Heap and Ternary Self-Distributive Cohomology

TL;DR

This paper develops cohomology theories for heaps and ternary self-distributive structures, revealing their connections to group cohomology and categorical properties, with implications for algebraic and topological applications.

Contribution

It introduces new cohomology theories for heaps and ternary self-distributive structures, establishing links to group cohomology and categorical frameworks.

Findings

Heap cohomology relates to group cohomology via a long exact sequence.

Maps between group and heap cohomology groups are constructed and injective.

Heap objects in monoidal categories are characterized as involutory Hopf monoids.

Abstract

Heaps are para-associative ternary operations bijectively exemplified by groups via the operation $(x,y,z) \mapsto x y^{-1} z$. They are also ternary self-distributive, and have a diagrammatic interpretation in terms of framed links. Motivated by these properties, we define para-associative and heap cohomology theories and also a ternary self-distributive cohomology theory with abelian heap coefficients. We show that one of the heap cohomologies is related to group cohomology via a long exact sequence. Moreover we construct maps between second cohomology groups of normalized group cohomology and heap cohomology, and show that the latter injects into the ternary self-distributive second cohomology group. We proceed to study heap objects in symmetric monoidal categories providing a characterization of pointed heaps as involutory Hopf monoids in the given category. Finally we prove that heap objects are also "categorically" self-distributive in an appropriate sense.