$\mathrm{\aleph }$-injective Banach spaces and $\mathrm{\aleph }$-projective compacta

A Banach space $E$ is said to be injective if for every Banach space $X$ and every subspace $Y$ of $X$ every operator $t:Y\to E$ has an extension $T:X\to E$\mathrm{.}Wesaythat$E$is$\aleph$-injective(respectively,universally$\aleph$-injective)iftheprecedingconditionholdsforBanachspaces$X$(respectively$Y$)withdensitylessthanagivenuncountablecardinal$\aleph$\mathrm{.}Weperformastudyof$\aleph$-injectiveanduniversally$\aleph$-injectiveBanachspaceswhichextendsthebasiccasewhere$\aleph=\aleph_1$ is the first uncountable cardinal. When dealing with the corresponding “isometric” properties we arrive to our main examples: ultraproducts and spaces of type $C(K)$. We prove that ultraproducts built on countably incomplete $\mathrm{\aleph }$-good ultrafilters are $(1,\mathrm{\aleph })$-injective as long as they are Lindenstrauss spaces. We characterize $(1,\mathrm{\aleph })$-injective $C(K)$ spaces as those in which the compact $K$ is an $F_{\mathrm{\aleph }}$-space (disjoint open subsets which are the union of less than $\mathrm{\aleph }$ many closed sets have disjoint closures) and we uncover some projectiveness properties of $F_{\mathrm{\aleph }}$-spaces.