of locally free sheaves does not even form a generator of the category of all quasi-coherent sheaves, so it clearly contains more information.
Secondly, Quiaochu Yuan's construction for the affine case does not work globally for most schemes. What he does is indeed freely adding cokernels (to get to coherent sheaves) or freely adding all colimits (to get to quasi-coherent sheaves). The free cocompletion under all colimits of an additive category is given by taking the category of additive presheaves on it. (The free cocompletion under cokerenels is simply the closure of the representables under cokernels.) So, if we do that to the category of vector bundles on a scheme, we obtain a category of presheaves. However, any category of presheaves has a projective generator, while the category of quasi-coherent sheaves rarely does.
Finally, something positive that can be said: If you do assume that your scheme satisfies the resolution property (and I'll assume it is quasi-compact, not sure if that's necessary), then the full subcategory of vector bundles is a dense subcategory of the category of quasi-coherent sheaves. This is actually a quite amazing result: in a Grothendieck abelian category, any strong generator is dense, see
Brian Day and Ross Street, Categories in which all strong generators are dense, J. Pure Appl. Algebra 43 (1986), no. 3, 235–242. MR 868984
Thus we know that the category of quasi-coherent sheaves is a reflective subcategory of the free cocompletion of the category of vector bundles. Any reflective subcategory is the localization of the surrounding category at the morphisms that the reflector inverts (that is, it can be obtained by formally inverting a class of morphisms). Since we're dealing with a locally finitely presentable category, this can be further reduced to inverting a generating set of these morphisms. In some sense this says that the category of quasi-coherent sheaves can be obtained by first freely adding colimits, and then imposing some relations (formally turn a certain set of morphisms into isomorphisms).
It seems however rather difficult to get an explicit such set of morphisms in general.
Edit: I noticed that you're also interested in algebraic spaces and algebraic stacks. The above argument about the category of quasi-coherent sheaves also works at that level of generality as long as the resolution property holds. Specifically, if you have a quasi-compact stack $X$ on the fpqc-site of affine schemes which has the resolution property (in algebraic topology these are sometimes called Adams stacks, since they are precisely the stacks associated to Adams Hopf algebroids), then the category of quasi-coherent sheaves on $X$ is given by a localization of the free cocompletion of the category of dualizable quasi-coherent sheaves on $X$ at a set of morphisms.
Note that it is not clear from this argument wether or not this set of morphisms is entirely determined by the subcategory of dualizable quasi-coherent sheaves. If that is not the case, there could be Adams stacks in the above sense with equivalent categories of dualizable sheaves but inequivalent categories of quasi-coherent sheaves.