Universal $\delta$-functors and relative injective objects in the category of $p$-Banach spaces

Abstract

Inspired by Grothendieck's foundational work, we establish that the sequence of functors $(Ext^n_{pBan}(E,-))_{n \ge 0}$ on the category of $p$-Banach spaces ($0<p<1$) forms a universal $\delta$-functor. To this end, for any pair of $p$-Banach spaces $E$ and $X$, we construct a $p$-Banach space $\mathcal N_E^X$ that renders each functor $Ext^n_{pBan}(E,-)$ effaceable. Furthermore, we show that this space serves as a relative injective object for $E$ and $X$, enabling the construction of relative injective presentations and proving that the functors $Ext^n_{pBan}(E,-)$ are the right derived functors of $\mathcal{L}(E,-)$. We will also examine the situation in the category of quasi-Banach spaces.