n-Butterflies: Algebraically Modeling Morphisms between Homotopy n-Types

TL;DR

This paper introduces n-butterflies, algebraic structures that generalize butterflies, to model morphisms between pointed connected homotopy n-types, extending algebraic modeling beyond homotopy 2-types.

Contribution

It generalizes the algebraic model of butterflies to n-butterflies, enabling algebraic modeling of morphisms between pointed connected homotopy n-types.

Findings

n-butterflies generalize butterflies for higher homotopy types

Provides an algebraic framework for morphisms of homotopy n-types

Extends algebraic modeling to a broader class of homotopy types

Abstract

Crossed modules are known to be a model of pointed connected homotopy 2-types; formally, the homotopy category of crossed modules is equivalent to the category of pointed connected homotopy 2-types. In forming the homotopy category of crossed modules, one must resort to computing derived morphisms using non-constructive topological methods, but Behrang Noohi was able to find an algebraic model of these derived morphisms called butterflies. The result is a completely algebraic model of pointed connected homotopy 2-types. Reduced crossed complexes are a generalization of crossed modules and model a subclass of pointed connected homotopy types. We will present algebraic objects called n-butterflies which satisfy similar properties to butterflies and begin to generalize the theory of butterflies to model morphisms of pointed connected homotopy n-types.