TY - JOUR

T1 - Enhanced 2-categories and limits for lax morphisms

AU - Lack, Stephen

AU - Shulman, Michael

PY - 2012/1/15

Y1 - 2012/1/15

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=80655124529&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2011.08.014

DO - 10.1016/j.aim.2011.08.014

M3 - Article

AN - SCOPUS:80655124529

SN - 0001-8708

VL - 229

SP - 294

EP - 356

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 1

ER -