Here C denotes a coalgebra with counit (C,Delta,varepsilon). A left Hopf truss consists of

(*) a coalgebra C with two algebra structures such that one makes C into a Hopf algebra (C,\cdot,1,\Delta,\varepsilon,S) and the other into a nonunital bialgebra (C,\circ,\Delta,\varepsilon),

(**) connected by a coalgebra endomorphism \sigma: C -> C such that x.(y.z)=\sum(x_1. y). S\sigma(x_2).(x_3. z)$ holds for all x,y,z in C.

An equivalent formulation of the statement (**) is

(***) Define a ternary operation [-,-,-]: C\otimes C^{cop}\otimes C -> C by [x,y,z]=x.S(y).z$ for all x,y,zin C. Then x\circ [y,z,t]=\sum[x_1\circ y,x_2\circ z,x_3\circ t] holds for all x,y,z,t\in C.

A ternary operation defined in the statement (***) makes C into a Hopf heap (C,\Delta,\varepsilon,[-,-,-]). 

A Hopf heap consists of a coalgebra C with a coalgebra homomorphism \chi: C\otimes C^{cop}\otimes C -> C, x\otimes y\otimes z |-> [x,y,z] such that [[x,y,z],t,u]=[x,y,[z,t,u]] and \sum[x_1,x_2,y]=\sum[y,x_1,x_2]=\varepsilon(x)y hold for all x,y,z,t,u in C.

This talk is intended as a discussion of Hopf trusses and Hopf heaps.