Homotopy spectral sequences

TL;DR

This paper develops a categorical framework for homotopy spectral sequences, generalizing classical theories by incorporating kernels, cokernels, and null morphisms within a broad categorical setting.

Contribution

It introduces a new categorical approach to spectral sequences based on kernels and cokernels relative to null morphisms, extending beyond abelian and Puppe-exact categories.

Findings

Established a categorical setting for spectral sequences

Generalized classical homotopy theory structures

Provided a foundation for further algebraic topology research

Abstract

In homotopy theory, exact sequences and spectral sequences consist of groups and pointed sets, linked by actions. We prove that the theory of such exact and spectral sequences can be established in a categorical setting which is based on the existence of kernels and cokernels with respect to an assigned ideal of null morphisms, a generalisation of abelian categories and Puppe-exact categories.