Heap and ternary self-distributive cohomology

Abstract

Heaps are algebraic structures endowed with para-associative ternary operations, bijectively exemplified by groups via the operation (x, y, z) → xy-1z. 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.