The existence of UFO implies projectively universal morphisms

Balcerzak, Marek; Kania, Tomasz

doi:10.1090/proc/16422

The existence of UFO implies projectively universal morphisms
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by Marek Balcerzak and Tomasz Kania

Proc. Amer. Math. Soc. 151 (2023), 3737-3742

DOI: https://doi.org/10.1090/proc/16422

Published electronically: May 5, 2023

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Abstract:

Let $\mathcal C$ be a concrete category. We prove that if $\mathcal {C}$ admits a universally free object $\mathsf F$, then there is a projectively universal morphism $u\colon \mathsf F\to \mathsf F$, i.e., a morphism $u$ such that for any $B\in \mathcal {C}$ and $\tau \in \operatorname {Mor}(B)$ there exists an epimorphism $\pi \in \operatorname {Mor}(\mathsf F, B)$ such that $\pi \tau = u \pi$. This builds upon and extends various ideas by Darji and Matheron [Proc. Amer. Math. Soc. 145 (2017), pp. 251–265] who proved such a result for the category of separable Banach spaces with contractive operators as well as certain classes of dynamical systems on compact metric spaces. Specialising from our abstract setting, we conclude that the result applies to various categories of Banach spaces/lattices/algebras, C*-algebras, etc.

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