# Has the cotangent complex been used in context other than morphism of schemes?

Here is what I know about the history of the cotangent complex:

Quillen did it over a point (i.e. for morphism of rings), Illusie did it in a topos (i.e. for sheaves of rings in a topos). And proved general theorems on deformation/obstructions. People have been using that to study various deformation problems of schemes. E.g. constructing obstruction theory. There are also generalization to 1-morphisms to stacks or to the logarithmic world. It seems to be a crucial object in the derived algebraic geometry setup.

My question is, has the cotangent complex used in other context with the full power of topoi? I understand that it is related to the solution of Serre's conjecture and the definition Andre-Quillen cohomology. I also understand to "glue" the affine construction is hard and you really need the power of topoi. I just want to know if this has been used in general on a morphism of rings in a topos. Or more generally, for a morphism of ringed topoi.

(I hope this question is not too vague, a more "concrete" example might be: does infinitesimal thickening of the first order of the fppf topos of a scheme $X$ (by a module in it) over that of a scheme $Y$ contains some interesting information?)

Just to be clear, by topos I always mean a Grothendieck topos, i.e. a category equivalent to the category of sheaves of sets over a site.

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M. André, Homologie des algèbres commutatives (SPringer) –  Buschi Sergio May 25 '12 at 12:59
@sergio: might I be so lazy to ask you to explain the idea you're thinking about when pointing out that paper (so that I don't have to look it up myself)? –  Yosemite Sam May 25 '12 at 13:14
@temp: Which of Serre's conjectures are you referring to? –  Dylan Wilson May 25 '12 at 16:05
In the book "Homologie des algèbres commutatives", M. Andre uses and develops the concept of cotangent complex (does emerges from the study of differential in modules) in the context of commutative and homological algebra. Although this study is strictly inherent to algebraic geometry. –  Buschi Sergio May 25 '12 at 16:48
this also might be relevant for mentioning here: arxiv.org/abs/1004.0096 –  Vladimir Dotsenko May 25 '12 at 20:25

I do not know if it would be useful to study deformations of the structure sheaf on the fppf topos of a scheme $S$. Such a deformation would correspond to a consistent collection of deformations of every scheme over $S$.