  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.
