Gray, J. R. A.Martins-Ferreira, Nelson2023-05-292023-05-292018Gray, J. R. A., & Martins-Ferreira, N. (2018). New exactness conditions involving split cubes in protomodular categories. Theory and Applications of Categories, 33(33)1201-561Xhttp://hdl.handle.net/10400.8/8535This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Number 112086) and supported by the Fundação para a Ciência e a Tecnologia (FCT) and Centro2020 through the Project references: UID/Multi/04044/2013; PAMI - ROTEIRO/0328/2013 (N 022158); Next.parts (17963); and also by CDRSP and ESTG from the Polytechnic Institute of Leiria.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.engProtomodular categorySplit cubeVan Kampen TheoremNon-pointed coincidence of commutatorsNormality inside unionsPartially multiplicative graphDescentjournal article