Factorization systems for restriction categories (Q6593822)

Latent fibrations [\textit{R. Cockett} et al., Theory Appl. Categ. 36, 423--491 (2021; Zbl 1467.18009)] are the analogue of fibrations for restriction categories [\textit{J. R. B. Cockett} and \textit{S. Lack}, Theor. Comput. Sci. 270, No. 1--2, 223--259 (2002; Zbl 0988.18003); Theor. Comput. Sci. 294, No. 1--2, 61--102 (2003; Zbl 1023.18005); Math. Struct. Comput. Sci. 17, No. 4, 775--817 (2007; Zbl 1123.18003)]. The papers [\textit{J. R. B. Cockett} et al., Theory Appl. Categ. 26, 412--452 (2012; Zbl 1252.18003); Theory Appl. Categ. 26, 453--500 (2012; Zbl 1252.18004)] provided a source of examples of restriction categories with a latent factorization. The principal objective in this paper is to get an analogous result for latent fibrations to that of ordinary fibrations. As restriction categories are not self-dual, the notion of orthogonality for restriction categories takes a correspondlingly non-self-dual character. The main result of the paper (Theorem 4.5) claims that to have an \textsf{M}-category with an \textsf{M}-stable factorrization system is precisely to have a split restriction category with a latent factorization.