Pseudo-algebraic 2-categories of structured categories

* Abstract:*

The 2-category of small pretoposes, functors preserving the
pretopos operations up to isomorphism, and arbitrary natural
transformations is a central example of what I call a pseudoalgebraic
2-category. The latter is defined as the 2-category
of models in the 2-category of small categories, of a small
finite-pseudo-limit sketch -- subject to an important
restriction given below --, with 1-cells pseudo-natural
transformations, and arbitrary modifications as 2-cells. The
restriction, roughly speaking, is the exclusion of specifying
two different pairs of 1-cells in the sketch resulting in the
same composite. The main result reported on is that a pseudoalgebraic
2-category has an underlying category that is
dualregular, that is, equivalent to the category of regular
functors from a small regular category to the category of
sets; in particular, the underlying category is accessible and
has filtered colimits. The present work uses the author's
earlier work on anafunctors and that on generalized sketches.
The author wishes to thank John Bourke for his interest in
this work, and for alerting the author to the fact that
without the restriction mentioned above, the main result
mentioned above would be false.