#
Categories of weak fractions

##
Pierre-Alain Jacqmin

Given a set $\Sigma$ of morphisms in a category C, we construct a
functor F which sends elements of
$\Sigma$ to split monomorphisms. Moreover, we prove that F
is weakly universal with that property when considered in the world of
locally posetal 2-categories. Besides, we also use locally posetal
2-categories in order to construct weak left adjoints to those
functors for which any object in the codomain admits a weak reflection.
We then apply these two results in order to restate the Injective
Subcategory Problem for $\Sigma$ into the existence of some kind of
weak right adjoint for F.

Keywords:
split monomorphism, locally posetal 2-category, category of fractions,
weak reflection, weak adjoint, injective subcategory problem

2010 MSC:
55P60, 18A40, 18G05 (primary); 18A20, 18D05, 18B35, 18A10
(secondary)

*Theory and Applications of Categories,*
Vol. 34, 2019,
No. 46, pp 1526-1551.

Published 2019-12-18.

http://www.tac.mta.ca/tac/volumes/34/46/34-46.pdf

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