I think there is a paper by Frédéric Paugam ("Histories and observables in covariant field theory") where this is discussed.

EDIT: There is also a book in progress by the same author where he discusses the notion of derived critical space for the Euler-Lagrange equation (this is in Chapter 10).

Update Sept. 27: Gabriele Vezzosi has just posted a preprint on the arXiv that could be of interest for that question (below is my original answer).

I think there is a paper by Frédéric Paugam ("Histories and observables in covariant field theory") where this is discussed.

EDIT: There is also a book in progress by the same author where he discusses the notion of derived critical space for the Euler-Lagrange equation (this is in Chapter 10).

I think there is a paper by Frédéric Paugam ("Histories and observables in covariant field theory") where this is discussed.

I think there is a paper by Frédéric Paugam ("Histories and observables in covariant field theory") where this is discussed.

EDIT: There is also a book in progress by the same author where he discusses the notion of derived critical space for the Euler-Lagrange equation (this is in Chapter 10).