Abstract Excision and ${\ell }^1$-Homology

Witzig, Johannes

Confluentes Mathematici, Tome 15 (2023), pp. 107-136

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.

Classification : 55N35, 55U35, 18F50, 18N60
Keywords: $\ell ^1$-homology, excision, excisive approximation, $\infty $-categories

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