• DOI:10.1007/s10485-022-09687-9
• Corpus ID: 245006296

Quotients of span categories that are allegories and the representation of regular categories

@article{Hosseini2022QuotientsOS,
  title={Quotients of span categories that are allegories and the representation of regular categories},
  author={S. N. Hosseini and Amir R. Shir Ali Nasab and Walter Tholen and Leila Yeganeh},
  journal={ArXiv},
  year={2022},
  volume={abs/2112.04599}
}

• S. N. Hosseini, Amir R. Shir Ali Nasab, +1 author L. Yeganeh
• Published 8 December 2021
• Mathematics
• ArXiv

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 pullback-stable, but not necessarily proper, (E ,M)factorization system, we establish a quotient category SpanE(C) that is isomorphic to the category RelM(C) ofM-relations in C, and show that it is a (unitary and tabular) allegory precisely whenM is a class of monomorphisms in C. Without this restriction, one can still find a least pullback-stable and composition-closed class E• containing E such that SpanE•(C) is a unitary and tabular allegory. In this way one obtains a left adjoint to the 2-functor that assigns to every unitary and tabular allegory the regular category of its Lawverian maps. With the Freyd-Scedrov Representation Theorem for regular categories, we conclude that every finitely complete category with a stable factorization system has a reflection into the huge 2-category of all regular categories.