Universal finite functorial semi-norms (Q6600689)

A \textit{functorial semi-norm} on a functor \(F:C\rightarrow\mathsf{Vect}_{K}\) to vector spaces over a normed field \(K\) is a lift of \(F\) to a functor \(C\rightarrow\mathsf{snVect}_{K}\) to the category of semi-normed vector spaces over \(K\). A functorial semi-norm on \ is called \textit{universal} if it vanishes on as few classes as possible among all functorial semi-norms on \(F\). A geometric example of a functorial semi-norm is the \(l^{1}\)-semi-norm on singular homology [\textit{M. Gromov}, Publ. Math., Inst. Hautes Étud. Sci. 56, 5--99 (1982; Zbl 0516.53046)], which measures the size of homology classes in terms of singular simplicies.\N\NIt is known that the \(l^{1}\)-semi-norm is not universal in high degrees [\textit{D. Fauser} and \textit{C. Löh}, Glasg. Math. J. 61, No. 2, 287--295 (2019; Zbl 1432.55005)], and this paper aims to affirmatively answer the question concerning existence of universal finite semi-norms over singular homology in case of the category of spaces homotopy equivalent to finite CW-complexes. The authors establish the following general theorem, using a suitable diagonalization technique.\N\NTheorem. Let \(C\) be\ a category admitting a skeleton with at most countably many objects. Let \(K\) be a normed field and let \(F:C\rightarrow\mathsf{Vect}_{K}\) be a functor.\N\begin{itemize}\N\item[(1)] If \(K\) is countable and if \(F\) maps to the category \(\mathsf{Vect}_{K}^{\omega}\) of \(K\)-vector space of countable dimensions, then \(F\) admits a universal finite functorial semi-norm.\N\item[(2)] If \(F\) maps to the category \(\mathsf{Vect}_{K}^{\mathrm{fin}}\) of \(K\)-vector space of finite dimensions, then \(F\) admits a universal finite functorial semi-norm.\N\end{itemize}