On the axioms for adhesive and quasiadhesive categories (Q2884483)

The manuscript continues to develop the theory of \textit{adhesive categories} introduced by the second author and \textit{P.~Sobo\-ciński} [``Adhesive categories'', Lect. Notes Comput. Sci. 2987, 273--288 (2004; Zbl 1126.68447)], following the earlier idea of extensive categories of \textit{A. Carboni}, \textit{S. Lack} and \textit{R. F. C. Walters} [``Introduction to extensive and distributive categories'', J. Pure Appl. Algebra 84, No. 2, 145--158 (1993; Zbl 0784.18001)]. More precisely, while the latter require coproducts to be stable under pullbacks, the former substitute coproducts by pushouts along monomorphisms. In particular, the second author and \textit{P. Sobociński} [``Toposes are adhesive'', Lect. Notes Comp. Sci. 4178, 184--198 (2006; Zbl 1157.18303)] show that every topos is adhesive. Moreover, the second author [``An embedding theorem for adhesive categories'', Theory Appl. Categ. 25, 180--188 (2011; Zbl 1247.18002)] proves that adhesive categories are precisely the ones, which admit a structure-preserving (called \textit{adhesive}) full embedding into topoi, thereby showing that adhesive categories capture the relationships between pushouts along monomorphisms and pullbacks in a topos.NEWLINENEWLINETo capture the respective relationships (for regular morphisms) in a quasitopos, the second author and \textit{P. Sobociński} [``Adhesive and quasiadhesive categories'', Theor. Inform. Appl. 39, No. 3, 511--545 (2005; Zbl 1078.18010)] presented the concept of quasiadhesive category (substitute monomorphisms with regular ones). Surprisingly enough, \textit{P. T. Johnstone}, \textit{S. Lack} and \textit{P. Sobociński} [``Quasitoposes, quasiadhesive categories and Artin glueing'', Lect. Notes Comp. Sci. 4624, 312--326 (2007; Zbl 1214.18003)] obtained that not every quasitopos is a quasiadhesive category. It is the purpose of the current manuscript, to provide a further relaxation of the requirements of adhesive categories, in order to restore the above-mentioned result of the second author in the setting of quasitopoi. More precisely, the paper now calls quasiadhesive categories rm-adhesive, and, moreover, introduces rm-quasiadhesive ones as the categories, having pushouts along regular monomorphisms, which are stable (see page 29 of the paper) and are pullbacks (i.e., are not necessarily \textit{van Kampen} as in the quasiadhesive framework). The authors then consider characterizations of adhesive (Theorem 3.2 on page 39), rm-adhesive (Theorem 3.3 on pages 39--40) and rm-quasiadhesive categories in terms of adhesive morphisms, introduced and studied in Section 2 of the manuscript. Additionally, they show that rm- (resp. quasi-)adhesive categories are precisely the ones, which admit a structure-preserving full embedding into topoi (resp. quasitopoi).NEWLINENEWLINEThe manuscript is well-written (almost no typos), contains a lot of easy-to-follow proofs, but requires from the reader a significant amount of knowledge on its topic and its many related notions, which are not provided in the paper.