Dear Professor Michael Barr, I hope all is well. If I may, Grothendieck redefines subobject as: a subobject of A is not an object, but an object B, together with a monomorphism u: B --> A (https://www.math.mcgill.ca/barr/papers/gk.pdf, p. 2). Your time permitting, would you be kind enough to shed some light on Grothendieck's conceptual trajectory, along with whatever it is that might have made it indispensable, from the then subset is a set to his subset is a 1-1 function, a conceptual refinement, whose mathematical value was more than adequately explained in Professor F. William Lawvere's geometry of figures (Lawvere & Schanuel, Conceptual Mathematics, pp. 370-371, https://conceptualmathematics.wordpress.com/wp-content/uploads/2013/02/conce...), which can be traced all the way back to Cayley's representation, not to forget Yoneda embedding or Professor's F. William Lawvere's functorial semantics, along the way. If I am not mistaken, the above Tohoku paper you translated is the first instance of a subobject reincarnating as monomorphism. When viewed from the perspective of: a part of a whole is both itself and its relationship to the whole (Lawvere, Objective Logic, p. 53, https://zenodo.org/records/7059109), shouldn't Grothendieck's 'subobject is monomorphism' have been readily incorporated into math (without having to go through the trials and tribulations of having to fight against the accusations of circumlocution, etc.). I look forward to your corrections and clarifications! Thanking you, Yours truly, posina P.S. Rational passage between concepts (https://conceptualmathematics.wordpress.com/2023/02/17/rational-passage/) is my Year 2025 project :) So, I'd appreciate any pointers to go from one concept to another. I could be wrong, but broadly it seems, speaking of the passage between concepts (GRAPH, MONOID, FUNCTION, etc.), since concepts are objectified as categories (the theory of irreflexive directed graphs as a category consisting of two parallel maps, i.e., source, target: Arrows --> Dots), it seems a passage between concepts must necessarily be functorial (mapping of objects must be respectful of their mutual relations, composition, etc). 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>