Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 24 1 2026 01 01 Category of $\mathcal{M}$-relations as a quotient of the span category 91 103 105134 10.48308/cgasa.2024.237620.1526 EN Naser Hosseini Department of Pure Mathematics, Faculty of Math and Computers, Shahid Bahonar University of Kerman, Kerman, Iran 0000-0001-8420-2061 Journal Article 2024 11 19 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. https://cgasa.sbu.ac.ir/article_105134_972ac73a3daf5f4adf01d720e40db91e.pdf