Dear All, For some time, I have been staring at the basic relation of algebraic geometry (https://zenodo.org/records/7079058, p. 2), which is an adjoint relation between geometry and algebra: V^Yop <===> (V^Y)op which Professor F. William Lawvere was kind enough to suggest as a framework to abstract the mathematical content of the fundamental dialectic of philosophy: ([epistemology vs. ontology] vs. reality) I think compounding epistemology and ontology into which reality is resolved is a major outstanding scientific program. Surely, Newton would be happy, having emphasized synthesis after analysis. Professor F. William Lawvere referred to the above adjointness as Isbell conjugacy (http://www.tac.mta.ca/tac/reprints/articles/8/tr8.pdf, p. 17). Simply put, Isbell conjugacy is about getting to know a category V in terms of geometric Y-shaped figures vs. algebraic Y-valued properties (https://zenodo.org/records/7059109, p. 49). Going by our experience with sets, a single-element set 1 = {*} is adequate enough to completely characterize every set and to test for the equality of functions, but we need a two-element coadequate set 2 = {false, true} to tell apart elements of any domain set. Before long I can't help but wonder if the relationship between V^Aop <=?=> (V^C)op (where A and C are adequate and coadequate subcategories, respectively, of a category V) would be relatively more informative than the above Isbell conjugacy. Furthermore, in a CatList post (06 March 2009), Professor F. William Lawvere points out that Isbell conjugacy is a special case of the construction of the total category with two descriptions which objectify adjointness (unfortunately I couldn't find any mention of 'total category' in a quick search of his Functorial Semantics of Algebraic Theories (http://www.tac.mta.ca/tac/reprints/articles/5/tr5.pdf, which he cites). I look forward to your corrections and suggestions! Happy Bakrid :) Thanking you, Yours truly, posina You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g>