Turing machines and spin models share a notion of universality according to which some simulate all others. Is there a theory of universality that captures this notion? We set up a categorical framework for universality which includes as instances universal Turing machines, universal spin models, NP completeness, top of a preorder, denseness of a subset, and more. By identifying necessary conditions for universality, we show that universal spin models cannot be finite. We also characterize when universality can be distinguished from a trivial one and use it to show that universal Turing machines are non-trivial in this sense. Our framework allows not only to compare universalities within each instance, but also instances themselves. We leverage a Fixed Point Theorem inspired by a result of Lawvere to establish that universality and negation give rise to unreachability (such as uncomputability). As such, this work sets the basis for a unified approach to universality and invites the study of further examples within the framework.

翻译：图灵机和自旋模型共享一种普适性概念，即某些系统能够模拟所有其他系统。是否存在一种理论能刻画这种普适性？我们建立了一个范畴化普适性框架，其实例涵盖通用图灵机、通用自旋模型、NP完全性、预序顶端、子集稠密性等。通过识别普适性的必要条件，我们证明通用自旋模型不能是有限的。我们还刻画了普适性何时可区别于平凡普适性，并据此表明通用图灵机在此意义上是非平凡的。本框架不仅允许在每个实例内部比较普适性，还能实现实例间的比较。我们利用受Lawvere结果启发的不动点定理，证明普适性与否定性结合会导出不可达性（如不可计算性）。因此，这项工作为普适性的统一处理奠定了基础，并鼓励对该框架内的更多实例进行研究。