Category of \mathcal{M}-relations as a quotient of the span category

We introduce \mathcal{M}-spans for a class \mathcal{M} of morphisms in a category \mathcal{C}. Using the equivalence class of \mathcal{M}-spans under a given equivalence relation, we give the notion of an \mathcal{M}-relation in \mathcal{C}. We first show under what conditions, \mathcal{C}-objects together with \mathcal{M}-relations form a category, called the category of \mathcal{M}-relations and we construct a quotient of the span category as a byproduct. Then we investigate the connection between \mathcal{M}-relation categories and quotient span categories. We establish when a category of \mathcal{M}-relations is isomorphic to a quotient span category. Finally several illustrative examples are given.

کلیدواژه ها:

\mathcal{M}-relation ، (quotient of) \mathcal{M}-span ، (compatible) equivalence relation ، isomorphism of categories