The existence of UFO implies projectively universal morphisms

TL;DR

The paper proves that in certain concrete categories with a universally free object, there exists a projectively universal morphism, extending previous results from Banach spaces to broader mathematical structures.

Contribution

It establishes the existence of projectively universal morphisms in categories with universally free objects, generalizing prior work on Banach spaces and dynamical systems.

Findings

Existence of projectively universal morphisms in categories with a universally free object.

Extension of previous results from Banach spaces to categories like C*-algebras.

Applicability to various categories including Banach spaces, lattices, and algebras.

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 $\tau\in {\rm Mor}(B)$ there exists an epimorphism $\pi\in {\rm Mor}(\mathsf F, B)$ such that $\pi \tau = u \pi$. This builds upon and extends various ideas by Darji and Matheron (Proc. Am. Math. Soc. 145 (2017)) 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.