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Span composition using fake pullbacks

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Ross Street

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.

Published 2021-03-01.

http://www.tac.mta.ca/tac/volumes/36/4/36-04.pdf

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