Span composition using fake pullbacks

Ross Street*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.’. 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.

Original languageEnglish
Pages (from-to)102-117
Number of pages16
JournalTheory and Applications of Categories
Issue number4
Publication statusPublished - 2021


  • span
  • partial map
  • factorization system
  • relation
  • Puppe exact category

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