\tableofcontents ## Idea In [[class theory]], a universal class is a "[[class]] of all [[sets]]". ## Definition ### In material class theory to be defined... ### In structural class theory A **universal class** is a [[class]] $U$ such that for all classes $C$, there is a [[monic]] class [[map]] $U \hookrightarrow C$. There should also be a definition from a hypothetical "[[division allegory]] with class structure" rather than a [[category with class structure]]. ## See also * [[class theory]] * [[category with class structure]] * [[universe]] ## References * [[Steve Awodey]]. *Notes on algebraic set theory*, Notes for lectures given at the Summer School on Topos Theory, Haute-Bodeux, Belgium. May 29 to June 5, 2005. Carnegie Mellon University Technical Report No. CMU-PHIL-170. June 2005. ([pdf](https://www.phil.cmu.edu/projects/ast/Papers/bnotes.pdf)) [[!redirects class of all sets]]