An algebraic quantum field theory (AQFT) presents a QFT on Lorentzian manifolds as an assignment of algebras to spacetimes, subject to physical axioms (e.g. Einstein causality). Such algebras are interpreted as quantizations of the function algebras on the moduli space of a classical field theory. In many cases, e.g. the stack of a gauge theory, moduli spaces encode "higher structures". As a consequence, functions on such spaces form "higher algebras", which can be analyzed by homotopical algebra (à la Quillen). Therefore, to investigate the quantization of such moduli spaces, one needs to infuse AQFT with homotopical algebra, resulting in "homotopical AQFT", i.e. the assignment of "higher algebras" to spacetimes. After motivating our approach with a concrete application of homotopical algebra to the Cauchy problem of the Yang-Mills stack, I will provide a "working definition" of homotopical AQFT, emphasize its role in relation to gauge theories and present two toy examples arising via homotopy Kan extensions. Based on [arXiv: 1503.08839, 1610.06071, 1704.01378].