Two-dimensional regularity and exactness

Article history: Received 19 April 2013 Received in revised form 7 August 2013 Available online 25 November 2013 Communicated by J. Adámek MSC: 18F10; 18A32; 18D05 We define notions of regularity and (Barr-)exactness for 2-categories. In fact, we define three notions of regularity and exactness, each based on one of the three canonical ways of factorising a functor in Cat: as (surjective on objects, injective on objects and fully faithful), as (bijective on objects, fully faithful), and as (bijective on objects and full, faithful). The correctness of our notions is justified using the theory of lex colimits [12] introduced by Lack and the second author. Along the way, we develop an abstract theory of regularity and exactness relative to a kernel–quotient factorisation, extending earlier work of Street and others [24,3]. © 2013 Elsevier B.V. All rights reserved.