symmetric monoidal (∞,1)-category of spectra
The axioms of traditional algebraic quantum field theory or locally covariant perturbative quantum field theory, where one assigns a plain algebra (e.g. a C*-algebra or formal power series algebra) of observables to a spacetime region, fail when one considers gauge theory on spacetimes more general than Minkowski spacetime (AQFT on curved spacetimes):
The nature of gauge fields implies that to re-construct their global configuration (including non-trivial topology as in instanton sectors) one needs to locally remember the groupoid of gauge transformations (see Eggertsson 14, Schenkel 14, Schreiber 14 for exposition), and hence, after quantization, some higher algebra of observables.
The program of homotopical algebraic quantum field theory (Benini-Schenkel 16, see Schenkel 17 for survey) is to lift the axioms of AQFT on curved spacetimes to certain local nets of homotopical algebras such as to properly capture gauge theory with topologically non-trivial gauge field configurations.
The axioms as considered in Schenkel 17 have some similarity with that of factorization homology, but also do take into account the causality condition for quantum fields on Lorentzian manifolds.
For review and exposition see
Alexander Schenkel, Towards Homotopical Algebraic Quantum Field Theory, talk at Foundational and Structural Aspects of Gauge Theories, Mainz Institute for Theoretical Physics, 29 May – 2 June 2017. (pdf)
Alexander Schenkel, On the problem of gauge theories in locally covariant QFT, talk at Operator and Geometric Analysis on Quantum Theory Trento, 2014 (pdf) (with further emphasis on this point in the companion talk Schreiber 14)
Alexander Schenkel, From Fredenhagen’s universal algebra to homotopy theory and operads, talk at Quantum Physics meets Mathematics, Hamburg, December 2017 (pdf slides)
Construction and axiomatization of gauge field AQFT via homotopy theory and homotopical algebra is being developed in
Marco Benini, Claudio Dappiaggi, Alexander Schenkel, Quantized Abelian principal connections on Lorentzian manifolds, Communications in Mathematical Physics 2013 (arXiv:1303.2515)
Marco Benini, Alexander Schenkel, Richard Szabo, Homotopy colimits and global observables in Abelian gauge theory (arXiv:1503.08839)
Marco Benini, Alexander Schenkel, Quantum field theories on categories fibered in groupoids, Communications in Mathematical Physics November 2017, Volume 356, Issue 1, pp 19–64 (arXiv:1610.06071)
Marco Benini, Alexander Schenkel, Lukas Woike, Operads for algebraic quantum field theory (arXiv:1709.08657)
The stack of Yang-Mills gauge fields is discussed in