Limit Sketches and the Universal Realization of a Limit Sketch
Johnathon Taylor
TL;DR
This paper constructs the universal realized limit sketch from a limit sketch, via a fibrant-replacement style process in the comma category $E/\mathsf{Cat}$, and demonstrates how this universal realization yields a strong factorization system through Kelly's Small Object Argument. It then elevates the construction to a functor $L$ that is left adjoint to the forgetful functor $U$ forgetting limits, i.e., $L\dashv U$. Using the universal property, models over a limit sketch are shown to be in one-to-one correspondence with models over its universal realization. The results provide a modern, universal realization of limit sketches with explicit adjunctions, enabling the transfer of model-theoretic data across base sketches.
Abstract
We provide a construction of the universal realized limit sketch from a limit sketch. Moreover, we show that the universal realization extends to a functor that is the left adjoint to the forgetful functor that forgets limits. We finish by showing that models over a limit sketch are in one-to-one correspondance to models over the universal realization.