I recently came across a proof which I'd never seen before that the associated sheaf functor for a Lawvere--Tierney local operator in a topos preserves finite limits. I'd be interested to know if anyone on this list has seen it. As is well known, the fact that the subcategory of sheaves is an exponential ideal easily implies that the reflector L: E --> sh_j(E) preserves finite products (4.3.1 -- all numbered references are to Part A of `Sketches of an Elephant'). So the problem is to prove that it preserves equalizers (at least of coreflexive pairs). There is a slick proof due to Peter Freyd (4.4.7), a more pedestrian proof due to me (4.4.6), and doubtless many others. But here is the `new' one: Suppose E --> A is the equalizer of a coreflexive pair A ==> B (I use ==> to denote a parallel pair of arrows). Pare's proof that E^op is monadic over E (2.2.7) shows that PB ==> PA --> PE has the structure of a split coequalizer; and an easy modification of his argument shows that P_jB ==> P_jA --> P_jE is also a split coequalizer, where P_jA denotes (\Omega_j)^A. But it also follows from 4.3.1 that P_jA is canonically isomorphic to P_j(LA), so we have a split coequalizer P_j(LB) ==> P_j(LA) --> P_j(LE). And as a functor sh_j(E)^op --> sh_j(E), P_j is monadic, so by the Precise Monadicity Theorem it creates coequalizers of P_j-split pairs. Hence LE --> LA ==> LB is an equalizer diagram in sh_j(E). Peter Johnstone 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/groups/groupsubscription?source=EscalatedMessage&action=files&smtp=categories%40mq.edu.au&bO=true&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Leave group<https://outlook.office365.com/groups/groupsubscription?source=EscalatedMessage&action=leave&smtp=categories%40mq.edu.au&bO=true&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g>