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Tim Campion
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I did this once with the category of schemes in response to this question, with help from Laurent Moret-Bailly. But then Zhen Lin Low pointed out there's an obvious concretizing functor. Maybe it wasn't so obvious until we were sure it was there, though. So I suppose this falls under the "useful heuristic" category. In practice, the Isbell-Freyd criterion translated the problem into something more concrete (pardon the pun!) which an algebraic geometer had a sense for how to answer. At the time, I didn't know enough algebraic geometry to answer this question on my own, so translating it into more geometric language which I could ask somebody else was an essential step for me.

It helped that, as Ivan Di Liberti points out in the comments, the criterion is especially simple in a finitely-complete category.

I did this once with the category of schemes in response to this question, with help from Laurent Moret-Bailly. But then Zhen Lin Low pointed out there's an obvious concretizing functor. Maybe it wasn't so obvious until we were sure it was there, though. So I suppose this falls under the "useful heuristic" category.

I did this once with the category of schemes in response to this question, with help from Laurent Moret-Bailly. But then Zhen Lin Low pointed out there's an obvious concretizing functor. Maybe it wasn't so obvious until we were sure it was there, though. So I suppose this falls under the "useful heuristic" category. In practice, the Isbell-Freyd criterion translated the problem into something more concrete (pardon the pun!) which an algebraic geometer had a sense for how to answer. At the time, I didn't know enough algebraic geometry to answer this question on my own, so translating it into more geometric language which I could ask somebody else was an essential step for me.

It helped that, as Ivan Di Liberti points out in the comments, the criterion is especially simple in a finitely-complete category.

Source Link
Tim Campion
  • 63.9k
  • 13
  • 143
  • 384

I did this once with the category of schemes in response to this question, with help from Laurent Moret-Bailly. But then Zhen Lin Low pointed out there's an obvious concretizing functor. Maybe it wasn't so obvious until we were sure it was there, though. So I suppose this falls under the "useful heuristic" category.