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Hi everyone. Let be a bifibration, i.e. a functor that is both a Grothendieck fibration and a Grothendieck opfibration. For any object of , I write for the fiber of , i.e. the subcategory of containing the objects such that and the morphisms such that .
Any morphism of the base induces an adjunction between fiber categories
given by the bifibration property. This yields a comonad on the fiber .
I say that is reflective when the adjunction is reflective, i.e. equivalently when is fully faithful or when the counit is an isomorphism. (The statement with full faithfulness shows that it makes sense to talk about a reflective morphism even when is merely a fibration, but I don't care about that for the applications I have in mind.)
If we take the bifibration which corresponds to the subset hyperdoctrine, then reflective morphisms are exactly the surjective functions, as in general sends a subset on , the intersection with the image of .
Is this notion of reflective morphisms of a base in a bifibration already in the literature? The reason I consider these morphisms is that I have found a little property relating the Frobenius condition for cartesian closed functors and natural transformations between change of base functors. I thought that it was probably already found.
This topic was moved here from #theory: philosophy > Surjectivity w.r.t a bifibratin by Morgan Rogers (he/him).
Oops, thanks Morgan :)
Small update: one can consider the (big) bifibration , in which case the fiber of a small category is its presheaf category. Apparently, the reflective morphisms for this bifibration are the absolutely dense functors, see the paper On Functors Which Are Lax Epimorphisms. Such functors seem to also arise in descent theory!
I'm surprised no one has picked this up. I expect someone like @Jonas Frey or someone working on doctrines should be able to answer it...
Interesting question! In case where the bifibration satifsfies the Beck-Chevalley condition, this reminds me of the notion of [[effective descent morphism]]: an effective descent morphism in wrt a fibration is an arrow in the base sth the canonical comparison arrow into the category of descent data for is an equivalence, and if satisfies BC then , where . In other words, is a descent morphism if the adjunction is monadic, which is the case in particular if the adjunction is reflective.
if is a fibration of preorders, then being monadic is equivalent to being a reflection, and we recover your notion of surjectivity eg for the subobject fibration of Set (or any other topos), which satisfies BC.
Another interesting example to look at is the indexed category of sheaves over toposes, which is the contravariant functor which sends every Grothendieck topos to its category of sheaves (ie the topos itself), and every geometric morphism to . This is (the indexed version of) a "co-bifibration", since reindexing maps have right adjoints rather than left adjoints like in a fibration, so the analogous conditions would be to ask when is comonadic or a coreflection. It turns out that comonadicity of is precisely the notion of the gm being surjective, whereas being a co-reflection is the stronger notion of being connected surjective.
Thus, here we see that in the non-posetal case, surjectivity is better captured by (co)monadicity rather than (co)reflections in the fibration.
I see, thanks for the answers Jonas!
The case of presheaves over Cat can be viewed as a special case of the case of sheaves over toposes by identifying small categories with their toposes of presheaves. As @Vincent Moreau pointed out, the "connected surjective" maps here are the absolutely dense ones, whereas the surjective ones are the "Cauchy surjective functors", ie an adjunction between presheaf toposes is monadic iff every object in the codomain of is a retract of one in the image.
I have not seen this sheaves over toposes co-bifibration before. If I am not mistaken, it amounts to see the category of toposes and geometric morphisms as a certain subcategory of Adj, the category of categories and adjunctions between them, right?
What I mean by that is that a bifibrations correspond through the Grothendieck construction to pseudofunctors , where the latter category is described in my previous message. Then, modulo a change of orientation (which gives the co-bifibration instead of the bifibration), it seems to me that toposes form such a .
Vincent Moreau said:
I have not seen this sheaves over toposes co-bifibration before. If I am not mistaken, it amounts to see the category of toposes and geometric morphisms as a certain subcategory of Adj, the category of categories and adjunctions between them, right?
Yes, that's a way of looking at it. Anyway, this fibration is crucial in topos theory, and used more or less explicitly in the elephant. The notion of descent morphism of toposes is what you get by iknstantiating the fibrational notion of descent morphism that I mentioned above for this fibration.
Interesting, thanks!
Vincent Moreau said:
If we take the bifibration which corresponds to the subset hyperdoctrine, then reflective morphisms are exactly the surjective functions, as in general sends a subset on , the intersection with the image of .
Is this notion of reflective morphisms of a base in a bifibration already in the literature? The reason I consider these morphisms is that I have found a little property relating the Frobenius condition for cartesian closed functors and natural transformations between change of base functors. I thought that it was probably already found.
Related but not the same: Check fully faithfulness with respect to a proarrow equipment.
I agree there is a similarity. In the case of fully faithfulness in a proarrow equipment, we ask the unit to be invertible, which seems to be oriented as in the sheaf *-fibration that Jonas was mentioning above.
Haha now it's getting really confusing since in the equipment world, the convention is that is right adjoint to , which is dual to the topos convention.
Yes, I was just looking at that!!!
But this contradiction disappears if you enhance the 2-category of Grothendieck toposes to an equipment via the functor as Wood does in "Abstract proarrows II".
Here (or rather ) is the 2-cat of Grothendieck toposes and lex functors, and the inclusion sends a gm to its direct image. The co comes in since -cells in are defined in terms of natural transformations between inverse image functors.
Direct images functors are right adoints in , and therefore left adjoints in , ie we get in the equipment as required.
I think that Ivan's suggestion made me understand the situation better: for , it seems to me that the composition represents the profunctor of morphisms of that factor through the image of . Now this is really getting closer to condition 3 of Theorem 1.1 of this paper.
Jonas Frey said:
Haha now it's getting really confusing since in the equipment world, the convention is that is right adjoint to , which is dual to the topos convention.
That's why I prefer to write in the equipment world.
The equipment perspective also points out that this is a "2-categorical" notion of surjectivity, being the dual of fully-faithfulness. So it's not surprising that in the topos world it corresponds to "connected surjective", i.e. 0-connected, which is a categorification of "surjective", i.e. (-1)-connected.
One more thing that is kind of related: if is a tripos on a topos , then the constant objects functor is regular (preserves epis) iff "recognizes epis in " in the sense that is a reflection for all epis in .
This is important when iterating tripos constructions : given another tripos on , we can in general only represent by a tripos on if the above condition holds for .
Vincent Moreau said:
I agree there is a similarity. In the case of fully faithfulness in a proarrow equipment, we ask the unit to be invertible, which seems to be oriented as in the sheaf *-fibration that Jonas was mentioning above.
Tight arrows in an equipment satisfying the tightwise dual condition of fully-faithfulness, which are called covers in an equipment (at least by @Keisuke Hoshino and myself in this paper), are always final and initial (internal to the equipment) when the equipment has a terminal. Here, by being final, I mean that (canonically) where is the unique arrow into the terminal. The converse is also true when a double category is "discrete" in the sense that is fully-faithful and the equipment satisfies the Frobenius axiom, (although the first condition might not be necessary.) If I follow the argument above, this fact says that several notions of "surjectiveness"(cover=absolutely dense, final, initial) are collapsed in discrete equipments.
I see, that's interesting!
Mike made it very clear in this paper that the double category of sets, functions and relations can be recovered from the application of a very general construction to the subset bifibration. When restricting to bifibrations with adequate properties, I wonder if his construction could be used to bridge the gap between my definition of reflective morphism (w.r.t. a bifibration) and the equipment-theoretic notion of co-fully faithful tight morphism.
In the same vein, perhaps that this notion of discrete double category you mentioned can be shown to correspond to some property on fibrations (or at least, be implied by a sufficient condition).
The Frobenius axiom on cartesian equipments (part of the discreteness condition) characterizes the cartesian equipments that come from elementary existential fibrations (= bifibrations with the Beck-Chevalley conditions for a certain class of pullback squares and the Frobenius reciprocity for a certain class of arrows) . This is a central topic of my recent thesis, so I was surprised by the coincidence that you replied to me in that way :joy:
Incredible, so this is solved already! One more reason to read your thesis :smile:
Did you also investigate the relation between covers / co-ff tight morphisms on the equipment side and specific morphisms of the base on the fibered side, in the non-discrete case?
Vincent Moreau said:
Incredible, so this is solved already! One more reason to read your thesis :smile:
If you believe me.
Vincent Moreau said:
Did you also investigate the relation between covers / co-ff tight morphisms on the equipment side and specific morphisms of the base on the fibered side, in the non-discrete case?
In that case, I have no idea. When you extract the presheaf fibration from the equipment of profunctors, you are focusing on the loose arrows into (or, from) the terminal object in the equipment. I would not be surprised if this process of extraction breaks down some data, including cover/co-ff.
Well, in this affair I am coming from the fibered side: I already have a bifibration (with BC, etc), which gives me an equipment, so I don't need to do any extraction. I wonder if, given any such bifibration as input, its reflective morphisms can be characterized in the language of equipments as cover / co-ff.
Hayato Nasu said:
The Frobenius axiom on cartesian equipments (part of the discreteness condition) characterizes the cartesian equipments that come from elementary existential fibrations (= bifibrations with the Beck-Chevalley conditions for a certain class of pullback squares and the Frobenius reciprocity for a certain class of arrows) . This is a central topic of my recent thesis, so I was surprised by the coincidence that you replied to me in that way :joy:
Nice! Is there a difference between cartesian equipments and cartesian bicategories? And I assume you're only considering the Pos-enriched case?
Vincent Moreau said:
Well, in this affair I am coming from the fibered side: I already have a bifibration (with BC, etc), which gives me an equipment, so I don't need to do any extraction. I wonder if, given any such bifibration as input, its reflective morphisms can be characterized in the language of equipments as cover / co-ff.
I see. The construction of a cartesian equipment from an elementary existential fibration gives a 2-functor which is locally an equivalence, so I guess it seems to work well in that direction. What I don't fully understand is that how is related to this construction.
Jonas Frey said:
Hayato Nasu said:
The Frobenius axiom on cartesian equipments (part of the discreteness condition) characterizes the cartesian equipments that come from elementary existential fibrations (= bifibrations with the Beck-Chevalley conditions for a certain class of pullback squares and the Frobenius reciprocity for a certain class of arrows) . This is a central topic of my recent thesis, so I was surprised by the coincidence that you replied to me in that way :joy:
Nice! Is there a difference between cartesian equipments and cartesian bicategories? And I assume you're only considering the Pos-enriched case?
I studied the connection between them in Section 2.4.2 and 2.5.3, so I would be happy if you consider reading that part. In short, the loose (horizontal) bicategory of a cartesian equipment is a cartesian bicategory (as shown in this paper and Theorem 2.4.8.), and the other direction is only possible within pseudo double categories for a certain class. And I am considering not only the Pos-enriched/ fiberwise-posetal case, but in a general case.