arXiv subjects

Johannes Witzig

Publications and source records attributed to Johannes Witzig.

2 recordsLinked to original sources

arXiv2024

Universal finite functorial semi-norms

Functorial semi-norms on singular homology measure the "size" of homology classes. A geometrically meaningful example is the $\ell^1$-semi-norm. However, the $\ell^1$-semi-norm is not universal in the sense that it does not vanish on as few classes as possible. We show that universal finite functorial semi-norms do exist on singular homology on the category of topological spaces that are homotopy equivalent to finite CW-complexes. Our arguments also apply to more general settings of functorial semi-norms.

math.CT↗

arXiv2024

Abstract Excision and $\ell^1$-Homology

We use the abstract setting of excisive functors in the language of $\infty$-categories to show that the best approximation to the $\ell^1$-homology functor by an excisive functor is trivial. Then we make an effort to explain the used language on a conceptual level for those who do not feel at home with $\infty$-categories, prove that the singular chain complex functor is indeed excisive in the abstract sense, and show how the latter leads to classical excision statements in the form of Mayer-Vietoris sequences.

math.AT↗