size structure

A **size structure** is a class of discrete morphisms, called **discrete-small** or just **small**, satisfying certain closure axioms that directly generalize the axioms for classes of small maps in algebraic set theory.

It improves on the notion of classifying discrete opfibration by giving a notion of smallness for maps that are not necessarily opfibrations. In particular, we can say that an object $A$ is *locally small* if $A^{\mathbf{2}} \to A\times A$ is small, though it is definitely not an opfibration. We may ask that the small maps in a size structure which *are* opfibrations have a classifier, but it is not clear whether knowing the small opfibrations suffices to determine all the small maps. It is possible, however, that by using the 2-internal logic, from a classifying discrete opfibration satisfying suitable axioms we could canonically define a general notion of “small” and construct a size structure.

(to be written…)

Revised on June 12, 2012 11:10:00
by Andrew Stacey?
(129.241.15.200)