Generalized Clifford algebras, weighted polynomial laws, and adjunction

In this paper, we study Clifford algebra construction from the perspective of adjunctions motivated by the general framework of Krashen and Lieblich. We introduce a category of weighted polynomial laws whose associated Clifford algebra functor is a left adjoint on this category. We introduce a notion of lifting weighted polynomial laws generalizing explicit constructions of Krashen and Lieblich. As another application, we construct derived Clifford algebras of derived homogeneous polynomial forms of arbitrary degree with noncommutative coefficients.