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：图灵机与自旋模型共享一种普适性概念，即某些模型可以模拟所有其他模型。是否存在能够捕捉这一概念的普适性理论？我们建立了一个普适性的范畴论框架，其具体实例包括通用图灵机、通用自旋模型、NP完全性、预序集的最大元、子集的稠密性等。通过识别普适性的必要条件，我们证明有限自旋模型不可能具有普适性。我们还刻画了何时能够区分平凡普适性与非平凡普适性，并借此证明通用图灵机在此意义下是非平凡的。我们的框架不仅支持在同一实例内比较普适性，还能对不同实例本身进行比较。借助受Lawvere定理启发的固定点定理，我们论证了普适性与否定性共同导致不可达性（如不可计算性）。因此，本研究为普适性的统一研究奠定了基础，并倡导在该框架下探索更多实例。