Geodesic
Span composition using fake pullbacks
Theory and applications of categories, The Rosebrugh Festschrift, Tome 36 (2021), pp. 102-117.

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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.
Publié le :
Classification : 18A32, 18C05, 18D30, 18E13, 08A30, 20J99
Keywords: span, partial map, factorization system, relation, Puppe exact category 
@article{TAC_2021_36_a3,
     author = {Ross Street},
     title = {Span composition using fake pullbacks},
     journal = {Theory and applications of categories},
     pages = {102--117},
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     volume = {36},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2021_36_a3/}
}
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Ross Street. Span composition using fake pullbacks. Theory and applications of categories, The Rosebrugh Festschrift, Tome 36 (2021), pp. 102-117. http://geodesic.mathdoc.fr/item/TAC_2021_36_a3/