Homotopy spectral sequences

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.

💡 Research Summary

The paper addresses a long‑standing structural mismatch in homotopy theory: exact sequences and spectral sequences traditionally involve both groups and pointed sets, yet the categorical frameworks that handle groups (abelian categories) and those that handle pointed sets (Puppe‑exact categories) are distinct and only partially overlap. The author introduces a unifying categorical setting based on an assigned ideal I of null morphisms. In an “I‑exact category” every morphism f admits an I‑kernel ker_I(f) and an I‑cokernel coker_I(f), and these fit together in I‑exact triangles that satisfy the usual three‑term and ladder axioms up to I‑isomorphism. This construction generalizes both abelian categories (take I to be the class of all zero morphisms) and Puppe‑exact categories (take I to be the class of regular zero morphisms).

The first major contribution is a systematic development of the basic properties of I‑exact categories. The author shows that I‑isomorphisms form a well‑behaved class of morphisms with inverses, that I‑kernels and I‑cokernels are unique up to unique I‑isomorphism, and that the presence of a zero object allows pointed sets to be regarded as special objects (the zero object itself) within the same categorical universe. Consequently, groups and pointed sets can be treated uniformly as objects of an I‑exact category, and actions of groups on pointed sets become ordinary morphisms in this setting.

Building on this foundation, the paper reconstructs the classical James–Pietsch (or more generally, the Adams) spectral sequence inside an I‑exact category. Each page of the spectral sequence is expressed as a composition of I‑cokernels followed by I‑kernels, forming a ladder of I‑exact triangles. The connecting morphisms that traditionally link homotopy groups π_n(X) with the set of components π_0(X) are identified with the I‑exact connecting maps. The author proves that the usual convergence, degree‑shift, and exactness properties of the spectral sequence hold verbatim in any I‑exact category, thereby removing the need to switch between group‑only and set‑only contexts.

To demonstrate the breadth of the theory, several concrete examples are provided. Ordinary abelian categories become I‑exact for the ideal of all zero morphisms; Puppe‑exact categories become I‑exact for the ideal of regular zero morphisms. Moreover, module categories, arrow categories (categories of morphisms), and other non‑abelian contexts admit natural choices of I that satisfy the kernel‑cokernel axioms, showing that the framework is not limited to classical homological algebra. These examples illustrate that the proposed categorical machinery can be applied to a wide variety of algebraic and topological settings.

The paper concludes with a discussion of future directions. One line of inquiry is the construction of new spectral sequences in non‑abelian contexts using I‑exactness, potentially yielding computational tools for higher homotopy groups of spaces with exotic algebraic structures. Another is the exploration of weakened or alternative ideals of null morphisms, which could broaden the class of categories where the theory applies. Finally, the author suggests concrete computational experiments, such as applying the I‑exact spectral sequence to calculate stable homotopy groups of spheres or to analyze group actions on highly connected spaces, thereby testing the practical impact of the theory. In sum, the work provides a robust categorical unification of exact and spectral sequences in homotopy theory, extending the reach of these fundamental tools beyond the confines of traditional abelian or Puppe‑exact settings.