# Higher Arity Self-Distributive Operations in Cascades and their Cohomology

## TL;DR

This paper explores higher arity self-distributive operations, their cohomology, and geometric interpretations, extending classical binary operations to n-ary cases with applications in topology and algebra.

## Contribution

It introduces mutually distributive n-ary operations and a corresponding cohomology theory, generalizing binary cases and linking algebraic structures to geometric link invariants.

## Key findings

- Defined mutually distributive n-ary operations
- Established relations between cohomology groups of different arities
- Provided examples from Lie algebras, coalgebras, and Hopf algebras

## Abstract

We investigate constructions of higher arity self-distributive operations, and give relations between cohomology groups corresponding to operations of different arities. For this purpose we introduce the notion of mutually distributive $n$-ary operations generalizing those for the binary case, and define a cohomology theory labeled by these operations. A geometric interpretation in terms of framed links is described, with the scope of providing algebraic background of constructing $2$-cocycles for framed link invariants. This theory is also studied in the context of symmetric monoidal categories. Examples from Lie algebras, coalgebras and Hopf algebras are given.