We introduce and compare several new exactness conditions involving what we call split cubes. These conditions are motivated by their special cases, some which become familiar, in the pointed context, once we reformulate them with split cubes replaced with split extensions.
Keywords: protomodular category, split cube, van Kampen Theorem, non-pointed coincidence of commutators, normality inside unions, partially multiplicative graph, descent
2010 MSC: 18A10, 18A25, 18A30
Theory and Applications of Categories, Vol. 33, 2018, No. 33, pp 1031-1058.