The construction of a category of spans can be made in some categories A which do not have pullbacks in the traditional sense. The PROP for monoids is a good example of such an A. The 2012 book concerning homological algebra by Marco Grandis gives the proof of associativity of relations in a Puppe-exact category based on a 1967 paper of M.S. Calenko. The proof here is a restructuring of that proof in the spirit of the first sentence of this Abstract. We observe that these relations are spans of EM-spans and that EM-spans admit fake pullbacks so that spans of EM-spans compose. Our setting is more general than Puppe-exact categories. We mention the formalism of distributive laws which, in a generalized form, would cover our setting.
Keywords: span, partial map, factorization system, relation, Puppe exact category
2020 MSC: 18A32, 18C05, 18D30, 18E13, 08A30, 20J99
Theory and Applications of Categories, Vol. 36, 2021, No. 4, pp 102-117.