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Stream: learning: questions

Topic: Ran along a final functor

Vincent Moreau (Oct 08 2024 at 15:44):

Dear category theorists,

I have found the following fact and I wonder if it is written somewhere in the litterature. If $K : \mathbf{C} \to \mathbf{D}$ is a final functor between small categories, then $\operatorname{Ran}_K : [\mathbf{C}, \mathbf{Set}] \to [\mathbf{D}, \mathbf{Set}]$ preserves coproducts.

Vincent Moreau (Oct 08 2024 at 15:44):

To show this, we first remark that the following conditions are equivalent on the functor $K$:

1. $K$ is a final functor
2. For every $X$ in $\mathbf{D}$, the comma $X \downarrow K$ is a connected category
3. For every $X$ in $\mathbf{D}$, the discrete opfibration $(X \downarrow K) \to \mathbf{C}$ is a connected object of the category of discrete opfibrations over $\mathbf{C}$
4. For every $X$ in $\mathbf{D}$, the functor $\mathbf{D}(X, K -)$ is a connected object of $[\mathbf{C}, \mathbf{Set}]$

where 1 <=> 2 is well-known (actually the definition on the nlab), 2 => 3 is direct and 3 <=> 4 comes from the fact that the Grothendieck construction is an equivalence of categories (and actually generalizes to any form of colimit, not only coproducts).

Vincent Moreau (Oct 08 2024 at 15:44):

The proof of 3 => 2 that I have found is based on the fact that a category $\mathbf{A}$ is connected iff it is inhabited and any functor $\mathbf{A} \to 1 + 1$ factorises through one of the two canonical injections $1 \to 1 + 1$. Let $p : \mathbf{A} \to \mathbf{C}$ be a discrete opfibration which is a connected object in the category of discrete opfibrations over $\mathbf{C}$, and let $F : \mathbf{A} \to 1 + 1$ be any functor. Then, there is a morphism $F'$ from $p : \mathbf{A} \to \mathbf{C}$ to $[\operatorname{Id}, \operatorname{Id}] : \mathbf{C} + \mathbf{C} \to \mathbf{C}$ which sends $a$ on the injection of $p(a)$ into the copy of $\mathbf{C}$ that corresponds to the copy of $1$ that $F(a)$ belongs to. This is well defined on morphisms as if $f : a \to a'$ in $\mathbf{A}$, then $F(a) = F(a')$. The situation is the following:

$$\begin{CD} \mathbf{A} @>>F'> \mathbf{C} + \mathbf{C} @>!+!>> 1+1 \\ @VpVV @V[\operatorname{Id}, \operatorname{Id}]VV \\ \mathbf{C} @= \mathbf{C} \end{CD}$$

Moreover, we have $F = (! + !) \circ F'$ where $! : \mathbf{C} \to 1$. We can then use the fact that $p$ is a connected object to obtain that $F'$ factors through a canonical injection $\mathbf{C} \to \mathbf{C} + \mathbf{C}$, and we then get that $\mathbf{A}$ is a connected category. We get the result for $\mathbf{A} := X \downarrow K$.

Vincent Moreau (Oct 08 2024 at 15:45):

Now, we are interested right Kan extending functors $F : \mathbf{C} \to \mathbf{Set}$ along a fixed $K : \mathbf{C} \to \mathbf{D}$. We can use the following formula

$$\operatorname{Ran}_K F (X) \quad\cong\quad [\mathbf{C}, \mathbf{Set}] \left( \mathbf{D}(X, K - ) \ ,\ F \right)$$

Now, if $K$ is a final functor, by 1 <=> 4 we get that for every object $X$ of $\mathbf{D}$, the functor $\mathbf{D}(X, K - )$ is a connected object of $[\mathbf{C}, \mathbf{Set}]$, and therefore $\operatorname{Ran}_K$ preserves coproducts.

Is this written somewhere?

Morgan Rogers (he/him) (Oct 08 2024 at 16:21):

Vincent Moreau said:

Dear category theorists,

You don't have to write so formally every time :joy:

Morgan Rogers (he/him) (Oct 08 2024 at 16:29):

To answer the question, Bunge and Funk are the names that spring to mind, but it's probably quickest to follow the reference trail backwards from my last paper with @Jens Hemelaer (I'm on a train, otherwise I would check it myself instead)

Vincent Moreau (Oct 08 2024 at 16:47):

Unfortunately it's not clear to me how to relate the work of Bunge and Funk to this observation

Ivan Di Liberti (Oct 08 2024 at 16:57):

I think this is Example 2.16 in Spreads and the symmetric topos, indeed by Bunge and Funk.

Vincent Moreau (Oct 08 2024 at 17:09):

I'm really sorry but I still don't understand the link with either the fact that 3 => 2 above or the fact that Ran along a final functor preserves coproducts. Perhaps I am missing some vocabulary from topos theory?

Vincent Moreau (Oct 08 2024 at 17:13):

While I am at it, a functor $K : \mathbf{C} \to 1$ is final iff $\mathbf{C}$ is connected, and we would thus obtain the intriguing fact that lim(F + G) = lim(F + G) for two connected diagrams of sets F and G.

In particular, when $\mathbf{C}$ is the connected category consisting of two parallel morphisms, a functor $F : \mathbf{C} \to \mathbf{Set}$ can be thought of as a graph, and then its limit can be thought of as the set of loops of the graph, i.e. edges with same source and target. And indeed, if we do the coproduct of two graphs, then its loops are always loops of one of the two graphs we started with!

Vincent Moreau (Oct 08 2024 at 17:17):

Actually this special case is mentioned on the nlab page about commutativity of limits and colimits

Ivan Di Liberti (Oct 08 2024 at 17:18):

Let me start by saying that I had not thought about this before and that I do not think your result is trivial, or somewhat trivially expected.

The way I personally found the reference was by thinking the following. For f a functor between categories, Ran_f is the direct image of a geometric morphism between presheaf categories, and for it to preserve coproducts is the same to be "pure" (see Sketches C3.4.12(i)). Then you start googling "final functor" and "pure" and eventually you get to the paper I cited, where Example 2.16 is a reshuffle of your statement (I believe).

I do not know what Morgan's mental path could have been.

Morgan Rogers (he/him) (Oct 08 2024 at 17:30):

Okay, I'm back. The section of the paper of mine I was referencing is Section 5 here; as Ivan surmised, "pure" is the essential keyword, but it turns out that a geometric morphism between locally connected toposes (in particular, presheaf toposes) being pure is equivalent to the direct image preserving small coproducts.
Up to adjusting for op (and hence expressing the result in terms of a final-fibration factorization rather than an initial-opfibration factorization), the result that's most relevant is Proposition 5.1.1 there, which shows that this factorization at the level of functors extends (by applying the presheaf construction) to the pure--complete spread factorization at the level of geometric morphisms

Morgan Rogers (he/him) (Oct 08 2024 at 17:32):

We attribute the observation of this coincidence to a paper of Bunge and Funk, although they work in enough generality that it isn't necessarily transparent that this boils down to the equivalence you've observed more directly in terms of Ran

Morgan Rogers (he/him) (Oct 08 2024 at 17:40):

Morgan Rogers (he/him) said:

Proposition 5.1.1 there shows that this factorization at the level of functors extends (by applying the presheaf construction) to the pure--complete spread factorization at the level of geometric morphisms

To make this more explicit, since I really can't take credit for the result you're actually interested in here: if I take any functor $F:C \to D$, it induces an essential geometric morphism $[C,\mathsf{Set}] \to [D,\mathsf{Set}]$ (which is just saying we have an adjoint triple $\mathrm{Lan}_F \dashv F^* \dashv \mathrm{Ran}_F$ (beware that I'm dealing with covariant functor categories rather than presheaves here, which accounts for the discrepancy with the paper I pointed to before). For any geometric morphism there is an orthogonal "pure, complete spread" factorization, where the pure part is defined by the property that the direct image preserves finite coproducts, and when the toposes involved are locally connected this turns out to extend to the preservation of all small coproducts. Now B&F observed that the comprehensive factorization of $F$ extends to the pure, complete spread factorization of the geometric morphism (that's 2=>3, I think), but the fact that the factorization system at the level of toposes is orthogonal means that the converse must also hold, since the geometric morphism induced by $F$ is pure iff the right factor in that factorization system is an equivalence (which means that the fibration factor in the comprehensive factorization of $F$ must be an equivalence).

Vincent Moreau (Oct 08 2024 at 18:17):

Ivan Di Liberti said:

For f a functor between categories, Ran_f is the direct image of a geometric morphism between presheaf categories, and for it to preserve coproducts is the same to be "pure" (see Sketches C3.4.12(i)).

Now that I have read the content of C3.4.12(i), I finally understand what you mean: indeed, if we start with an initial functor, i.e. a final functor between the opposite categories, then the associated geometric morphism between presheaf categories is already pure dense, and so its direct image, which is Ran, therefore preserves coproducts!!!

Ivan Di Liberti (Oct 08 2024 at 18:19):

You were looking at it from the nose, but the statement looked better from the tail. :elephant:

Vincent Moreau (Oct 08 2024 at 18:19):

Hahahaha, indeed!!!

Vincent Moreau (Oct 08 2024 at 18:36):

I'm trying to understand your messages now Morgan. You say that every presheaf topos is locally connected, but when I see the definition on the nlab, I cannot help but think about the family construction i.e. the free coproduct completion. Is a locally connected topos a topos which is the coproduct completion of its full subcategory of connected objects, or is there a subtlety here?

Vincent Moreau (Oct 08 2024 at 18:41):

and by the way, unlike for the colimit completion which has a rich Morita theory, the coproduct completion does not as one can recover the "generator". What it means is that "being a free coproduct completion" and "being the free coproduct completion of its connected objects" is the same.

Morgan Rogers (he/him) (Oct 08 2024 at 18:54):

It's exactly that; check out section C3.3 of the Elephant (iirc) for that characterization.

Vincent Moreau (Oct 08 2024 at 19:11):

Morgan Rogers (he/him) said:

Now B&F observed that the comprehensive factorization of $F$ extends to the pure, complete spread factorization of the geometric morphism (that's 2=>3, I think), but the fact that the factorization system at the level of toposes is orthogonal means that the converse must also hold, since the geometric morphism induced by $F$ is pure iff the right factor in that factorization system is an equivalence (which means that the fibration factor in the comprehensive factorization of $F$ must be an equivalence).

I think I now get it a bit more. Indeed, 2 => 3 is fairly direct, but the converse is somehow a bit technical. In the Elephant just before C3.4.12, it is written

We define a geometric morphism $f : \mathcal{F} \to \mathcal{E}$ to be pure if $f_*$ preserves 2; more explicitly, if the canonical morphism $2_{\mathcal{E}} \to f_*(2_{\mathcal{F}})$ (the unit of ($f^* \dashv f_*$) at the object 2) si an isomorphism.

The fact that I use the opfibration $\mathbf{C} + \mathbf{C} \to \mathbf{C}$ is somehow reminiscent of that, because it is precisely the binary coproduct of the terminal object of discrete opfibrations. Perhaps this is a possible intuition of the bits of topos theory at play here that I am discovering on the fly?

Morgan Rogers (he/him) (Oct 08 2024 at 19:16):

I expect that another way to derive this result would be in terms of Beck-Chevalley conditions, but the calculus of those takes some practice to work with (roughly: you can characterize both pure morphisms and final functors in terms of a collection of Beck-Chevalley conditions, and you should be able to show that they are equivalent). This could either be a distraction or an interesting further point of view, depending on what you're planning to do with these facts ;)

Vincent Moreau (Oct 08 2024 at 19:19):

I see!

Vincent Moreau (Oct 08 2024 at 19:21):

Well, I should be sending my thesis in a few weeks so I think I am perhaps not going to think too much about that for the moment, but thanks to both you and Ivan for the great explanations! :star_struck:

Kevin Carlson (Oct 08 2024 at 20:22):

Vincent Moreau said:

and by the way, unlike for the colimit completion which has a rich Morita theory, the coproduct completion does not as one can recover the "generator". What it means is that "being a free coproduct completion" and "being the free coproduct completion of its connected objects" is the same.

That's a good point. I think it's basically because there aren't any absolute coproducts when enriching in Set; certainly the Ab-enriched coproduct completion has interesting Morita theory!