Enhanced 2-categories and limits for lax morphisms

Details Enhanced 2-categories and limits for lax morphisms Enhanced 2-categories and limits for lax morphisms

2011-04-12

arxiv.org/abs/1104.2111v2

Advances in Mathematics 229 (1):294-356, 2012

Mathematics

Category Theory

77 pages; v2 minor changes only, to appear in Advances

Scientific paper

10.1016/j.aim.2011.08.014

We study limits in 2-categories whose objects are categories with extra structure and whose morphisms are functors preserving the structure only up to a coherent comparison map, which may or may not be required to be invertible. This is done using the framework of 2-monads. In order to characterize the limits which exist in this context, we need to consider also the functors which do strictly preserve the extra structure. We show how such a 2-category of weak morphisms which is "enhanced", by specifying which of these weak morphisms are actually strict, can be thought of as category enriched over a particular base cartesian closed category F. We give a complete characterization, in terms of F-enriched category theory, of the limits which exist in such 2-categories of categories with extra structure.

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