I will explain the fundamental role in cyclic cohomology of two concepts:

1) the first is the quantized calculus which transposes the ordinary calculus in the quantum framework and provides immediate access to the notion of cycle, the three connecting maps of the SBI long exact sequence, and the chern character in K-homology. The integrality and locality of the character will be discussed.

2) the second is the cyclic category Lambda and the embedding it provides of the category of noncommutative algebras in the abelian category of cyclic vector spaces by the functor $ A \to A^\natural$. I will discuss the potential role of the Clausen-Scholze abelian category of topological vector spaces and the question of the extension of the above functor to the framework of S-algebras.