Has the Isbell–Freyd criterion ever been used to check that a category is concretisable?
$begingroup$
Isbell gave, in Two set-theoretic theorems in categories (1964), a necessary criterion for categories to be concretisable (i.e. to admit some faithful functor into sets). Freyd, in Concreteness (1973), showed that Isbell’s criterion is also sufficient.
My question is: Has anyone ever used Isbell’s criterion to check that a category is concretisable?
I’m interested not only in seeing the theorem is formally invoked in print, to show some category is concretisable — though of course that would be a perfect answer, if it’s happened. What I’m also interested in, and suspect is more likely to have occurred, is if anyone’s found the criterion useful as a heuristic for checking whether a category is concretisable, in a situation where one wants it to be concrete but finding a suitable functor is not totally trivial. (I’m imagining a situation similar to the adjoint functor theorems: they give very useful quick heuristics for guessing whether adjoints exist, but if they suggest an adjoint does exist, usually there’s an explicit construction as well, so they’re used as heuristics much more often than they’re formally invoked in print.)
What I’m not so interested in is uses of the criterion to confirm that an expected non-concretisable category is indeed non-concretisable — I’m after cases where it’s used in expectation of a positive answer.
ct.category-theory
asked 9 hours ago
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,67413869
$endgroup$
add a comment |
$begingroup$
Isbell gave, in Two set-theoretic theorems in categories (1964), a necessary criterion for categories to be concretisable (i.e. to admit some faithful functor into sets). Freyd, in Concreteness (1973), showed that Isbell’s criterion is also sufficient.
My question is: Has anyone ever used Isbell’s criterion to check that a category is concretisable?
I’m interested not only in seeing the theorem is formally invoked in print, to show some category is concretisable — though of course that would be a perfect answer, if it’s happened. What I’m also interested in, and suspect is more likely to have occurred, is if anyone’s found the criterion useful as a heuristic for checking whether a category is concretisable, in a situation where one wants it to be concrete but finding a suitable functor is not totally trivial. (I’m imagining a situation similar to the adjoint functor theorems: they give very useful quick heuristics for guessing whether adjoints exist, but if they suggest an adjoint does exist, usually there’s an explicit construction as well, so they’re used as heuristics much more often than they’re formally invoked in print.)
What I’m not so interested in is uses of the criterion to confirm that an expected non-concretisable category is indeed non-concretisable — I’m after cases where it’s used in expectation of a positive answer.
ct.category-theory
asked 9 hours ago
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,67413869
$endgroup$
4
$begingroup$
I really like this question, let me just say that when the category has finite limits the criterion simplifies to "the category is regular-well-powered".
$endgroup$
– Ivan Di Liberti
8 hours ago
1
$begingroup$
I'm intrigued by your comments about the adjoint functor theorems. I would have said that they are invoked quite often in print, especially when dealing with locally presentable categories whose adjoint functor theorem is particularly simple (any cocontinuous functor has a right adjoint, and any continuous accessible functor has a left adjoint).
$endgroup$
– Mike Shulman
7 hours ago
add a comment |
$begingroup$
Isbell gave, in Two set-theoretic theorems in categories (1964), a necessary criterion for categories to be concretisable (i.e. to admit some faithful functor into sets). Freyd, in Concreteness (1973), showed that Isbell’s criterion is also sufficient.
My question is: Has anyone ever used Isbell’s criterion to check that a category is concretisable?
I’m interested not only in seeing the theorem is formally invoked in print, to show some category is concretisable — though of course that would be a perfect answer, if it’s happened. What I’m also interested in, and suspect is more likely to have occurred, is if anyone’s found the criterion useful as a heuristic for checking whether a category is concretisable, in a situation where one wants it to be concrete but finding a suitable functor is not totally trivial. (I’m imagining a situation similar to the adjoint functor theorems: they give very useful quick heuristics for guessing whether adjoints exist, but if they suggest an adjoint does exist, usually there’s an explicit construction as well, so they’re used as heuristics much more often than they’re formally invoked in print.)
What I’m not so interested in is uses of the criterion to confirm that an expected non-concretisable category is indeed non-concretisable — I’m after cases where it’s used in expectation of a positive answer.
ct.category-theory
asked 9 hours ago
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,67413869
$endgroup$
Isbell gave, in Two set-theoretic theorems in categories (1964), a necessary criterion for categories to be concretisable (i.e. to admit some faithful functor into sets). Freyd, in Concreteness (1973), showed that Isbell’s criterion is also sufficient.
My question is: Has anyone ever used Isbell’s criterion to check that a category is concretisable?
I’m interested not only in seeing the theorem is formally invoked in print, to show some category is concretisable — though of course that would be a perfect answer, if it’s happened. What I’m also interested in, and suspect is more likely to have occurred, is if anyone’s found the criterion useful as a heuristic for checking whether a category is concretisable, in a situation where one wants it to be concrete but finding a suitable functor is not totally trivial. (I’m imagining a situation similar to the adjoint functor theorems: they give very useful quick heuristics for guessing whether adjoints exist, but if they suggest an adjoint does exist, usually there’s an explicit construction as well, so they’re used as heuristics much more often than they’re formally invoked in print.)
What I’m not so interested in is uses of the criterion to confirm that an expected non-concretisable category is indeed non-concretisable — I’m after cases where it’s used in expectation of a positive answer.
ct.category-theory
ct.category-theory
asked 9 hours ago
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,67413869
asked 9 hours ago
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,67413869
asked 9 hours ago
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,67413869
asked 9 hours ago
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,67413869
asked 9 hours ago
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,67413869
8,67413869
4
$begingroup$
I really like this question, let me just say that when the category has finite limits the criterion simplifies to "the category is regular-well-powered".
$endgroup$
– Ivan Di Liberti
8 hours ago
1
$begingroup$
I'm intrigued by your comments about the adjoint functor theorems. I would have said that they are invoked quite often in print, especially when dealing with locally presentable categories whose adjoint functor theorem is particularly simple (any cocontinuous functor has a right adjoint, and any continuous accessible functor has a left adjoint).
$endgroup$
– Mike Shulman
7 hours ago
add a comment |
4
$begingroup$
I really like this question, let me just say that when the category has finite limits the criterion simplifies to "the category is regular-well-powered".
$endgroup$
– Ivan Di Liberti
8 hours ago
1
$begingroup$
I'm intrigued by your comments about the adjoint functor theorems. I would have said that they are invoked quite often in print, especially when dealing with locally presentable categories whose adjoint functor theorem is particularly simple (any cocontinuous functor has a right adjoint, and any continuous accessible functor has a left adjoint).
$endgroup$
– Mike Shulman
7 hours ago
4
4
$begingroup$
I really like this question, let me just say that when the category has finite limits the criterion simplifies to "the category is regular-well-powered".
$endgroup$
– Ivan Di Liberti
8 hours ago
$begingroup$
I really like this question, let me just say that when the category has finite limits the criterion simplifies to "the category is regular-well-powered".
$endgroup$
– Ivan Di Liberti
8 hours ago
1
1
$begingroup$
I'm intrigued by your comments about the adjoint functor theorems. I would have said that they are invoked quite often in print, especially when dealing with locally presentable categories whose adjoint functor theorem is particularly simple (any cocontinuous functor has a right adjoint, and any continuous accessible functor has a left adjoint).
$endgroup$
– Mike Shulman
7 hours ago
$begingroup$
I'm intrigued by your comments about the adjoint functor theorems. I would have said that they are invoked quite often in print, especially when dealing with locally presentable categories whose adjoint functor theorem is particularly simple (any cocontinuous functor has a right adjoint, and any continuous accessible functor has a left adjoint).
$endgroup$
– Mike Shulman
7 hours ago
add a comment |
1 Answer
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votes
$begingroup$
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.
answered 8 hours ago
Tim CampionTim Campion
14.1k355125
$endgroup$
add a comment |
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$begingroup$
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.
answered 8 hours ago
Tim CampionTim Campion
14.1k355125
$endgroup$
add a comment |
$begingroup$
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.
answered 8 hours ago
Tim CampionTim Campion
14.1k355125
$endgroup$
add a comment |
$begingroup$
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.
answered 8 hours ago
Tim CampionTim Campion
14.1k355125
$endgroup$
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.
answered 8 hours ago
Tim CampionTim Campion
14.1k355125
edited 8 hours ago
answered 8 hours ago
Tim CampionTim Campion
14.1k355125
answered 8 hours ago
Tim CampionTim Campion
14.1k355125
answered 8 hours ago
Tim CampionTim Campion
14.1k355125
14.1k355125
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$begingroup$
I really like this question, let me just say that when the category has finite limits the criterion simplifies to "the category is regular-well-powered".
$endgroup$
– Ivan Di Liberti
8 hours ago
1
$begingroup$
I'm intrigued by your comments about the adjoint functor theorems. I would have said that they are invoked quite often in print, especially when dealing with locally presentable categories whose adjoint functor theorem is particularly simple (any cocontinuous functor has a right adjoint, and any continuous accessible functor has a left adjoint).
$endgroup$
– Mike Shulman
7 hours ago