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Some remarks on 2-categorical algebra. I. (English) Zbl 0687.18006

This paper presents a development of the elementary properties of some of the 2-categorical limits which are uniquely determined only up to equivalence, and some of their exactness properties. To provide for a wider audience, abstract 2-categories are not considered, just certain subcategories of Cat called kits. The motivating example is the kit of categories which admits pullbacks, and pullback preserving functors. Some of the results here are folklore, but unpublished. In fact, the reviewer (Sydney Category Seminar, 1979) has presented (unpublished) a notion of regular 2-category closely related to the author’s regular kits; for the reviewer the “surjections” were taken to form a “2-factorization” system with the conservative arrows (defined representably) and proved to be “quotients”. Other exactness conditions on 2-categories have been considered by the reviewer in several articles [e.g., “Fibrations in bicategories”, Cah. Topologie Géom. Différ. 21, 111-159 (1980; Zbl 0436.18005); Correction, ibid. 28, 53-56 (1987; Zbl 0622.18005)].
Reviewer: R.H.Street

MSC:

18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18A35 Categories admitting limits (complete categories), functors preserving limits, completions
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