We give several reformulations of action representability of a category as well as action representability of its category of morphisms. In particular we show that for a semi-abelian category C, its category of morphisms is action representable if and only if the functor from the category of split extensions in C to C, sending a split extension to its kernel, is a prefibration. To obtain these reformulations we show that certain conditions are equivalent for right regular spans of categories.
Keywords: action representable, semi-abelian, split extension, normalizer, prefibration, regular span
2010 MSC: 18A05, 18A22, 18A25, 18A40, 18A99, 18B99, 18D99
Theory and Applications of Categories, Vol. 32, 2017, No. 43, pp 1501-1521.