Quotients of span categories that are allegories and the representation of regular categories
Abstract
We consider the ordinary category Span(C) of (isomorphism classes of) spans of morphisms in a category C with finite limits as needed, composed horizontally via pullback, and give a general criterion for a quotient of Span(C) to be an allegory. In particular, when C carries a pullbackstable, but not necessarily proper, (E, M)factorization system, we establish a quotient category Span_E(C) that is isomorphic to the category Rel_M(C) of Mrelations in C, and show that it is a (unitary and tabular) allegory precisely when M is a class of monomorphisms in C. Without this restriction, one can still find a least pullbackstable and compositionclosed class E. containing E such that Span_E.(C) is a unitary and tabular allegory. In this way one obtains a left adjoint to the 2functor that assigns to every unitary and tabular allegory the regular category of its Lawverian maps. With the FreydScedrov Representation Theorem for regular categories, we conclude that every finitely complete category with a stable factorization system has a reflection into the huge 2category of all regular categories.
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 arXiv:
 arXiv:2112.04599
 Bibcode:
 2021arXiv211204599N
 Keywords:

 Mathematics  Category Theory;
 Computer Science  Logic in Computer Science;
 18B10;
 18A32;
 18E08