Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lax Additivity

Merlin Christ, Tobias Dyckerhoff, Tashi Walde

TL;DR

This work develops a calculus of lax additivity for $(\infty,2)$-categories, enriching the classical theory of additive categories to a higher-categorical context. The authors introduce lax matrices and show that lax limits and lax colimits coincide and are absolute in locally cocomplete settings, enabling a robust categorified homological algebra. They categorify fundamental constructions such as mapping complexes and mapping cones, establishing their basic properties and universal characterizations within this framework. The results provide a unified, higher-categorical foundation for homological techniques in areas like Fukaya-type categories and categorified perverse theory, with potential applications to gluing, recollements, and higher Morita theory.

Abstract

We introduce notions of lax semiadditive and lax additive $(\infty,2)$-categories, categorifying the classical notions of semiadditive and additive 1-categories. To establish a well-behaved axiomatic framework, we develop a calculus of lax matrices and use it to prove that in locally cocomplete $(\infty,2)$-categories lax limits and lax colimits agree and are absolute. In the lax additive setting, we categorify fundamental constructions from homological algebra such as mapping complexes and mapping cones and establish their basic properties.

Lax Additivity

TL;DR

This work develops a calculus of lax additivity for -categories, enriching the classical theory of additive categories to a higher-categorical context. The authors introduce lax matrices and show that lax limits and lax colimits coincide and are absolute in locally cocomplete settings, enabling a robust categorified homological algebra. They categorify fundamental constructions such as mapping complexes and mapping cones, establishing their basic properties and universal characterizations within this framework. The results provide a unified, higher-categorical foundation for homological techniques in areas like Fukaya-type categories and categorified perverse theory, with potential applications to gluing, recollements, and higher Morita theory.

Abstract

We introduce notions of lax semiadditive and lax additive -categories, categorifying the classical notions of semiadditive and additive 1-categories. To establish a well-behaved axiomatic framework, we develop a calculus of lax matrices and use it to prove that in locally cocomplete -categories lax limits and lax colimits agree and are absolute. In the lax additive setting, we categorify fundamental constructions from homological algebra such as mapping complexes and mapping cones and establish their basic properties.
Paper Structure (20 sections, 42 theorems, 185 equations, 3 tables)

This paper contains 20 sections, 42 theorems, 185 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Rules of categorification
  3. Lax additivity
  4. Mapping cones and mapping complexes
  5. Acknowledgements
  6. Additive 1-categories
  7. Preliminaries
  8. (infinity,2)-categories
  9. Lax limits and colimits
  10. Lax matrices
  11. Lax additivity
  12. Oplax additivity
  13. Coordinate change for Delta1-matrices
  14. Chain complexes and chain maps
  15. The oplax mapping cone construction
  16. ...and 5 more sections

Key Result

Theorem 1

In locally cocompleteAn $(\infty,2)$-category is locally cocomplete if all its hom-categories have colimits and if composition of $1$-arrows preserves colimits in each variable.$(\infty,2)$-categories

Theorems & Definitions (136)

  • Theorem : \ref{['cor:lax-co-limits-are-bilimits']} and \ref{['thm:lax-lims-absolute']}
  • Definition : \ref{['defi:lax-additive']}
  • Theorem : \ref{['cor:univ-prop-lax-cof-fib']}
  • Definition 2.1
  • Lemma 2.6
  • Corollary 2.9
  • proof : Proof of \ref{['lem:prod-coprod-equations']}
  • Definition 3.12
  • Remark 3.15
  • Example 3.16
  • ...and 126 more