The following: "In 1981 I [Lawvere] visited him [Grothendieck] in his stone hut, in the middle of a lavender field in the south of France [...] subobject classifier which, as he said, is one of the few ingredients of topos theory that he had not foreseen. Later in his work on homotopy he kindly referred to that object as the "Lawvere element"." --Picado, J. (2007) An interview with F. William Lawvere, Bulletin of the International Center for Mathematics may be of interest in the context of: On Tue, Nov 18, 2014 at 4:02 PM, Michael Barr <barr@math.mcgill.ca> wrote:

So at the end of his talk I asked him if he was familiar with the Lawvere-Tierney axioms for a topos, which looked a lot more like set theory than the Giraud axioms. He said that >he didn't know what they were and asked me to come to the board and explain them. Which I did (I added complete and cocomplete to recover the original definition that G. had >used). He agreed that looked a lot more like set theory.

Bill can correct me if I am wrong, but I recall that at the Nice meeting a year earlier, Bill had tried to tell him about elementary toposes, but G. wasn't interested in anything >mathematical.

Michael

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