Finite functorial semi-norms and representability

TL;DR

This paper explores how various forms of representability influence the existence and triviality of finite functorial semi-norms on different functors, including cohomology theories and homology classes.

Contribution

It provides a detailed analysis of the impact of representability on the non-triviality of functorial semi-norms across multiple homological and cohomological contexts.

Findings

Representability conditions affect semi-norm triviality.

Weakly flexible classes relate to semi-norm behavior.

Results apply to singular and l1-homology.

Abstract

Functorial semi-norms are semi-normed refinements of functors such as singular (co)homology. We investigate how different types of representability affect the (non-)triviality of finite functorial semi-norms on certain functors or classes. In particular, we consider representable functors, generalised cohomology theories, and so-called weakly flexible homology classes in singular homology and l1-homology.