Protoreflections, relational algebras and topology.
Stephen H. Kamnitzer · Open University of Cape Town (University of Cape Town) · 1974
I. RELATIONS 1.N REGULAR CATEGORIES Relations in categories have bsen ~iscussed by various ,• authors, among them Mac.Lane• [19()1], Klein [1970], Gri I let [1971], Fay [1973] ~nd Mei sen [1973a, b].Mac Lan~ is concern~d with abel ian categories, Klein and Mei sen with relati6ns in a category with a given bicategory structure, while Grili'.etas well.as Meisen d•iscuss relations in regular categorles.In this work .wewil I fol low Gri I let by restricting our attention to r~lations .inregular categories.• This rest~iction appears to tie justified for the fol lo.wing reasons: The axioms• for a regular category are weak enough to include as examples (I) sets, (2) abel ian categories, (3) varieties of universal algebras, (4) ~lgebras of monads in regular categories, (5) functor categories of the form Funct (~,A) with~ small and ~regular, (6) every partially ordered s~t whtch has inf ima of finite subsets, considered as a category, (7) compact Hausdorff spaces and (8) compact Hausdorff zero d~mensional ~paces~The axioms are also strong' enough to give relati.ons in regulc;:ir categories most of the properties required for the proof of the 'topological type' results which we present in later sections .. Our starting point is thus Gri I let's paper.•In this section we summarise the results of Gril let which we wi I I use later on and a I so derive a number of.add it i ona I properties of re-I at i ans.diagrams /Bl~ commute.