It is known that one can define a smooth structure on a manifold using a sheaf-theoretic formulation via defining the algebra of the (a fortiori) smooth functions on it (which satisfies the usual equalizer condition), therefore defining the manifold by its maps instead of the regular charts.
I'm wondedring whether one can similarly define a structure sheaf using Morse functions instead.
Clearly, the set of Morse functions on a manifold does not have an algebra structure (or even a group structure) with the usual operations. So can we still define the necessary group/algebra structure on Morse functions, so that the resulting sheaf would make sense? In this case, will we get some meaningful structure on the manifold?