  In this paper we define and compare several new Quillen model structures
which present the homotopy theory of algebraic quantum field theories. In this
way, we expand foundational work of Benini et al. by providing a richer
framework to detect and treat homotopical phenomena in quantum field theory.
Our main technical tool is a new extension model structure on operadic algebras
which is constructed via (right) Bousfield localization. We expect that this
tool is useful in other contexts.
