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Stream: theory: category theory

Topic: Surjectivity w.r.t a bifibration

Vincent Moreau (Feb 12 2025 at 14:18):

Hi everyone. Let $p : \mathbf{E} \to \mathbf{B}$ be a bifibration, i.e. a functor that is both a Grothendieck fibration and a Grothendieck opfibration. For any object $X$ of $\mathbf{B}$, I write $\mathbf{E}_{X}$ for the fiber of $X$, i.e. the subcategory of $\mathbf{E}$ containing the objects $U$ such that $p(U) = X$ and the morphisms $u : U \to V$ such that $p(u) = \mathrm{Id}_{X}$.

Vincent Moreau (Feb 12 2025 at 14:26):

Any morphism $f : X \to Y$ of the base $\mathbf{B}$ induces an adjunction $f_! \dashv f^*$ between fiber categories

$$f_! \quad:\quad \mathbf{E}_{X} \leftrightarrows \mathbf{E}_{Y} \quad:\quad f^*$$

given by the bifibration property. This yields a comonad on the fiber $\mathbf{E}_{Y}$.
I say that $f$ is reflective when the adjunction $f_! \dashv f^*$ is reflective, i.e. equivalently when $f^*$ is fully faithful or when the counit $f_! \circ f^* \to \mathrm{Id}$ is an isomorphism. (The statement with full faithfulness shows that it makes sense to talk about a reflective morphism even when $p$ is merely a fibration, but I don't care about that for the applications I have in mind.)

Vincent Moreau (Feb 12 2025 at 14:28):

If we take the bifibration $p : \mathbf{SubSet} \to \mathbf{Set}$ which corresponds to the subset hyperdoctrine, then reflective morphisms are exactly the surjective functions, as in general $f_! \circ f^*$ sends a subset $S \subseteq Y$ on $S \cap f(X)$, the intersection with the image of $f$.

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.

Notification Bot (Feb 12 2025 at 14:52):

This topic was moved here from #theory: philosophy > Surjectivity w.r.t a bifibratin by Morgan Rogers (he/him).

Vincent Moreau (Feb 12 2025 at 14:58):

Oops, thanks Morgan :)

Vincent Moreau (Feb 16 2025 at 18:28):

Small update: one can consider the (big) bifibration $p : \mathbf{PSh} \to \mathbf{Cat}$, 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!

Morgan Rogers (he/him) (Feb 18 2025 at 15:29):

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...

Jonas Frey (Feb 18 2025 at 15:47):

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 $\mathbb C$ wrt a fibration $p:\mathbb E\to \mathbb C$ is an arrow $f:X\to Y$ in the base sth the canonical comparison arrow $p_X\to \mathrm{Desc}(f)$ into the category of descent data for $f$ is an equivalence, and if $p$ satisfies BC then $\mathrm{Desc}_f\simeq\mathrm{Alg}(T_f)$, where $T_f=f^*\circ f_!$. In other words, $f$ is a descent morphism if the adjunction $f_!\dashv f^*$ is monadic, which is the case in particular if the adjunction is reflective.

Jonas Frey (Feb 18 2025 at 15:50):

if $p$ 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.

Jonas Frey (Feb 18 2025 at 15:58):

Another interesting example to look at is the indexed category of sheaves over toposes, which is the contravariant functor $Sh(-):Top^{op}\to Cat$ which sends every Grothendieck topos to its category of sheaves (ie the topos itself), and every geometric morphism $(f^*\dashv f_* : E\to F)$ to $f^*:F\to E$. 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 $(f^*\dashv f_*)$ is comonadic or a coreflection. It turns out that comonadicity of $f^*\dashv f_*$ 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.

Vincent Moreau (Feb 18 2025 at 16:07):

I see, thanks for the answers Jonas!

Jonas Frey (Feb 18 2025 at 16:08):

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 $f_!\dashv f^*$ between presheaf toposes is monadic iff every object in the codomain of $f$ is a retract of one in the image.

Vincent Moreau (Feb 18 2025 at 16:09):

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?

Vincent Moreau (Feb 18 2025 at 16:11):

What I mean by that is that a bifibrations correspond through the Grothendieck construction to pseudofunctors $\mathbf{B}^{\mathrm{op}} \to \mathbf{Adj}$, 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 $\mathbf{B}$.

Jonas Frey (Feb 18 2025 at 16:12):

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.

Vincent Moreau (Feb 18 2025 at 16:12):

Interesting, thanks!

Ivan Di Liberti (Feb 18 2025 at 16:26):

Vincent Moreau said:

If we take the bifibration $p : \mathbf{SubSet} \to \mathbf{Set}$ which corresponds to the subset hyperdoctrine, then reflective morphisms are exactly the surjective functions, as in general $f_! \circ f^*$ sends a subset $S \subseteq Y$ on $S \cap f(X)$, the intersection with the image of $f$.

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.

Vincent Moreau (Feb 18 2025 at 16:39):

I agree there is a similarity. In the case of fully faithfulness in a proarrow equipment, we ask the unit $1 \to f^* f_* := B(f, f)$ to be invertible, which seems to be oriented as in the sheaf *-fibration that Jonas was mentioning above.

Jonas Frey (Feb 18 2025 at 16:46):

Haha now it's getting really confusing since in the equipment world, the convention is that $f^*$ is right adjoint to $f_*$, which is dual to the topos convention.

Vincent Moreau (Feb 18 2025 at 16:46):

Yes, I was just looking at that!!!

Jonas Frey (Feb 18 2025 at 16:47):

But this contradiction disappears if you enhance the 2-category $Top$ of Grothendieck toposes to an equipment via the functor $Top\to Toplex^{co}$ as Wood does in "Abstract proarrows II".

Jonas Frey (Feb 18 2025 at 16:49):

Here $Toplex$ (or rather $TOPLEX$) is the 2-cat of Grothendieck toposes and lex functors, and the inclusion $TOP\to TOPLEX^{co}$ sends a gm to its direct image. The co comes in since $2$-cells in $TOP$ are defined in terms of natural transformations between inverse image functors.

Jonas Frey (Feb 18 2025 at 16:55):

Direct images functors are right adoints in $TOPLEX$, and therefore left adjoints in $TOPLEX^{co}$, ie we get $f_*\dashv f^*$ in the equipment as required.

Vincent Moreau (Feb 18 2025 at 16:58):

I think that Ivan's suggestion made me understand the situation better: for $f : A \to B$, it seems to me that the composition $f_* f^*$ represents the profunctor of morphisms of $B$ that factor through the image of $f$. Now this is really getting closer to condition 3 of Theorem 1.1 of this paper.

Mike Shulman (Feb 18 2025 at 17:02):

Jonas Frey said:

Haha now it's getting really confusing since in the equipment world, the convention is that $f^*$ is right adjoint to $f_*$, which is dual to the topos convention.

That's why I prefer to write $f_\bullet \dashv f^\bullet$ in the equipment world.

Mike Shulman (Feb 18 2025 at 17:03):

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.

Jonas Frey (Feb 18 2025 at 17:18):

One more thing that is kind of related: if $P:E^{op}\to Pos$ is a tripos on a topos $E$, then the constant objects functor $\Delta : E \to E[P]$ is regular (preserves epis) iff $P$ "recognizes epis in $E$" in the sense that $e_!\dashv e^*$ is a reflection for all epis $e$ in $E$.

This is important when iterating tripos constructions : given another tripos $Q$ on $E[P]$, we can in general only represent $E[P][Q]$ by a tripos on $E$ if the above condition holds for $P$.

Hayato Nasu (Feb 20 2025 at 05:20):

Vincent Moreau said:

I agree there is a similarity. In the case of fully faithfulness in a proarrow equipment, we ask the unit $1 \to f^* f_* := B(f, f)$ 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 $f$ being final, I mean that $f^*!_*\cong !_*$ (canonically) where $!$ is the unique arrow into the terminal. The converse is also true when a double category is "discrete" in the sense that $\Delta:A\to A\times A$ 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.

Vincent Moreau (Feb 20 2025 at 09:49):

I see, that's interesting!

Vincent Moreau (Feb 20 2025 at 09:55):

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.

Vincent Moreau (Feb 20 2025 at 09:59):

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).

Hayato Nasu (Feb 20 2025 at 11:41):

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:

Vincent Moreau (Feb 20 2025 at 11:44):

Incredible, so this is solved already! One more reason to read your thesis :smile:

Vincent Moreau (Feb 20 2025 at 11:46):

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?

Hayato Nasu (Feb 20 2025 at 12:15):

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.

Vincent Moreau (Feb 20 2025 at 12:27):

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.

Jonas Frey (Feb 20 2025 at 15:25):

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?

Hayato Nasu (Feb 21 2025 at 03:13):

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 $\mathbb{P}\mathrm{rof}$ 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.