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What is the geometric significance of fibered category theory in topos theory?
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Often in topos theory, one starts with a geometric morphism $f: mathcal Y to mathcal X$, but quickly passes to the Grothendieck fibration $U_f: mathcal Y downarrow f^ast to mathcal X$, which is "$mathcal Y$ regarded as an $mathcal X$-topos". This maneuver takes one outside of the category of toposes and geometric morphisms, and thus tends to remove me from thinking of toposes geometrically, as generalized spaces.
It seems there should be some geometric interpretation lurking here -- after all, Grothendieck fibrations are "categories varying over a base", and a topos is "a base that can be varied over". But although $mathcal Y downarrow f^ast$ is a topos, the fibration $U_f: mathcal Y downarrow f^ast to mathcal X$ is not (the direct image of) a geometric morphism! In fact it has a right adjoint $F^ast$ which has a further right adjoint $F_ast: mathcal Y downarrow f^ast to mathcal X$. We in fact have a totally connected geometric morphism $F: mathcal Y downarrow f^ast to mathcal X$.
Moreover, the fibers of $U_f$ are toposes, but the reindexing functors are most naturally viewed as the inverse images of étale geometric morphisms. So we don't naturally have a "topos fibered in toposes" it seems.
For one more perspective, $mathcal Y downarrow f^ast$ is a cocomma object in the 2-category of toposes. But again, the functor $U_f$ doesn't even live in the category of toposes.
I'm not really sure what to make of this. So here are some
Questions: Let $f: mathcal Y to mathcal X$ be a geometric morphism.
Is there a geometric interpretation of the fibration $U_f: mathcal Y downarrow f^ast to mathcal X$?
Does the totally connected geometric morphism $F: mathcal Y downarrow f^ast to mathcal X$ enjoy some sort of universal property with respect to $f$? (for example, does the inclusion of the category of toposes and totally connected geometric morphisms into toposes and all geometric morphisms have a left adjoint which is being applied here?)
Is there a fibration in the 2-category of toposes lurking around here somewhere (see the Elephant B4.4 for some discussion of these)?
Is there a general geometric interpretation of an opfibration over a topos whose fibers are toposes, with étale geometric morphisms for reindexing?
Addendum: Here are a few places where "indexed" notions come up where the viewpoint that "everything is indexed over the codomain" seems kind of inflexible to me:
A proper geometric morphism $f: mathcal Y to mathcal X$ is one such that $f_ast$ preserves $mathcal X$-indexed filtered colimits of subterminal objects. It's a bit awkward, for example, to even discuss the composition of proper morphisms if you're tied to the view that $mathcal Y$ is internal to $mathcal X$.
A separated geometric morphism is one such that the diagonal is proper; in particular, a Hausdorff topos is one such that $mathcal X to mathcal X times mathcal X$ is proper. It seems very unnatural to me in this context to think of $mathcal X$ as being primarily an object "internal to $mathcal X times mathcal X$".
Locally connected geometric morphisms also use indexing in one form of their definition. So do local geometric morphisms.
Let me also point out there there is at least one place in Sketches of an Elephant where $mathcal Y downarrow f^ast$ plays a role qua topos -- in Ch C3.6 on local geometric morphisms, where it's referred to as a "scone". See in particular Example 3.6.3(f), Lemma 3.6.4, and Corollary 3.6.13.
ct.category-theory topos-theory fibered-categories
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Tim Campion
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Often in topos theory, one starts with a geometric morphism $f: mathcal Y to mathcal X$, but quickly passes to the Grothendieck fibration $U_f: mathcal Y downarrow f^ast to mathcal X$, which is "$mathcal Y$ regarded as an $mathcal X$-topos". This maneuver takes one outside of the category of toposes and geometric morphisms, and thus tends to remove me from thinking of toposes geometrically, as generalized spaces.
It seems there should be some geometric interpretation lurking here -- after all, Grothendieck fibrations are "categories varying over a base", and a topos is "a base that can be varied over". But although $mathcal Y downarrow f^ast$ is a topos, the fibration $U_f: mathcal Y downarrow f^ast to mathcal X$ is not (the direct image of) a geometric morphism! In fact it has a right adjoint $F^ast$ which has a further right adjoint $F_ast: mathcal Y downarrow f^ast to mathcal X$. We in fact have a totally connected geometric morphism $F: mathcal Y downarrow f^ast to mathcal X$.
Moreover, the fibers of $U_f$ are toposes, but the reindexing functors are most naturally viewed as the inverse images of étale geometric morphisms. So we don't naturally have a "topos fibered in toposes" it seems.
For one more perspective, $mathcal Y downarrow f^ast$ is a cocomma object in the 2-category of toposes. But again, the functor $U_f$ doesn't even live in the category of toposes.
I'm not really sure what to make of this. So here are some
Questions: Let $f: mathcal Y to mathcal X$ be a geometric morphism.
Is there a geometric interpretation of the fibration $U_f: mathcal Y downarrow f^ast to mathcal X$?
Does the totally connected geometric morphism $F: mathcal Y downarrow f^ast to mathcal X$ enjoy some sort of universal property with respect to $f$? (for example, does the inclusion of the category of toposes and totally connected geometric morphisms into toposes and all geometric morphisms have a left adjoint which is being applied here?)
Is there a fibration in the 2-category of toposes lurking around here somewhere (see the Elephant B4.4 for some discussion of these)?
Is there a general geometric interpretation of an opfibration over a topos whose fibers are toposes, with étale geometric morphisms for reindexing?
Addendum: Here are a few places where "indexed" notions come up where the viewpoint that "everything is indexed over the codomain" seems kind of inflexible to me:
A proper geometric morphism $f: mathcal Y to mathcal X$ is one such that $f_ast$ preserves $mathcal X$-indexed filtered colimits of subterminal objects. It's a bit awkward, for example, to even discuss the composition of proper morphisms if you're tied to the view that $mathcal Y$ is internal to $mathcal X$.
A separated geometric morphism is one such that the diagonal is proper; in particular, a Hausdorff topos is one such that $mathcal X to mathcal X times mathcal X$ is proper. It seems very unnatural to me in this context to think of $mathcal X$ as being primarily an object "internal to $mathcal X times mathcal X$".
Locally connected geometric morphisms also use indexing in one form of their definition. So do local geometric morphisms.
Let me also point out there there is at least one place in Sketches of an Elephant where $mathcal Y downarrow f^ast$ plays a role qua topos -- in Ch C3.6 on local geometric morphisms, where it's referred to as a "scone". See in particular Example 3.6.3(f), Lemma 3.6.4, and Corollary 3.6.13.
ct.category-theory topos-theory fibered-categories
asked 1 hour ago
Tim Campion
11.7k348110
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
Often in topos theory, one starts with a geometric morphism $f: mathcal Y to mathcal X$, but quickly passes to the Grothendieck fibration $U_f: mathcal Y downarrow f^ast to mathcal X$, which is "$mathcal Y$ regarded as an $mathcal X$-topos". This maneuver takes one outside of the category of toposes and geometric morphisms, and thus tends to remove me from thinking of toposes geometrically, as generalized spaces.
It seems there should be some geometric interpretation lurking here -- after all, Grothendieck fibrations are "categories varying over a base", and a topos is "a base that can be varied over". But although $mathcal Y downarrow f^ast$ is a topos, the fibration $U_f: mathcal Y downarrow f^ast to mathcal X$ is not (the direct image of) a geometric morphism! In fact it has a right adjoint $F^ast$ which has a further right adjoint $F_ast: mathcal Y downarrow f^ast to mathcal X$. We in fact have a totally connected geometric morphism $F: mathcal Y downarrow f^ast to mathcal X$.
Moreover, the fibers of $U_f$ are toposes, but the reindexing functors are most naturally viewed as the inverse images of étale geometric morphisms. So we don't naturally have a "topos fibered in toposes" it seems.
For one more perspective, $mathcal Y downarrow f^ast$ is a cocomma object in the 2-category of toposes. But again, the functor $U_f$ doesn't even live in the category of toposes.
I'm not really sure what to make of this. So here are some
Questions: Let $f: mathcal Y to mathcal X$ be a geometric morphism.
Is there a geometric interpretation of the fibration $U_f: mathcal Y downarrow f^ast to mathcal X$?
Does the totally connected geometric morphism $F: mathcal Y downarrow f^ast to mathcal X$ enjoy some sort of universal property with respect to $f$? (for example, does the inclusion of the category of toposes and totally connected geometric morphisms into toposes and all geometric morphisms have a left adjoint which is being applied here?)
Is there a fibration in the 2-category of toposes lurking around here somewhere (see the Elephant B4.4 for some discussion of these)?
Is there a general geometric interpretation of an opfibration over a topos whose fibers are toposes, with étale geometric morphisms for reindexing?
Addendum: Here are a few places where "indexed" notions come up where the viewpoint that "everything is indexed over the codomain" seems kind of inflexible to me:
A proper geometric morphism $f: mathcal Y to mathcal X$ is one such that $f_ast$ preserves $mathcal X$-indexed filtered colimits of subterminal objects. It's a bit awkward, for example, to even discuss the composition of proper morphisms if you're tied to the view that $mathcal Y$ is internal to $mathcal X$.
A separated geometric morphism is one such that the diagonal is proper; in particular, a Hausdorff topos is one such that $mathcal X to mathcal X times mathcal X$ is proper. It seems very unnatural to me in this context to think of $mathcal X$ as being primarily an object "internal to $mathcal X times mathcal X$".
Locally connected geometric morphisms also use indexing in one form of their definition. So do local geometric morphisms.
Let me also point out there there is at least one place in Sketches of an Elephant where $mathcal Y downarrow f^ast$ plays a role qua topos -- in Ch C3.6 on local geometric morphisms, where it's referred to as a "scone". See in particular Example 3.6.3(f), Lemma 3.6.4, and Corollary 3.6.13.
ct.category-theory topos-theory fibered-categories
asked 1 hour ago
Tim Campion
11.7k348110
Often in topos theory, one starts with a geometric morphism $f: mathcal Y to mathcal X$, but quickly passes to the Grothendieck fibration $U_f: mathcal Y downarrow f^ast to mathcal X$, which is "$mathcal Y$ regarded as an $mathcal X$-topos". This maneuver takes one outside of the category of toposes and geometric morphisms, and thus tends to remove me from thinking of toposes geometrically, as generalized spaces.
It seems there should be some geometric interpretation lurking here -- after all, Grothendieck fibrations are "categories varying over a base", and a topos is "a base that can be varied over". But although $mathcal Y downarrow f^ast$ is a topos, the fibration $U_f: mathcal Y downarrow f^ast to mathcal X$ is not (the direct image of) a geometric morphism! In fact it has a right adjoint $F^ast$ which has a further right adjoint $F_ast: mathcal Y downarrow f^ast to mathcal X$. We in fact have a totally connected geometric morphism $F: mathcal Y downarrow f^ast to mathcal X$.
Moreover, the fibers of $U_f$ are toposes, but the reindexing functors are most naturally viewed as the inverse images of étale geometric morphisms. So we don't naturally have a "topos fibered in toposes" it seems.
For one more perspective, $mathcal Y downarrow f^ast$ is a cocomma object in the 2-category of toposes. But again, the functor $U_f$ doesn't even live in the category of toposes.
I'm not really sure what to make of this. So here are some
Questions: Let $f: mathcal Y to mathcal X$ be a geometric morphism.
Is there a geometric interpretation of the fibration $U_f: mathcal Y downarrow f^ast to mathcal X$?
Does the totally connected geometric morphism $F: mathcal Y downarrow f^ast to mathcal X$ enjoy some sort of universal property with respect to $f$? (for example, does the inclusion of the category of toposes and totally connected geometric morphisms into toposes and all geometric morphisms have a left adjoint which is being applied here?)
Is there a fibration in the 2-category of toposes lurking around here somewhere (see the Elephant B4.4 for some discussion of these)?
Is there a general geometric interpretation of an opfibration over a topos whose fibers are toposes, with étale geometric morphisms for reindexing?
Addendum: Here are a few places where "indexed" notions come up where the viewpoint that "everything is indexed over the codomain" seems kind of inflexible to me:
A proper geometric morphism $f: mathcal Y to mathcal X$ is one such that $f_ast$ preserves $mathcal X$-indexed filtered colimits of subterminal objects. It's a bit awkward, for example, to even discuss the composition of proper morphisms if you're tied to the view that $mathcal Y$ is internal to $mathcal X$.
A separated geometric morphism is one such that the diagonal is proper; in particular, a Hausdorff topos is one such that $mathcal X to mathcal X times mathcal X$ is proper. It seems very unnatural to me in this context to think of $mathcal X$ as being primarily an object "internal to $mathcal X times mathcal X$".
Locally connected geometric morphisms also use indexing in one form of their definition. So do local geometric morphisms.
Let me also point out there there is at least one place in Sketches of an Elephant where $mathcal Y downarrow f^ast$ plays a role qua topos -- in Ch C3.6 on local geometric morphisms, where it's referred to as a "scone". See in particular Example 3.6.3(f), Lemma 3.6.4, and Corollary 3.6.13.
ct.category-theory topos-theory fibered-categories
ct.category-theory topos-theory fibered-categories
asked 1 hour ago
Tim Campion
11.7k348110
asked 1 hour ago
Tim Campion
11.7k348110
edited 8 mins ago
asked 1 hour ago
Tim Campion
11.7k348110
asked 1 hour ago
Tim Campion
11.7k348110
asked 1 hour ago
Tim Campion
11.7k348110
11.7k348110
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The following might not qualifies as an answer to the question: I'm more trying to say that in my opinion these questions are not completely meaningful. But it was way to long for a comment anyway.
Roughly, I think it is a mistake to see this $mathcalY/f^*$ as a topos. it is not meant to be a topos. It is meant to be a $mathcalX$-indexed topos.
A 'category' over $mathcalX$ is just a category object in $mathcalX$, which is the same as a sheaf of category over $mathcalX$ which up to some $2$-categorical details can be seen as a fibered category over $mathcalX$.
Following general indexed category theory, one can formulate what it means for a $mathcalX$-indexed category to have finite limits (it has objectwise finite limits and they are preserved by transition) to have arbitrary colimits (it has preserved finite colimits as before, and the reindexing functor have left adjoint satisfying a Beck-Chevaley condition).
And one can also define what it means to be internally a topos, it is a type of indexed category satisfies the condition above together with some relative presentability condition are relative exactness condition that I will not go into. (See Sketches of an elephant for the details I guess...)
That is the kind of object that $mathcalY/f^*$ is. Now there is a not completely trivial theorem which says that the category of Bounded geometric morphism to a topos $mathcalT$ (not neccesarilly Grothendieck actually) is equivalent to the category of indexed Grothendieck $mathcalT$ topos. The equivalence being given in one direction by the construction you are talking about.
Now, yes indexed categories can be seen as fibration, and you can look at the total category of that fibration. But it has no reason to have any kind a geometric meaning. For example if $mathcalX$ is the category of sets, then you take a general topos $mathcalY$, you see it as set indexed (with the canonical indexing) and you can look at the total category of that fibration, but I don't see any reason it should have a geometric interpretation.
Regarding the precise questions you ask:
For 1), maybe, but not necessarily an interesting one. For example, when you apply it to an identity map it takes the product of the topos with the Sierpinski space.
For 2) you can translate the definition in terms of the indexed category, but that is not going to give you much, and you can do that with most property of geometric morphisms (open, proper, atomic and so one...) so I'm not exactly sure what you have in mind here.
for 3) "Fibrations" is a model dependant notion. You get a notion of fibrations from the fact that the category of toposes is a strict $2$-category, but I'm not really sure this is an interesting notion (I've never really looked at it).
Though I have always thought that the appropriate notion of fibrations for toposes are the map $f :mathcalY rightarrow mathcalX$ where $mathcalY$ is represented by an $mathcalX$-indexed site. Any morphisms can be put in this form up to equivalence, and that is what you need to do if you want to compute a pullback for example. And this is clearly closely related to what you are mentioning.
For 4) If you add a few conditions (it is a stack, it satisfies Beck Chevaley) then you basically get the definition of an indexed topos, (see the discussion above). Without theses conditions, I don't think there is a nice interpretation though.
answered 35 mins ago
Simon Henry
13.4k14477
Thanks. I've tried to clarify some of the reasons why I'd still find a more "geometric" interpretation desirable. I've also tried to clarify what I mean in (2) and added an Elephant reference for (3).
– Tim Campion
15 mins ago
Then I'm affraid I don't understand what you are asking for. The point is that moving from the picture "morphisms $mathcalY rightarrow mathcalX$ to the picture of $mathcalX$-indexed topos, geometrically corresponds to the change from an application (i.e. something sending a point for $mathcalY$ to a point of $mathcalX$ vs a fibration/bundle (i.e. something attaching to each point of $mathcalX$ a topos ). But geometrically, it is the same object as before.
– Simon Henry
9 mins ago
Another to explain my point of view: to me the fact that $mathcalY/f^*$ is a topos is an accident and is not relevant to the picture. The theory of proper map, separated maps, connected maps etc... never makes any use of the fact that $mathcalY/f^*$ is a topos. (and yes the theory of local morphisms is to some extent an exception)
– Simon Henry
5 mins ago
add a comment |Â
1 Answer
1
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1 Answer
1
active
oldest
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active
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active
oldest
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up vote
2
down vote
The following might not qualifies as an answer to the question: I'm more trying to say that in my opinion these questions are not completely meaningful. But it was way to long for a comment anyway.
Roughly, I think it is a mistake to see this $mathcalY/f^*$ as a topos. it is not meant to be a topos. It is meant to be a $mathcalX$-indexed topos.
A 'category' over $mathcalX$ is just a category object in $mathcalX$, which is the same as a sheaf of category over $mathcalX$ which up to some $2$-categorical details can be seen as a fibered category over $mathcalX$.
Following general indexed category theory, one can formulate what it means for a $mathcalX$-indexed category to have finite limits (it has objectwise finite limits and they are preserved by transition) to have arbitrary colimits (it has preserved finite colimits as before, and the reindexing functor have left adjoint satisfying a Beck-Chevaley condition).
And one can also define what it means to be internally a topos, it is a type of indexed category satisfies the condition above together with some relative presentability condition are relative exactness condition that I will not go into. (See Sketches of an elephant for the details I guess...)
That is the kind of object that $mathcalY/f^*$ is. Now there is a not completely trivial theorem which says that the category of Bounded geometric morphism to a topos $mathcalT$ (not neccesarilly Grothendieck actually) is equivalent to the category of indexed Grothendieck $mathcalT$ topos. The equivalence being given in one direction by the construction you are talking about.
Now, yes indexed categories can be seen as fibration, and you can look at the total category of that fibration. But it has no reason to have any kind a geometric meaning. For example if $mathcalX$ is the category of sets, then you take a general topos $mathcalY$, you see it as set indexed (with the canonical indexing) and you can look at the total category of that fibration, but I don't see any reason it should have a geometric interpretation.
Regarding the precise questions you ask:
For 1), maybe, but not necessarily an interesting one. For example, when you apply it to an identity map it takes the product of the topos with the Sierpinski space.
For 2) you can translate the definition in terms of the indexed category, but that is not going to give you much, and you can do that with most property of geometric morphisms (open, proper, atomic and so one...) so I'm not exactly sure what you have in mind here.
for 3) "Fibrations" is a model dependant notion. You get a notion of fibrations from the fact that the category of toposes is a strict $2$-category, but I'm not really sure this is an interesting notion (I've never really looked at it).
Though I have always thought that the appropriate notion of fibrations for toposes are the map $f :mathcalY rightarrow mathcalX$ where $mathcalY$ is represented by an $mathcalX$-indexed site. Any morphisms can be put in this form up to equivalence, and that is what you need to do if you want to compute a pullback for example. And this is clearly closely related to what you are mentioning.
For 4) If you add a few conditions (it is a stack, it satisfies Beck Chevaley) then you basically get the definition of an indexed topos, (see the discussion above). Without theses conditions, I don't think there is a nice interpretation though.
answered 35 mins ago
Simon Henry
13.4k14477
Thanks. I've tried to clarify some of the reasons why I'd still find a more "geometric" interpretation desirable. I've also tried to clarify what I mean in (2) and added an Elephant reference for (3).
– Tim Campion
15 mins ago
Then I'm affraid I don't understand what you are asking for. The point is that moving from the picture "morphisms $mathcalY rightarrow mathcalX$ to the picture of $mathcalX$-indexed topos, geometrically corresponds to the change from an application (i.e. something sending a point for $mathcalY$ to a point of $mathcalX$ vs a fibration/bundle (i.e. something attaching to each point of $mathcalX$ a topos ). But geometrically, it is the same object as before.
– Simon Henry
9 mins ago
Another to explain my point of view: to me the fact that $mathcalY/f^*$ is a topos is an accident and is not relevant to the picture. The theory of proper map, separated maps, connected maps etc... never makes any use of the fact that $mathcalY/f^*$ is a topos. (and yes the theory of local morphisms is to some extent an exception)
– Simon Henry
5 mins ago
add a comment |Â
up vote
2
down vote
The following might not qualifies as an answer to the question: I'm more trying to say that in my opinion these questions are not completely meaningful. But it was way to long for a comment anyway.
Roughly, I think it is a mistake to see this $mathcalY/f^*$ as a topos. it is not meant to be a topos. It is meant to be a $mathcalX$-indexed topos.
A 'category' over $mathcalX$ is just a category object in $mathcalX$, which is the same as a sheaf of category over $mathcalX$ which up to some $2$-categorical details can be seen as a fibered category over $mathcalX$.
Following general indexed category theory, one can formulate what it means for a $mathcalX$-indexed category to have finite limits (it has objectwise finite limits and they are preserved by transition) to have arbitrary colimits (it has preserved finite colimits as before, and the reindexing functor have left adjoint satisfying a Beck-Chevaley condition).
And one can also define what it means to be internally a topos, it is a type of indexed category satisfies the condition above together with some relative presentability condition are relative exactness condition that I will not go into. (See Sketches of an elephant for the details I guess...)
That is the kind of object that $mathcalY/f^*$ is. Now there is a not completely trivial theorem which says that the category of Bounded geometric morphism to a topos $mathcalT$ (not neccesarilly Grothendieck actually) is equivalent to the category of indexed Grothendieck $mathcalT$ topos. The equivalence being given in one direction by the construction you are talking about.
Now, yes indexed categories can be seen as fibration, and you can look at the total category of that fibration. But it has no reason to have any kind a geometric meaning. For example if $mathcalX$ is the category of sets, then you take a general topos $mathcalY$, you see it as set indexed (with the canonical indexing) and you can look at the total category of that fibration, but I don't see any reason it should have a geometric interpretation.
Regarding the precise questions you ask:
For 1), maybe, but not necessarily an interesting one. For example, when you apply it to an identity map it takes the product of the topos with the Sierpinski space.
For 2) you can translate the definition in terms of the indexed category, but that is not going to give you much, and you can do that with most property of geometric morphisms (open, proper, atomic and so one...) so I'm not exactly sure what you have in mind here.
for 3) "Fibrations" is a model dependant notion. You get a notion of fibrations from the fact that the category of toposes is a strict $2$-category, but I'm not really sure this is an interesting notion (I've never really looked at it).
Though I have always thought that the appropriate notion of fibrations for toposes are the map $f :mathcalY rightarrow mathcalX$ where $mathcalY$ is represented by an $mathcalX$-indexed site. Any morphisms can be put in this form up to equivalence, and that is what you need to do if you want to compute a pullback for example. And this is clearly closely related to what you are mentioning.
For 4) If you add a few conditions (it is a stack, it satisfies Beck Chevaley) then you basically get the definition of an indexed topos, (see the discussion above). Without theses conditions, I don't think there is a nice interpretation though.
answered 35 mins ago
Simon Henry
13.4k14477
Thanks. I've tried to clarify some of the reasons why I'd still find a more "geometric" interpretation desirable. I've also tried to clarify what I mean in (2) and added an Elephant reference for (3).
– Tim Campion
15 mins ago
Then I'm affraid I don't understand what you are asking for. The point is that moving from the picture "morphisms $mathcalY rightarrow mathcalX$ to the picture of $mathcalX$-indexed topos, geometrically corresponds to the change from an application (i.e. something sending a point for $mathcalY$ to a point of $mathcalX$ vs a fibration/bundle (i.e. something attaching to each point of $mathcalX$ a topos ). But geometrically, it is the same object as before.
– Simon Henry
9 mins ago
Another to explain my point of view: to me the fact that $mathcalY/f^*$ is a topos is an accident and is not relevant to the picture. The theory of proper map, separated maps, connected maps etc... never makes any use of the fact that $mathcalY/f^*$ is a topos. (and yes the theory of local morphisms is to some extent an exception)
– Simon Henry
5 mins ago
add a comment |Â
up vote
2
down vote
up vote
2
down vote
The following might not qualifies as an answer to the question: I'm more trying to say that in my opinion these questions are not completely meaningful. But it was way to long for a comment anyway.
Roughly, I think it is a mistake to see this $mathcalY/f^*$ as a topos. it is not meant to be a topos. It is meant to be a $mathcalX$-indexed topos.
A 'category' over $mathcalX$ is just a category object in $mathcalX$, which is the same as a sheaf of category over $mathcalX$ which up to some $2$-categorical details can be seen as a fibered category over $mathcalX$.
Following general indexed category theory, one can formulate what it means for a $mathcalX$-indexed category to have finite limits (it has objectwise finite limits and they are preserved by transition) to have arbitrary colimits (it has preserved finite colimits as before, and the reindexing functor have left adjoint satisfying a Beck-Chevaley condition).
And one can also define what it means to be internally a topos, it is a type of indexed category satisfies the condition above together with some relative presentability condition are relative exactness condition that I will not go into. (See Sketches of an elephant for the details I guess...)
That is the kind of object that $mathcalY/f^*$ is. Now there is a not completely trivial theorem which says that the category of Bounded geometric morphism to a topos $mathcalT$ (not neccesarilly Grothendieck actually) is equivalent to the category of indexed Grothendieck $mathcalT$ topos. The equivalence being given in one direction by the construction you are talking about.
Now, yes indexed categories can be seen as fibration, and you can look at the total category of that fibration. But it has no reason to have any kind a geometric meaning. For example if $mathcalX$ is the category of sets, then you take a general topos $mathcalY$, you see it as set indexed (with the canonical indexing) and you can look at the total category of that fibration, but I don't see any reason it should have a geometric interpretation.
Regarding the precise questions you ask:
For 1), maybe, but not necessarily an interesting one. For example, when you apply it to an identity map it takes the product of the topos with the Sierpinski space.
For 2) you can translate the definition in terms of the indexed category, but that is not going to give you much, and you can do that with most property of geometric morphisms (open, proper, atomic and so one...) so I'm not exactly sure what you have in mind here.
for 3) "Fibrations" is a model dependant notion. You get a notion of fibrations from the fact that the category of toposes is a strict $2$-category, but I'm not really sure this is an interesting notion (I've never really looked at it).
Though I have always thought that the appropriate notion of fibrations for toposes are the map $f :mathcalY rightarrow mathcalX$ where $mathcalY$ is represented by an $mathcalX$-indexed site. Any morphisms can be put in this form up to equivalence, and that is what you need to do if you want to compute a pullback for example. And this is clearly closely related to what you are mentioning.
For 4) If you add a few conditions (it is a stack, it satisfies Beck Chevaley) then you basically get the definition of an indexed topos, (see the discussion above). Without theses conditions, I don't think there is a nice interpretation though.
answered 35 mins ago
Simon Henry
13.4k14477
The following might not qualifies as an answer to the question: I'm more trying to say that in my opinion these questions are not completely meaningful. But it was way to long for a comment anyway.
Roughly, I think it is a mistake to see this $mathcalY/f^*$ as a topos. it is not meant to be a topos. It is meant to be a $mathcalX$-indexed topos.
A 'category' over $mathcalX$ is just a category object in $mathcalX$, which is the same as a sheaf of category over $mathcalX$ which up to some $2$-categorical details can be seen as a fibered category over $mathcalX$.
Following general indexed category theory, one can formulate what it means for a $mathcalX$-indexed category to have finite limits (it has objectwise finite limits and they are preserved by transition) to have arbitrary colimits (it has preserved finite colimits as before, and the reindexing functor have left adjoint satisfying a Beck-Chevaley condition).
And one can also define what it means to be internally a topos, it is a type of indexed category satisfies the condition above together with some relative presentability condition are relative exactness condition that I will not go into. (See Sketches of an elephant for the details I guess...)
That is the kind of object that $mathcalY/f^*$ is. Now there is a not completely trivial theorem which says that the category of Bounded geometric morphism to a topos $mathcalT$ (not neccesarilly Grothendieck actually) is equivalent to the category of indexed Grothendieck $mathcalT$ topos. The equivalence being given in one direction by the construction you are talking about.
Now, yes indexed categories can be seen as fibration, and you can look at the total category of that fibration. But it has no reason to have any kind a geometric meaning. For example if $mathcalX$ is the category of sets, then you take a general topos $mathcalY$, you see it as set indexed (with the canonical indexing) and you can look at the total category of that fibration, but I don't see any reason it should have a geometric interpretation.
Regarding the precise questions you ask:
For 1), maybe, but not necessarily an interesting one. For example, when you apply it to an identity map it takes the product of the topos with the Sierpinski space.
For 2) you can translate the definition in terms of the indexed category, but that is not going to give you much, and you can do that with most property of geometric morphisms (open, proper, atomic and so one...) so I'm not exactly sure what you have in mind here.
for 3) "Fibrations" is a model dependant notion. You get a notion of fibrations from the fact that the category of toposes is a strict $2$-category, but I'm not really sure this is an interesting notion (I've never really looked at it).
Though I have always thought that the appropriate notion of fibrations for toposes are the map $f :mathcalY rightarrow mathcalX$ where $mathcalY$ is represented by an $mathcalX$-indexed site. Any morphisms can be put in this form up to equivalence, and that is what you need to do if you want to compute a pullback for example. And this is clearly closely related to what you are mentioning.
For 4) If you add a few conditions (it is a stack, it satisfies Beck Chevaley) then you basically get the definition of an indexed topos, (see the discussion above). Without theses conditions, I don't think there is a nice interpretation though.
answered 35 mins ago
Simon Henry
13.4k14477
answered 35 mins ago
Simon Henry
13.4k14477
answered 35 mins ago
Simon Henry
13.4k14477
answered 35 mins ago
Simon Henry
13.4k14477
13.4k14477
Thanks. I've tried to clarify some of the reasons why I'd still find a more "geometric" interpretation desirable. I've also tried to clarify what I mean in (2) and added an Elephant reference for (3).
– Tim Campion
15 mins ago
Then I'm affraid I don't understand what you are asking for. The point is that moving from the picture "morphisms $mathcalY rightarrow mathcalX$ to the picture of $mathcalX$-indexed topos, geometrically corresponds to the change from an application (i.e. something sending a point for $mathcalY$ to a point of $mathcalX$ vs a fibration/bundle (i.e. something attaching to each point of $mathcalX$ a topos ). But geometrically, it is the same object as before.
– Simon Henry
9 mins ago
Another to explain my point of view: to me the fact that $mathcalY/f^*$ is a topos is an accident and is not relevant to the picture. The theory of proper map, separated maps, connected maps etc... never makes any use of the fact that $mathcalY/f^*$ is a topos. (and yes the theory of local morphisms is to some extent an exception)
– Simon Henry
5 mins ago
add a comment |Â
Thanks. I've tried to clarify some of the reasons why I'd still find a more "geometric" interpretation desirable. I've also tried to clarify what I mean in (2) and added an Elephant reference for (3).
– Tim Campion
15 mins ago
Then I'm affraid I don't understand what you are asking for. The point is that moving from the picture "morphisms $mathcalY rightarrow mathcalX$ to the picture of $mathcalX$-indexed topos, geometrically corresponds to the change from an application (i.e. something sending a point for $mathcalY$ to a point of $mathcalX$ vs a fibration/bundle (i.e. something attaching to each point of $mathcalX$ a topos ). But geometrically, it is the same object as before.
– Simon Henry
9 mins ago
Another to explain my point of view: to me the fact that $mathcalY/f^*$ is a topos is an accident and is not relevant to the picture. The theory of proper map, separated maps, connected maps etc... never makes any use of the fact that $mathcalY/f^*$ is a topos. (and yes the theory of local morphisms is to some extent an exception)
– Simon Henry
5 mins ago
Thanks. I've tried to clarify some of the reasons why I'd still find a more "geometric" interpretation desirable. I've also tried to clarify what I mean in (2) and added an Elephant reference for (3).
– Tim Campion
15 mins ago
Thanks. I've tried to clarify some of the reasons why I'd still find a more "geometric" interpretation desirable. I've also tried to clarify what I mean in (2) and added an Elephant reference for (3).
– Tim Campion
15 mins ago
Then I'm affraid I don't understand what you are asking for. The point is that moving from the picture "morphisms $mathcalY rightarrow mathcalX$ to the picture of $mathcalX$-indexed topos, geometrically corresponds to the change from an application (i.e. something sending a point for $mathcalY$ to a point of $mathcalX$ vs a fibration/bundle (i.e. something attaching to each point of $mathcalX$ a topos ). But geometrically, it is the same object as before.
– Simon Henry
9 mins ago
Then I'm affraid I don't understand what you are asking for. The point is that moving from the picture "morphisms $mathcalY rightarrow mathcalX$ to the picture of $mathcalX$-indexed topos, geometrically corresponds to the change from an application (i.e. something sending a point for $mathcalY$ to a point of $mathcalX$ vs a fibration/bundle (i.e. something attaching to each point of $mathcalX$ a topos ). But geometrically, it is the same object as before.
– Simon Henry
9 mins ago
Another to explain my point of view: to me the fact that $mathcalY/f^*$ is a topos is an accident and is not relevant to the picture. The theory of proper map, separated maps, connected maps etc... never makes any use of the fact that $mathcalY/f^*$ is a topos. (and yes the theory of local morphisms is to some extent an exception)
– Simon Henry
5 mins ago
Another to explain my point of view: to me the fact that $mathcalY/f^*$ is a topos is an accident and is not relevant to the picture. The theory of proper map, separated maps, connected maps etc... never makes any use of the fact that $mathcalY/f^*$ is a topos. (and yes the theory of local morphisms is to some extent an exception)
– Simon Henry
5 mins ago
add a comment |Â
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