Parametrised functor calculus: excision, spheres, and semiadditivity 2410 22327v1 скачать в хорошем качестве

Parametrised functor calculus: excision, spheres, and semiadditivity 2410 22327v1

Potcast by Google NotebookLM(20241031목) Understanding Parametrized Functor Calculus Parametrised functor calculus: excision, spheres, and semiadditivity by Kaif Hilman and Sil Linskens 1. Introduction This section introduces the concept of parametrized functor calculus, a generalization of Goodwillie's functor calculus aimed at understanding polynomial phenomena within multiplicative structures in higher algebra. It highlights the importance of this calculus and its relation to previous work by Blumberg, Dotto, Moi, and Stoll. The authors present the key questions driving their research: How can Goodwillie calculus be adapted to multiplicative settings? Can parameterized stability be characterized by the invertibility of specific spheres? 2. Parametrised excision for posets This section delves into the foundational concepts of parameterized excisiveness. 2.1 Excisable Posets This subsection defines key terms related to posets in a parameterized setting, including: Distributive parameterized posets with finite products and coproducts. Complementable posets where each object has a complement object. Excisable posets, which are complementable parameterized posets with a specific downward-closed subposet, forming the basis for excisive approximations. It explores properties of these posets, particularly how complementable posets can be decomposed into smaller ones. This subsection concludes by defining an excisable poset, a concept inspired by Goodwillie's work on cocartesian cubes. 2.2 Face Decompositions This subsection explores how to decompose excisable posets based on "triple decompositions" of its elements. It introduces the concept of "face posets" and shows how the excisable structure of the original poset induces an excisable structure on these face posets. This decomposition is crucial for analyzing the behavior of functors on these posets. 2.3 Excision This subsection formally defines the notion of σ-excisive functors. It introduces the concept of a σ-extensible category and establishes the fundamental Theorem A: For a σ-extensible category C with a final object and a σ-differentiable category D, the fully faithful inclusion of σ-excisive functors from C to D admits a left adjoint. This theorem provides a framework for approximating functors by their σ-excisive counterparts. 3. Parametrised cubical excision This section specializes the theory of parameterized excisiveness developed in Section 2 to atomic orbital base categories. 3.1 Cubes This subsection introduces the concepts of atomic and orbital categories and defines parameterized cubes within this context. It explores the structure of these cubes, noting they are parameterized posets with fibers given by ordinary cubes. This subsection concludes by demonstrating that external indexed products of singleton cocartesian cubes are again singleton cocartesian. 3.2 Suspensions and spheres This subsection uses the parameterized cubes defined in 3.1 to construct parameterized suspension and loop functors, generalizing the standard notions from homotopy theory. It introduces the concept of parameterized spheres, which play a crucial role in characterizing parameterized stability. This subsection concludes by defining w-spherically faithful and w-spherically invertible categories based on the properties of their suspension and loop functors. 3.3 Cubical excision This subsection demonstrates that the parameterized cubes constructed in 3.1 fit into the framework of excisable posets introduced in Section 2. It proves that specific subcategories of these cubes are downward-closed, thus providing them with an excisable poset structure. This connection allows applying the results from Section 2 to the specific case of parameterized cubes. 3.4 Semiadditivity and stability This subsection connects the theory of excisive functors with the concepts of parameterized semiadditivity and stability. It establishes Theorem B, proving that under certain conditions, if a functor is singleton w-excisive for all morphisms w, then it is parameterized semiadditive. This theorem links the invertibility of specific spheres to the concept of parameterized semiadditivity. The authors then present Theorem C, demonstrating that for pointed parameterized presentable categories over locally short atomic orbital categories, parameterized stability is equivalent to spherical invertibility. A. Decomposition of colimit diagrams This appendix provides a technical result used within the paper, showing how colimits over a parameterized category admitting a decomposition into a colimit can be computed in two steps. This result simplifies the computation of colimits in specific cases.