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Stream: theory: category theory

Topic: presheaves whose restrictions are all invertible

Matteo Capucci (he/him) (Jan 27 2025 at 12:16):

Suppose $P$ is a presheaf on $C$ such that $Pf$ is iso for every $f:c \to c'$ in $C$. These are not constant presheaves (because $P$ can assume different values on different connected components of $C$), are they locally constant though?
In order to conclude that I should exhibit, for each connected component $U$ of $C$, a natural iso $P\vert_U \cong \Delta A_U$ where $A_U$ is some set. However, I'm not sure how to do that when $C$ is not a preorder or without appealing to arbitrary choices. Intuitively, $P\vert_U$ is hitting an iso class and we want $A_U$ to be a representative, but it seems I'm forced to choose an arbitrary one per each connected component $U$? Or is there a nicer way to construct it?
One idea is to basically construct $A_U$ as the étale space of $P\vert_U$, so summing up all the $Pc$, $c \in U$ and then identify those elements which are related by a restriction map. I do end up with a colimiting cocone $P\vert_U \to \Delta A_U$ but I'm not sure it is iso :thinking:

Martti Karvonen (Jan 27 2025 at 13:33):

Surely they're not locally constant without further assumptions on the underlying category? Take e.g., the delooping of a group and look at the only representable: in general a group acts on itself in a way that is not isomorphic to the trivial action.

Morgan Rogers (he/him) (Jan 27 2025 at 13:35):

Consider a presheaf on the integers qua one-object category and you'll see that indeed you cannot conclude that they are locally constant!

Morgan Rogers (he/him) (Jan 27 2025 at 13:35):

Ah Martti was faster than me

Martti Karvonen (Jan 27 2025 at 13:36):

Essentially the same example

Matteo Capucci (he/him) (Jan 27 2025 at 15:27):

Ah yes good point!!

Matteo Capucci (he/him) (Jan 27 2025 at 15:46):

So is there a name for these things? Are such presheaves special in anyway within the larger category of presheaves?

Matteo Capucci (he/him) (Jan 27 2025 at 15:49):

I look at these as a generalization of local objects with respect to a class of maps $P \subseteq C$. Wlog suppose $P$ is a category, one has indeed a functor $C \to {\bf Set}^{P^{op}}$ given by restriction of the Yoneda embedding. Now an object $X \in C$ is $P$-local when the associated presheaf $C(-,X) \in {\bf Set}^{P^{op}}$ has all restrictions invertible, essentially by definition.

Mike Shulman (Jan 27 2025 at 16:28):

So don't you mean a "special case of" rather than a "generalization of", then?

John Baez (Jan 27 2025 at 16:42):

So is there a name for these things?

I don't know a general name for them, but the example Martti gave, where you take a nontrivial presheaf on the delooping of a group $G$, is called 'a nontrivial covering space of the classifying space $BG$'. Sitting over each point of $BG$ there's a fiber that's a set, and and you march around a nontrivial loop in $BG$, the points overhead get permuted.

This is just to provide some topological / visual picture of what's going on.

Matteo Capucci (he/him) (Jan 27 2025 at 16:49):

Mike Shulman said:

So don't you mean a "special case of" rather than a "generalization of", then?

I'd say a presheaf over $P$ with all invertible restrictions is a 'generalized local object', and '$P$-local objects' are a special instance of this more general definition?

Mike Shulman (Jan 27 2025 at 17:09):

By your description, when $P\subseteq C$, a $P$-local object is not a presheaf on $P$ with invertible restrictions, but a presheaf on $C$ that restricts to a presheaf on $P$ with invertible restrictions. On the other hand, a presheaf on $P$ with invertible restrictions is exactly the special case of a $P$-local object when $C=P$.

Matteo Capucci (he/him) (Jan 27 2025 at 17:21):

Uhm maybe I didn't explain myself well. What I wanted to say is, one can generalize $P$-locality by replacing (the restriction of) $C(-,=)$ with some other profunctor $H:C \to P$. Say then that $X$ is $P$-local with respect to $H$ when $H(-,X)$ has invertible restrictions. At this point, there is no need that $P$ and $C$ are related at all except through $H$, so one might as well say that a presheaf over $P$ is a 'generalized local object' when it has invertible restrictions, and the classical definition arises by choosing $P \subseteq C$ and mapping objects of $C$ to their representables.

Mike Shulman (Jan 27 2025 at 17:49):

Okay, but then it isn't "having invertible restrictions" that is a generalization of $P$-locality, since the one is a property of $H(-,X)$ whereas the other is a property of $X$.

Matteo Capucci (he/him) (Jan 27 2025 at 18:00):

Yeah that's right. I think I said it's a generalization of P-local objects (as a functor into $Set^{P^{op}}$), not P-locality per se. Does this make more sense?

Mike Shulman (Jan 27 2025 at 18:02):

Not really, I don't see the difference between "a generaliation of P-local objects" and "a generalization of P-locality".

Mike Shulman (Jan 27 2025 at 18:02):

It's a generalization of the presheaves induced by P-local objects...

Matteo Capucci (he/him) (Jan 27 2025 at 18:04):

Ok, point taken

Kevin Carlson (Jan 27 2025 at 19:49):

Not exactly a name, but as a description, these are exactly presheaves on the groupoid reflection $C[\mathrm{everything}^{-1}].$ That at least characterizes the categories they form and provides an adjoint triple between them and arbitrary presheaves on $C.$

John Baez (Jan 27 2025 at 21:55):

I'm not following carefully, but if Kevin is right and y'all are talking about presheaves on the groupoid reflection of a category, then my topological digression was not so digressive, since those are really the same as covering spaces of a certain space: the classifying space of that groupoid (which you build by putting in an n-simplex for each composable n-tuple of morphisms).

If I'm not confused, presheaves on a groupoid form a Boolean topos. But [[Boolean topos]] doesn't list this as an example, so maybe I'm wrong. But that article doesn't list any examples, so maybe I'm right.)

Maybe the process of going from the topos of presheaves on a category $C$ to the topos presheaves on its groupoid reflection $C[\text{everything}^{-1}]$ is some sort of 'Booleanization' of the original topos?

For some reason the phrase 'double negation topology' bubbles to the top of my mind.

Kevin Carlson (Jan 27 2025 at 22:47):

I see an examples section in that article!

Kevin Carlson (Jan 27 2025 at 22:54):

That said, yes, I think toposes of $G$-sets are Boolean. If $X<Y$ is a sub-$G$-set, then so is $Y\setminus X,$ since if $g\cdot y \in X$ then $g^{-1}\cdot (g\cdot y)= y \in X$ too. Thus the subobject lattices in $G$-set categories have complements in the Boolean algebra sense.

Kevin Carlson (Jan 27 2025 at 22:54):

Notice this won't fly from toposes of $M$-sets with $M$ just a monoid! If I have an action of the walking-idempotent monoid, then that's just a set with a retraction onto a subset, and the Heyting complement of the retract will always be empty, showing complementation isn't an involution there.

Kevin Carlson (Jan 27 2025 at 22:55):

I'm also curious how the topos of double negation sheaves relates to the topos of groupoid-reflection-presheaves.

Kevin Carlson (Jan 27 2025 at 22:57):

Another way of seeing the result about $G$-set categories is that sub-$G$-sets are always disjoint unions of sets of complete orbits.

John Baez (Jan 27 2025 at 22:57):

Kevin Carlson said:

I see an examples section in that article!

True! I think the examples need examples, for a cave man like me to consider them "examples".

Kevin Carlson (Jan 27 2025 at 22:58):

I added $G$-sets :grinning_face_with_smiling_eyes:

Kevin Carlson (Jan 27 2025 at 22:58):

Oh no, that links to $G$-sets for a topological group $G$, and now I have to wonder about that...

Kevin Carlson (Jan 27 2025 at 22:58):

I think the same argument works, though.

John Baez (Jan 27 2025 at 22:59):

Kevin Carlson said:

I'm also curious how the topos of double negation sheaves relates to the topos of groupoid-reflection-presheaves.

In a recent conversation I think James Dolan nodded to my guess that if you take the category of presheaves on $\mathrm{B}\mathbb{N}$ and look at sheaves in the double negation topology you get the category of presheaves on $\mathrm{B}\mathbb{Z}$. $\mathbb{Z}$ is what you get when you take the monoid $\mathbb{N}$ and throw in inverses. But I'm not going to blame him if this guess of mine was incorrect!

Kevin Carlson (Jan 27 2025 at 23:26):

Cool! Presumably it can't always be that simple, though; I feel like somebody would've told me if the topos of double negation sheaves is always of presheaf type.

John Baez (Jan 27 2025 at 23:44):

Nobody would have told me. :crying_cat:

Ryuya Hora (Jan 28 2025 at 01:29):

Kevin Carlson said:

I'm also curious how the topos of double negation sheaves relates to the topos of groupoid-reflection-presheaves.

Though I don't know the relationship with the subtopos of sheaves of the double negation topology,
the topos of groupoid-reflection-presheaves is naturally a quotient topos of the original presheaf topos. In fact, the functor $C\to C[\text{everything}^{-1}]$ induces a connected (=lax-epi) geometric morphism $\mathbf{PSh}(C) \to \mathbf{PSh}(C[\text{everything}^{-1}])$. (On Functors Which Are Lax Epimorphisms)

Ryuya Hora (Jan 28 2025 at 01:36):

I feel this is similar to the theory of covering spaces. For a very nice space $X$, we have the full embedding functor between two topoi $\mathbf{Sh}(X) \hookleftarrow \mathbf{Cov}(X) \cong \mathbf{PSh}(\Pi_1(X))$ (and its right adjoint?).

Mike Shulman (Jan 28 2025 at 02:10):

I think a more common and concise notation for $C[\text{everything}^{-1}]$ is $C[C^{-1}]$. I don't think this can be the double-negation sheaves in general for the reason Ryuya gives: the functor goes the wrong direction.

Ryuya Hora (Jan 28 2025 at 02:25):

Mike Shulman said:

I don't think this can be the double-negation sheaves in general for the reason Ryuya gives: the functor goes the wrong direction.

I don't intend to claim that the double-negation sheaves are in this form. (In fact, there are lots of presheaf topoi whose booleanization is not a presheaf topos. Examples include the Schanuel topos, and the Cohen topos.)

Mike Shulman (Jan 28 2025 at 02:25):

Yeah, I was saying you said that they aren't. (-:

Mike Shulman (Jan 28 2025 at 02:26):

But now I see that my sentence could have been parsed the other way too. (-:O

Mike Shulman (Jan 28 2025 at 02:27):

That is, I meant "(I don't think this can be the double-negation sheaves in general) for the reason Ryuya gives", not "I don't think this can (be the double-negation sheaves in general for the reason Ryuya gives)".

Ryuya Hora (Jan 28 2025 at 02:29):

Mike Shulman said:

That is, I meant "(I don't think this can be the double-negation sheaves in general) for the reason Ryuya gives", not "I don't think this can (be the double-negation sheaves in general for the reason Ryuya gives)".

!! now I understand my misunderstanding!

Morgan Rogers (he/him) (Jan 28 2025 at 10:09):

Kevin Carlson said:

Oh no, that links to $G$-sets for a topological group $G$, and now I have to wonder about that...

Don't worry, those toposes are still Boolean

Morgan Rogers (he/him) (Jan 28 2025 at 10:24):

John Baez said:

Kevin Carlson said:

I'm also curious how the topos of double negation sheaves relates to the topos of groupoid-reflection-presheaves.

In a recent conversation I think James Dolan nodded to my guess that if you take the category of presheaves on $\mathrm{B}\mathbb{N}$ and look at sheaves in the double negation topology you get the category of presheaves on $\mathrm{B}\mathbb{Z}$. $\mathbb{Z}$ is what you get when you take the monoid $\mathbb{N}$ and throw in inverses. But I'm not going to blame him if this guess of mine was incorrect!

This does happen to be correct. In a conversation with @Jérémie Marquès recently I mistakenly generalized this, so I should explain what is special about this case. In general we have a functor $F: C \to C[C^{-1}]$ which induces an essential, connected geometric morphism between the corresponding presheaf toposes, as Ryuya and Mike were discussing; that is, the functor induced by the restriction along $F$ is fully faithful and has adjoints on each side given by Kan extension. Considering the left Kan extension, we have an adjunction between the presheaf toposes in which the right adjoint is fully faithful. But in order for that to be an inclusion of toposes, we need the left Kan extension to preserve finite limits, and it usually does not. This happens if $C[C^{-1}]$ is flat as a $C$-set, which is the case with $\mathbb{Z}$ over $\mathbb{N}$ but nothing general.

What is the relation to double negation? Sheaves for double negation form the largest Boolean subtopos and the smallest dense subtopos. A subtopos is dense iff it includes the initial object, in this case the empty presheaf, which is preserved by the inclusion. Thus if the left Kan extension preserves finite limits, then it presents the presheaves over $C[C^{-1}]$ as a dense Boolean subtopos, which must coincide with the double negation sheaves!

Fernando Yamauti (Jan 28 2025 at 15:22):

The left Kan extension is left exact when $C$ has a right calculus of fractions in the sense of Gabriel-Zisman. That happens, for instance, when one takes the localisation at the cartesian arrows of a site fibration over a filtered category. But in all those cases, the presheaf topos is already De Morgan. So that's not very interesting since the topos is already almost Boolean. Perhaps someone here know interesting examples where $C$ does not satisfy the Ore condition and, hence, the respective presheaf topos is not De Morgan...(?)

Matteo Capucci (he/him) (Jan 28 2025 at 16:25):

Kevin Carlson said:

Not exactly a name, but as a description, these are exactly presheaves on the groupoid reflection $C[\mathrm{everything}^{-1}].$ That at least characterizes the categories they form and provides an adjoint triple between them and arbitrary presheaves on $C.$

Good observation! This nicely motivates their 'local' feeling... (In fact, for P-local objects, that's exactly the same kind of local: an object $X$ is P-local if the restriction of $y^X$ over $C[P^{-1}]$ is again represented by $X$).

Matteo Capucci (he/him) (Jan 30 2025 at 08:48):

I found a characterization/name I'm happy with: these are presheaves for which the terminal map is étale in the sense of Joyal-Moerdiijk (nat squares are pullbacks)

Nathanael Arkor (Jan 30 2025 at 09:17):

I.e. [[cartesian natural transformations]]?

Matteo Capucci (he/him) (Jan 30 2025 at 09:40):

yep

Matteo Capucci (he/him) (Jan 30 2025 at 09:41):

John Baez said:

those are really the same as covering spaces of a certain space: the classifying space of that groupoid (which you build by putting in an n-simplex for each composable n-tuple of morphisms).

I wonder if it is the same as this observation?

Fernando Yamauti (Jan 30 2025 at 16:35):

Matteo Capucci (he/him) said:

John Baez said:

those are really the same as covering spaces of a certain space: the classifying space of that groupoid (which you build by putting in an n-simplex for each composable n-tuple of morphisms).

I wonder if it is the same as this observation?

Everything is about locally constant sheaves (aka local systems). The case of $G$-sets actually doesn't fail. It's just that the definition of locally constant was not the correct one or, rather, the site being considered was to small. From the point of view of topos theory, every object of $G$-sets is actually locally constant (in the sense that there is a covering of the terminal such that pulling back by that covering, gives us a disjoint union of constant sheaves).

The relation to topology appears when one considers things up to shape equivalence and, then, the sheaves over the topological $BG$ coincides with $G$-sets in the sense that they have the same category of local systems (that means they are shape equivalent).

In that case of sheaves over the topological $BG$, that initial naive definition of locally constant sheaves coincides with the correct one, because, now, the site is fat enough so that we can actually consider any $g \in G$ as a path which can be covered by a bunch of open sets (so that a path can be identified with a zig-zag of inclusions $U \cap V \to U$). That I think was your initial intuition: cover a space by opens and consider the (nerve of the) Cech groupoid.

That all work for presheaves over arbitrary small cats (in the higher sense also). In such a case, the presheaves in $\widehat{C}$ defined over the groupoid completion $\widehat{C [C^{-1}]}$ are exactly the locally constant objects of $\widehat{C}$.

Actually, all that also works to some extent for arbitrary topoi except that we actually get a pro-groupoid when your topos is not locally connected (also presheaves on that pro-groupoid will usually fail to embed fully faithfully as the topos of locally constant objects).

John Baez (Jan 30 2025 at 21:02):

Ryuya Hora said:

Kevin Carlson said:

I'm also curious how the topos of double negation sheaves relates to the topos of groupoid-reflection-presheaves.

Though I don't know the relationship with the subtopos of sheaves of the double negation topology, the topos of groupoid-reflection-presheaves is naturally a quotient topos of the original presheaf topos. In fact, the functor $C\to C[\text{everything}^{-1}]$ induces a connected (=lax-epi) geometric morphism $\mathbf{PSh}(C) \to \mathbf{PSh}(C[\text{everything}^{-1}])$. (On Functors Which Are Lax Epimorphisms)

I'm still curious about this. To keep things simple suppose $C = BM$ for some monoid $M$. Then $C[\text{everything}^{-1}]$, the category where we formally invert all morphisms in $C$, is $BG$ where $G$ is the [[group completion]] of the monoid $M$. (That's the group where we formally invert all elements of $M$, so what I'm saying is trivial.)

So, I'm curious about how double negation sheaves on $BM$ compare to presheaves on $BG$. Since I don't understand double negation sheaves very well, let's look at a very small example.

Take $M$ to be the 2-element monoid that's not a group: this is $\{F,T\}$ with "or" as its monoid operation, or $\{0,1\}$ with $0 + x = x$, $1 + x = 1$. Let's use the latter description to reduce confusion when we start doing a bit of logic.

Group completion of $M$ is a destructive process since $1 + x = 1$: when we give $1$ a formal inverse $-1$ we get

$1 = 0 + 1 = (-1 + 1) + 1 = (-1) + 1 + 1 = (-1) + 1 = 0$

and $G$ is the trivial group. So, the category of presheaves on $G$ is just $\mathsf{Set}$.

How does this compare with the category of double negation sheaves on $M$?
My intuition, and hand-wavy calculations, suggest that this category is also just $\mathsf{Set}$. But I'm not sure that's true.

Fernando Yamauti (Jan 30 2025 at 21:26):

John Baez said:

I'm still curious about this. To keep things simple suppose $C = BM$ for some monoid $M$. Then $C[\text{everything}^{-1}]$, the category where we formally invert all morphisms in $C$, is $BG$ where $G$ is the [[group completion]] of the monoid $M$. (That's the group where we formally invert all elements of $M$, so what I'm saying is trivial.)

So, I'm curious about how double negation sheaves on $BM$ compare to presheaves on $BG$. Since I don't understand double negation sheaves very well, let's look at a very small example.

Take $M$ to be the 2-element monoid that's not a group: this is $\{F,T\}$ with "or" as its monoid operation, or $\{0,1\}$ with $0 + x = x$, $1 + x = 1$. Let's use the latter description to reduce confusion when we start doing a bit of logic.

Any commutative monoid will work. That's because the site given by the delooping of $M$ satisfies the the conditions of a right calculus of fractions. As I'had mentioned above, that implies the left adjoint is left exact (just write down what sheaffication means using the formula for the Hom's of the localisation and notice that the formula is about filtered colimits).

But I would like to see a noncommutative example doing the job...

John Baez (Jan 30 2025 at 21:35):

Any commutative monoid will work.

By "work", do you mean the category of double negation sheaves on $BM$ is equivalent to the category of presheaves on $BG$?

Fernando Yamauti (Jan 30 2025 at 21:48):

John Baez said:

Any commutative monoid will work.

By "work", do you mean the category of double negation sheaves on $BM$ is equivalent to the category of presheaves on $BG$?

I think so. It defines a Boolean dense subtopos, so it has to be the one you get by the double negation topology.

Actually in the commutative case, the dense topology (or, equivalently the double negation one) coincides with the atomic topology. That's why I thought finding $C$'s that do the job and yet fail to satisfy the Ore condition would be interesting (not sure if they do exist though).

John Baez (Jan 30 2025 at 22:22):

That would be interesting. Being less knowledgeable about this stuff, I'd even like to see some simple examples of noncommutative $M$ that don't "work".

Fernando Yamauti (Jan 30 2025 at 22:51):

John Baez said:

That would be interesting. Being less knowledgeable about this stuff, I'd even like to see some simple examples of noncommutative $M$ that don't "work".

A case that fails (though it's not a monoid) was mentioned above by @Ryuya Hora, the cat of finite sets with monos. In that case also the atomic top is the double negation top (since the site satisfies the Ore condition) and we get the Schanuel topos. But I wonder if one can still think about the Schanuel topos as something codifying the homotopy type of the respective presheaf topos...

Perhaps someone here knows what's the étale homotopy type of the Schanuel topos (?). That could possibly clarify the connection to group completion.

Morgan Rogers (he/him) (Jan 31 2025 at 07:17):

John Baez said:

That would be interesting. Being less knowledgeable about this stuff, I'd even like to see some simple examples of noncommutative $M$ that don't "work".

The free monoid on two generators already doesn't work! If you want to check @Fernando Yamauti 's condition, you can check that it doesn't have a right calculus of fractions, but actually that boils down to the free monoid failing to satisfy an Ore condition (you can't multiply the generators on the same side by any element to make them equal).

Fernando Yamauti (Jan 31 2025 at 12:43):

Morgan Rogers (he/him) said:

John Baez said:

That would be interesting. Being less knowledgeable about this stuff, I'd even like to see some simple examples of noncommutative $M$ that don't "work".

The free monoid on two generators already doesn't work! If you want to check Fernando Yamauti 's condition, you can check that it doesn't have a right calculus of fractions, but actually that boils down to the free monoid failing to satisfy an Ore condition (you can't multiply the generators on the same side by any element to make them equal).

Nice! Does that answer my question negatively in general, then? I mean is it really impossible that any cat not satisfying the Ore condition to have the groupoid completion as the booleanisation?

I think I can prove that using a more general notion of calculus of fraction (the one in Cisinski's HA book) such that the colimit is not necessarily filtered anymore, but I didn't check it. Perhaps you have a more direct proof...

Maybe it's even the case a non De Morgan topos cannot have a Boolean subtopos that is also a quotient topos (i.e., the inclusion of double negation sheaves into sheaves has a right adjoint)?

Morgan Rogers (he/him) (Jan 31 2025 at 15:50):

I think it should be possible to make this concrete enough. The colimit used to calculate the left Kan extension, evaluated at an object $c$ of $C[C^{-1}]$, is indexed by the comma category $F^{\mathrm{op}}/c$, for $F: C \to C[C^{-1}]$ the localisation functor. Limits and colimits being computed pointwise, it is necessary for that category to be filtered for the left Kan extension to preserve finite limits.
A square in this category must in particular commute in $C$. Given a span $c \leftarrow d \to c'$ in $C$ (let's call those morphisms $f,f'$), we get a cospan in $F^{\mathrm{op}}/c$ of the form $\mathrm{id}_c \to f \leftarrow {f'}^{-1}\circ f$. The variance here is confusing, but the point is: since a square in $F^{\mathrm{op}}/c$ must have an underlying square in $C^{\mathrm{op}}$, if the original span in $C$ can't be completed to a square then this cospan can't either, so the diagram is not filtered! So for the double negation subtopos to coincide with presheaves on the localisation, the Ore condition is indeed necessary (and by extension $[C^{\mathrm{op}},\mathrm{Set}]$ is de Morgan).

Morgan Rogers (he/him) (Jan 31 2025 at 15:52):

Fernando Yamauti said:

Maybe it's even the case a non De Morgan topos cannot have a Boolean subtopos that is also a quotient topos (i.e., the inclusion of double negation sheaves into sheaves has a right adjoint)?

I have an open Masters project proposal on my website about de Morgan toposes; I should make note of this as a question to explore.

John Baez (Jan 31 2025 at 16:50):

Given your name I think you should write a paper about it. We could call you "de Morgan" Rogers.

John Baez (Jan 31 2025 at 16:55):

But seriously, while all this is over my head, I'm really glad to hear there's a necessary and sufficient condition for figuring out when booleanization can be done using a group completion!

I should attempt to get some intuition for the Ore condition. I'd seen it in ring theory once, but avoided learning about it. I see now that Wikipedia gives a simple explanation in the "General Idea" section of its article Ore condition. It's interesting that this explanation argues that the Ore condition is necessary to localize a ring at some multiplicative subset, not just sufficient. Is that related to your necessity argument, de @Morgan Rogers (he/him)?

John Baez (Jan 31 2025 at 17:02):

Actually, reading this article is suddenly transforming my attitude from "this is some high-powered mysterious topos theory stuff I'll never understand" to "maybe I could understand this: the analogue in ring theory actually makes sense".

Morgan Rogers (he/him) (Jan 31 2025 at 17:06):

The category theory version is even simpler, it's just about being able to complete spans or cospans to squares! [[Ore condition]]

Morgan Rogers (he/him) (Jan 31 2025 at 17:10):

I would have to work a bit harder to figure out if the Ore condition is sufficient; I expect the gap to be quite small if there is one.

John Baez (Jan 31 2025 at 21:42):

So are the left and right Ore conditions in ring theory 'just' the Ore conditions in category theory (every span or cospan can be extended to a commutative square) generalized slightly to Ab-enriched categories (where it takes the exact same form) and then restricted to one-object Ab-enriched categories (which are rings)?

(This is an instance of the category theorist's 'just', as in: "the thing other mathematicians understand is 'just' a special case of something category theorists understand".)

Fernando Yamauti (Feb 01 2025 at 03:01):

Morgan Rogers (he/him) said:

I would have to work a bit harder to figure out if the Ore condition is sufficient; I expect the gap to be quite small if there is one.

Wait. Isn't the Schanuel topos a counter-example? I mean it does satisfy the right Ore condition, right? And I'm not sure it can be a presheaf topos (is that even $\omega$-accessible?)...

Ryuya Hora (Feb 01 2025 at 04:24):

Fernando Yamauti said:

Wait. Isn't the Schanuel topos a counter-example? I mean it does satisfy the right Ore condition, right? And I'm not sure it can be a presheaf topos (is that even $\omega$-accessible?)...

The Shanuel topos is not a presheaf topos, due to its description as a topological group action topos. In fact, for a topological group $G$, the topos of continuous $G$-actions is a presheaf topos if and only if $G$ has the minimum open subgroup.

Ryuya Hora (Feb 01 2025 at 05:57):

In extreme cases where $C[C^{-1}]$ is chaotic, Morgan's comma category argument implies that the condition
・$C$ is cofiltered．
is sufficient and necessary.

In the case of the Schanuel topos, $C^{\mathrm{op}}= \mathbf{FinSet}_{\text{mono}}$ has a chaotic full-localization, is not filtered, and satisfies the Ore condition).

Morgan Rogers (he/him) (Feb 01 2025 at 08:36):

Nice!

Fernando Yamauti (Feb 01 2025 at 15:12):

Ryuya Hora said:

In extreme cases where $C[C^{-1}]$ is chaotic, Morgan's comma category argument implies that the condition
・$C$ is cofiltered．
is sufficient and necessary.

In the case of the Schanuel topos, $C^{\mathrm{op}}= \mathbf{FinSet}_{\text{mono}}$ has a chaotic full-localization, is not filtered, and satisfies the Ore condition).

Hmm... are you claiming that $\widehat{C [C^{-1}]}$ is the cat of sheaves for the double negation iff $C$ is cofiltered? If so, I don't think that can be the case, because there are several examples of $C$ not cofiltered that admits a right calculus of fractions. Extreme example: just take a discrete category.

Actually, when $C$ is (co)filtered, $C[C^{-1}]$ is a point. What we need is to require, at least, every $C / c$ to be cofiltered (not $c \setminus C$ as a confusingly had written before **), which, in this particular case, is the same thing as having cocones for finite discrete diagrams.

**PS: How do I cross out a text in math mode here?

Fernando Yamauti (Feb 01 2025 at 15:14):

Btw, that nlab article answers my other question about the shape (or étale hom type) of the Schanuel topos: it's contractible. Perhaps it's always true that the booleanisation preserves the shape...(?)

Fernando Yamauti (Feb 01 2025 at 17:40):

Fernando Yamauti said:

[...]? If so, I don't think that can be the case, because there are several examples of $C$ not cofiltered that admits a right calculus of fractions. Extreme example: just take a discrete category.

Actually, when $C$ is (co)filtered, $C[C^{-1}]$ is a point. What we need is to require, at least, every $C / c$ to be cofiltered (not $c \setminus C$ as a confusingly had written before **), which, in this particular case, is the same thing as having cocones for finite discrete diagrams.

**PS: How do I cross out a text in math mode here?

Let me rewrite the conclusion since I had messed up the variance here before. In order to avoid further confusions, let's instead use $\widehat{C^{o}}$ and talk about left calculus of fractions instead.

A left calculus of fractions (for $C$ the weak equivalences) means exactly that $c \setminus C$ is filtered. On the other side left exactness of the left adjoint means exactly that $L/ c$ is filtered for $L \colon C \to C[C^{-1}]$. Now we just have to notice that $L/ c$ is filtered iff $c \setminus C$ is filtered.

For any finite diagram $D$ in $c \setminus C$ there exists a corresponding finite diagram $D'$ in $L / c$ given by taking every $f \colon c \to d$ in $C$ and sending it to $L(f)^{-1}$. A cocone of $D'$ is a cocone of $D$ and, hence, if $L/ c$ is filtered $c \setminus C$ is also filtered. That means a left calculus of fractions is a also a necessary condition.

That it is a sufficient condition is Gabriel-Zisman formula for the left adjoint (aka right derived functor) using filtered colimits. So the Ore condition is not enough and we need exactly a right calculus of fractions on $C$ (when we are talking about $\widehat{C}$).

Dammit! I have to stop insisting on using opposite categories. I can't sustain my attention for long enough while iterating the opposition operation.

Ryuya Hora (Feb 02 2025 at 01:38):

Fernando Yamauti said:

Hmm... are you claiming that $\widehat{C [C^{-1}]}$ is the cat of sheaves for the double negation iff $C$ is cofiltered?

No, sorry for making confusion. I claim it only for the extreme cases where $C[C^{-1}]$ is chaotic, i.e., $C[C^{-1}]\simeq 1$. This suffices for the Schanuel topos, since $\mathbf{FinSet}_{\text{mono}}$ satisfies this "extreme" condition. (A discrete category $C$ does not satisfy it unless $C=1$.)

In this extreme cases, the condition `$L/c$ is filtered' (as you mentioned in)
Fernando Yamauti said:

left exactness of the left adjoint means exactly that $L/ c$ is filtered for $L \colon C \to C[C^{-1}]$.

becomes a bit easier; We need to check only one object $c \in \mathrm{ob}(C[C^{-1}])$ and the comma category $L/c$ is equivalent to $C$. (We need to put $\mathrm{op}$ somewhere...)

Fernando Yamauti (Feb 02 2025 at 01:58):

Ryuya Hora said:

No, sorry for making confusion. I claim it only for the extreme cases where $C[C^{-1}]$ is chaotic, i.e., $C[C^{-1}]\simeq 1$. This suffices for the Schanuel topos, since $\mathbf{FinSet}_{\text{mono}}$ satisfies this "extreme" condition. (A discrete category $C$ does not satisfy it unless $C=1$.)

Ah!Ok. My bad. I've never heard anyone call that chaotic :sweat_smile: . I'd thought you were referring to the chaotic site or chaotic topology.

But, anyways, I guess everything is settled and $\widehat{C}$ is such that the booleanisation is the groupoid completion iff $C$ has a right calculus of fractions (for $C$ as the weak equivalences).

Matteo Capucci (he/him) (Feb 03 2025 at 19:28):

That's so interesting!