Cospan construction of the graph category of Borisov and Manin (Q1660526)

The objects of the graph category defined by \textit{D. V. Borisov} and \textit{Y. I. Manin} [Prog. Math. 265, 247--308 (2008; Zbl 1180.18004)] are graphs with open-ended edges (also called tails or legs); there are two families of special morphisms, graftings and compressions, and it is proved that any morphism can be essentially uniquely factorized as a grafting followed by a compression. In this paper, the same category is studied in the converse way, starting from the unique decomposition of morphisms, with the help of distributive laws of \textit{R. Rosebrugh} and \textit{R. J. Wood} [J. Pure Appl. Algebra 175, No. 1--3, 327--353 (2002; Zbl 1023.18002)]: considering two classes of morphisms \(G\) and \(C\) on a common objects saw, a composition is defined on the pairs \((c,g)\), with \(c\in C\) and \(g\in G\), with the help of the distributive law. This formalism is used for graphs by \textit{A. Joyal} and the author [Electron. Notes Theor. Comput. Sci. 270, No. 2, 105--113 (2011; Zbl 1348.81242)]: their reduced covers are essentially Borisov and Manin's graftings, and their refinement morphisms are the opposite of compressions. Moreover, the distributive law is given by a pushout. Finally, it is obtained that the Borisov and Manin's morphisms are seen as cospans of reduced covers and refinement morphisms.