For (quasi-compact and quasi-separated) schemes there is a categorical way to characterise quasi-coherent sheaves of finite type using purely the abelian category $\operatorname{QCoh}(X)$. In an abelian category one can define a *categorically finitely generated* object as being an object M such that for any directed family of subobjects $M\_\alpha \subset M$, such that the sum $\sum\_\alpha M\_\alpha$ is equal to the original $M$, then there exists an index $\alpha\_0$ such that $M\_{\alpha\_0} = M$.


Now, in the category of $R$-modules this is easy to see, and the argument essentially relies on the fact that any module is the colimit of its finitely generated submodules.
For quasi-compact and quasi-separated schemes the only reference I could find is Daniel Murfet's excellent notes. This is Corollary 64 from [here](http://therisingsea.org/notes/ModulesOverAScheme.pdf).


I would like to know whether this result still holds for algebraic spaces or more generally for algebraic stacks. (I imagine that knowing this is essentially equivalent to proving Lemma 61 in Murfet's notes for a scheme but using the étale toplogy)


EDIT: Perhaps one cook up a variant of Proposition 15.4 of Laumon-Moret-Bailly. Unfortunately the argument there seems to use at the very end that a submodule of a finite type module is of finite type, which does not hold in the non noetherian case.


EDIT2: By the way, for qcqs algebraic spaces this was already known to Raynaud-Gruson [RG Proposition 5.7.8].



Is every module the colimit of its finitely generated submodules? (for algebraic spaces or stacks)
Note that on any qcqs algebraic space or Artin stack, a quasi-coherent sheaf is finite type if its pullback to any fppf scheme cover is finite type in the usual sense on schemes. We can similarly define the notion of "finite type" for a quasi-coherent sheaf of modules over any quasi-coherent sheaf of algebras on such an algebraic space or stack in the same way, and this is the notion that will be used below; it satisfies the categorical condition you want, so that is good enough.


In this [paper](http://arxiv.org/abs/0910.5008), absolute noetherian approximation is proved for qcqs algebraic spaces, building on the version for schemes proved by Thomason and Trobaugh: according to Theorem 1.2.2, every qcqs algebraic space $X$ is an inverse limit (with affine transition maps) of finitely presented algebraic spaces over $\mathbf{Z}$. In particular, $X$ is *affine* over an algebraic space $X\_0$ of finite presentation over $\mathbf{Z}$. 


Hence, we have an affine morphism $f:X \rightarrow X\_0$ to a noetherian algebraic space, so quasi-coherent $O\_X$-modules are the "same" as quasi-coherent sheaves of modules over $X\_0$ over the quasi-coherent $O\_{X\_0}$-algebra $A := f\_{\ast}(O\_X)$. Consequently, it suffices to show that for any *noetherian* algebraic space $X\_0$ and quasi-coherent $O\_{X\_0}$-module $A$, every quasi-coherent $A$-module $F$ is the direct limit of its directed system quasi-coherent $A$-submodules of finite type. The underlying $O\_{X\_0}$-module $F'$ of $F$ is the direct limit of its directed system of coherent $O\_{X\_0}$-submodules $F'\_i$. Hence, the image $F\_i$ of the natural $A$-linear map $A \otimes\_{O\_{X\_0}} F'\_i \rightarrow F$ is a quasi-coherent $A$-submodule of $F$ of finite type (over $A$), and it contains $F'\_i$, so clearly the inclusion $\varinjlim F\_i \rightarrow F$ is an isomorphism.


That settles the case of qcqs algebraic spaces, and to prove the same for qcqs Artin stacks you just need absolute approximation for qcqs Artin stacks (which would give that any such stack is affine over a noetherian one, so you can bootstrap from the known noetherian case exactly as above). In the case that the Artin stack is qcqs with quasi-finite diagonal (e.g., any Deligne--Mumford stack), this is proved in this [paper](http://arxiv.org/abs/0904.0227v3) (which includes as a special case qcqs algebraic spaces, so it is an alternative to the initial reference at the top).


