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5.3 Thehomogeneoussplitextensionclassifier . . . . . . . . . . . . 52 5.4 Centralizers of Schreier reflexive relations . . . . . . . . . . . . 54 5.5 Thecaseofcommutativemonoids ................ 55
6 Semirings
6.1 Properties of Schreier split epimorphisms of semirings
6.2 ThefibrationofSchreierpointsinsemirings . . . . . .
6.3 Schreierinternalstructuresinsemirings . . . . . . . .
6.4 Mal’tsev aspects of SRng related to Schreier split epimorphisms 69
6.5 Schreieraccessibility ........................ 70
6.6 Centralizers of Schreier reflexive relations . . . . . . . . . . . . 76
6.7 SemidirectproductsinSRng ................... 77
7 Special Schreier and special homogeneous surjections 79
7.1 SpecialSchreiersurjectionsinMon. . . . . . . . . . . . . . . . 79
7.2 SpecialSchreiershortfivelemma ................. 81
7.3 Special Schreier extensions with abelian kernel . . . . . . . . . 83
7.4 ThedirectionfunctorinMon................... 86
7.5 BaersumsinMon ......................... 89
7.6 SpecialSchreiersurjectionsinSRng . . . . . . . . . . . . . . . 90
7.7 Special Schreier extensions with trivial kernel . . . . . . . . . . 91
7.8 BaersumsinSRng......................... 95
8 Conclusion 97
8.1 Partialprotomodularity ...................... 98 8.2 BacktoC′-unitalcategories.................... 99 8.3 Leftexactconservativeforgetfulfunctors. . . . . . . . . . . . . 100
9 Appendix 101
9.1 Pullbacks .............................. 101 9.1.1 Leftexactconservativefunctors. . . . . . . . . . . . . . 102
9.2 Regularepimorphisms ....................... 102
9.3 Fibrations.............................. 103
9.4 Simplicialobjects.......................... 104
9.5 Internal preorders, categories and monoids . . . . . . . . . . . . 106
9.6 Internal groupoids, equivalence relations, groups . . . . . . . . 108
9.7 Theshiftfunctor .......................... 110
Bibliography 111 Index 115
..... 59 . . . . . 61 . . . . . 64
57
