2019/02/20 by Nelson Martins-Ferreira, Martins-Ferreira, Nelson
Mathematics · #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1902.07703
A classification theorem for three different sorts of Mal'tsev categories is\nproven. The theorem provides a classification for Mal'tsev category, naturally\nMalt'sev category, and weakly Mal'tsev category in terms of classifying classes\nof spans. The class of all spans characterizes naturally Mal'tsev categories.\nThe class of relations (i.e. jointly monomorphic spans) characterizes Mal'tsev\ncategories. The class of strong relations (i.e. jointly strongly monomorphic\nspans) characterizes weakly Mal'tsev categories. The result is based on the\nuniqueness of internal categorical structures such as internal category and\ninternal groupoid (Lawvere condition). The uniqueness of these structures is\nviewed as a property on their underlying reflexive graphs, restricted to the\nclassifying spans. The class of classifying spans is combined, via a new\ncompatibility condition, with split squares. This is analogous to orthogonality\nbetween spans and cospans. The result is a general classifying scheme which\ncovers the main characterizations for Mal'tsev like categories. The class of\npositive relations has recently been shown to characterize Goursat categories\nand hence it is a new example that fits in this general scheme.\n