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  • $\begingroup$ In particular, André and Quillen take a free or projective resolution $B$ in the category of $A$-algebras (it's slightly more subtle than an ordinary projective resolution, but you get the picture), but I have heard many times that in Grothendieck toposes, you very often may fail to have any sort of nontrivial projective resolution (with respect to the global sections functor (which is the unenriched $Hom(\mathcal{O}_X,-)$)). $\endgroup$ Commented Nov 13, 2010 at 16:46
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    $\begingroup$ The whole point of the cotangent complex is its applications to deformation theory of objects and morphisms, and these are often not affine (and in the case of group schemes can involve rather intricate diagrams). Gluing is also a problem in the derived category, as you must know, so it's a highly nontrivial matter to adapt the "affine" theory to the global case. Those problems have to be overcome already for schemes, regardless of any bells & whistles like algebraic spaces. Perhaps it can be done more easily nowadays with Lurie's stuff, but that may be just moving the hard work around. $\endgroup$ Commented Nov 13, 2010 at 17:03
  • $\begingroup$ Actually, I didn't realize until you just said it right now that gluing is a problem in the derived category (although it obviously is, now that I think about it). To glue in the derived category correctly, you need to look at hocolims instead of ordinary colims. Thanks for the answer! $\endgroup$ Commented Nov 13, 2010 at 17:29
  • $\begingroup$ Oh, by the way, BCnrd, I think that Voevodsky-Morel deals with the issue of gluing together simplicial sheaves, but that is still pretty recent compared to Illusie's work. $\endgroup$ Commented Nov 13, 2010 at 18:34