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Generalized Clifford algebras, weighted polynomial laws, and adjunction

Nguyen Xuan Bach

TL;DR

This work reframes Clifford algebra constructions through adjunctions in a broad ∞-categorical setting, introducing weighted polynomial laws and proving that the associated Clifford algebra functor Cl is a left adjoint. By developing lifting results and a derived version of Clifford algebras, the authors unify classical generalized Clifford algebras with Krashen–Lieblich’s framework and extend to derived contexts, animated rings, and derived Artin stacks. The approach yields explicit descriptions via colimits, base-change compatibilities, and functorial constructions that recast geometric and derived structures (e.g., relative spectra and projective bundles) within a coherent adjunction-theoretic paradigm. The results advance both the algebraic and geometric understanding of generalized Clifford algebras, enabling systematic derivations and applications to derived categories, quasi-coherent sheaves, and higher-stack geometry, while recapturing known constructions as special cases.

Abstract

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.

Generalized Clifford algebras, weighted polynomial laws, and adjunction

TL;DR

This work reframes Clifford algebra constructions through adjunctions in a broad ∞-categorical setting, introducing weighted polynomial laws and proving that the associated Clifford algebra functor Cl is a left adjoint. By developing lifting results and a derived version of Clifford algebras, the authors unify classical generalized Clifford algebras with Krashen–Lieblich’s framework and extend to derived contexts, animated rings, and derived Artin stacks. The approach yields explicit descriptions via colimits, base-change compatibilities, and functorial constructions that recast geometric and derived structures (e.g., relative spectra and projective bundles) within a coherent adjunction-theoretic paradigm. The results advance both the algebraic and geometric understanding of generalized Clifford algebras, enabling systematic derivations and applications to derived categories, quasi-coherent sheaves, and higher-stack geometry, while recapturing known constructions as special cases.

Abstract

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.

Paper Structure

This paper contains 14 sections, 38 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Constructions of left adjoints
  3. Adjunctions between symmetric monoidal $\infty$-categories
  4. Adjunctions between $\infty$-categories of algebras
  5. Applications to quasi-coherent derived categories
  6. A category of weighted polynomial laws
  7. Polynomial laws over schemes
  8. Weighted polynomia laws
  9. Lifting of weighted polynomial laws
  10. Comparison with Krashen and Lieblich's explicit construction
  11. Derived Clifford algebras
  12. Derived Clifford algebras over commutative rings
  13. Derived Clifford algebras over animated commutative rings
  14. Derived Clifford algebras over derived Artin stacks

Key Result

Lemma 1.1.1

Let $F:\mathcal{C} \to \mathcal{D}$ be a monoidal functor between closed monoidal $\infty$-categories with a right adjoint $G$. Assume that Then $F$ admits a left adjoint and this left adjoint is given on $\mathcal{C}^{\omega}$ by $\mathrm{D}_\mathcal{C}^{\mathrm{op}}G^{\mathrm{op}}\mathrm{D}_{\mathcal{D}^\omega}$.

Theorems & Definitions (100)

  • Lemma 1.1.1
  • proof
  • Lemma 1.2.1
  • proof
  • Corollary 1.2.2
  • proof
  • Remark 1.2.3
  • Lemma 1.2.4
  • proof
  • Lemma 1.2.5
  • ...and 90 more