I'm trying to understand cotangent complexes and their role in deformation theory, and later the statement that they're somehow natural in a derived scheme/stack.


* I understand that the standard reference is Illusie's two books, unfortunately my French abilities are lacking
* I'm aware of the stacks project treatment. It looks quite terse. Is there a more elementary treatment?
* I think the [nlab article](https://ncatlab.org/nlab/show/cotangent+complex) is amazing, shame it's rather brief.


People who understand the cotangent complex: how did you first learn it?



Elementary (English) reference for the cotangent complex?
The homotopy-theoretic way to look at the cotangent complex is as the left derived functor of the Kahler differentials functor. Now:


* This statement makes sense just in the context of homological algebra. i.e. in some category of chain complexes of modules. This route to constructing the cotangent complex is the subject of section 8.8 in Weibel's homological algebra book.
* More generally we have notions of derived functors in any model category, a good reference is Dwyer and Spalinski. Constructing the cotangent complex at this level of generality is the subject of [a very nice note by Zhou](https://web.archive.org/web/20180304232621/http://www.math.harvard.edu/~rzhou/minorthesis.pdf).


