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Stream: deprecated: topos theory

Topic: Factorization system on points

Tim Campion (Nov 04 2022 at 18:58):

Is there written anywhere an account of the structure needed on a topos $\mathcal B$ in order for the category of $\mathcal X$-points of $\mathcal B$ to acquire the structure of an orthogonal factorization system (naturally in $\mathcal X$... in some sense)?

Tim Campion (Nov 04 2022 at 18:58):

I'd be happy to restrict attention to the case where $\mathcal B$ classifies an essentially algebraic theory.

Tim Campion (Nov 04 2022 at 19:02):

Maybe this is a reasonable enough thing to ask for without a specific motivation, but my motivating example would be $\mathcal B = BCRing$, the classifying topos for commutative rings. Here, the $\mathcal X$-points of $\mathcal B$ have a (localization, conservative) factorization system, which should be functorial in $\mathcal X$ in the sense which is appropriate.

Reid Barton (Nov 04 2022 at 19:25):

Have you looked at Lurie's "fractured $\infty$-topoi" (SAG chapters 20, 21)?

Reid Barton (Nov 04 2022 at 19:27):

& specifically Theorem 21.3.0.1 here

Reid Barton (Nov 04 2022 at 19:32):

Mathieu Anel and I thought about this topic a while back but I would have to refresh my memory--you might write to him and see what he has to say.

Tim Campion (Nov 04 2022 at 20:32):

Oh nice! I was looking there, but had not yet come across that theorem, thanks! (Although now looking at the title of Section 21.3 it's clear I wasn't looking hard enough!)

Tim Campion (Nov 04 2022 at 20:33):

So Lurie shows that a fractured $\infty$-topos gives rise to such factorization systems (where reindexing preserves both classes)

Tim Campion (Nov 04 2022 at 20:34):

I wonder if a converse is to be expected...

Morgan Rogers (he/him) (Nov 05 2022 at 07:59):

@Axel Osmond do you have some sufficient conditions in the 1-topos case?

Axel Osmond (Nov 05 2022 at 13:27):

thanks for the notification @Morgan Rogers (he/him) !
If you have an essentially algebraic theory $\mathbb{T}$, any left generated factorization system on $\mathbb{T}[\mathcal{S}]$ will extend to a factorization system in the category of models of any Grothendieck topos, and both the left and right classes will be stable under inverse image (and also actually direct image). A left generated factorization system is just a factorization system where the left class is obtained as the inductive completion of a set of finitely presented maps, which by the way can always be chosen to be saturated under composition, right cancellation, and finite colimits. You then obtain the factorization of a map $f : B \rightarrow A$ under a given object by taking the filtered colimit of all its possible factorizations through a pushout under $B$ of finitely presented left maps. This is done in detail in Anel "Grothendieck topologies from factorization systems" and Coste "Topos, Tripes and Spectra". You can find it also in detail in the first chapter of my thesis (and the connection with geometries in the 3rd chapter).

Tim Campion (Nov 05 2022 at 14:51):

Axel Osmond said:

thanks for the notification Morgan Rogers (he/him) !
If you have an essentially algebraic theory $\mathbb{T}$, any left generated factorization system on $\mathbb{T}[Set]$ will extend to a factorization system in the category of models of any Grothendieck topos, and both the left and right classes will be stable under inverse image (and also actually direct image). A left generated factorization system is just a factorization system where the left class is obtained as the inductive completion of a set of finitely presented maps, which by the way can always be chosen to be saturated under composition, right cancellation, and finite colimits. You then obtain the factorization of a map $f : B \rightarrow A$ under a given object by taking the filtered colimit of all its possible factorization through a pushout under $B$ of a finitely presented left map. This is done in detail in Anel "Grothendieck topology from factorization systems" and Coste "Topos, Tripes and spectra". You can find it also in detail in the first chapter of my thesis (and the connection with geometries in the 3rd chapter).

Thanks, that's great! So in the case where $\mathcal B$ classifies an essentially-algebraic theory, you can induce a factorization system on points by starting from a factorization system on $\mathcal S$-points which is generated by finitely-presentable maps.

Now let me get greedy. I wonder if there is a way of packaging this data which I can more easily make sense of from a "toposes as generalized spaces" sort of perspective?

Toward that end, here's a wildly speculative analogy. Let's think of a category $C$ as being like a "directed space" $X$. The objects of $C$ correspond to points of $X$, and the morphisms of $C$ correspond to "paths" in $X$. Suppose that $C$ is equipped with an orthogonal factorization system $(\mathcal L, \mathcal R)$. Then I claim that what $(\mathcal L, \mathcal R)$ is giving you is a notion of "shortness" or "boundedness" of paths, in the sense that a morphism in $\mathcal L$ should be thought of as a path which is "short"/ "bounded". The closure properties of $\mathcal L$ are consistent with this: identity paths are "short", a composite of "short" paths is "short", a small colimit of "short" paths is "short". The cobase-change of a "short" path is some kind of "translation" of that path, and so also "short"; right cancellation also kind of makes sense.

One sort of intuition I want to have is that "short" might mean "of length less than $r$" for some fixed $r \geq 0$ -- but if this intuition is to be used, we should think of the notion of "length" as coming from some kind of "ultrametric" rather than an "archimedean" metric (hence the closure under composition). In this sense, maybe a notion of "boundedness" is a better intuition.

So from this perspective, putting a factorization system on $C = Geom(\mathcal X, \mathcal B)$ is endowing the points of this mapping "space" with some kind of binary notion of "closeness". In the case at hand, we have that a notion of "closeness" on the $\mathcal S$-points $Geom(\mathcal S, \mathcal B)$ induces such a notion on all mapping spaces -- presumably the resulting factorization systems are all kind of "levelwise"?

Ok, I realize now I don't know where I'm going with this, but I wonder if that sort of picture makes sense to anyone? (And optimistically, does it maybe even tie in with these "theories of spectra" somehow?)

Morgan Rogers (he/him) (Nov 05 2022 at 16:05):

I like the picture, and it can probably be interpreted literally in something like the Butz-Moerdijk construction of a topological groupoid from the category of points (although I don't know of any resource for understanding those groupoids concretely enough to make such a picture "useful")

On the logical side, one could probably come up with sufficient structural criteria on a theory to derive factorization systems of a given form, especially when a factorization system exists already at the level of finitely presentable models. Lots of theories have an "injection-surjection" factorization, for instance, which must surely be definable under mild conditions.

Axel Osmond (Nov 05 2022 at 22:17):

I never thought of those factorization systems as expressing some notion of closeness, so before discussing this, I would rather recall a bit how those factorizations involved in the theory of spectrum can be understood either from a logical and a geometric point of view :

• geometrically, the left maps are dual to some saturated compact inclusions (amongst which the finitely presented one code for compact opens), while the right maps encode some notion of connectedness. This is related to the (connected, truncated) pattern that encompasses a lot of examples of factorization systems in categories of spaces, in particular various factorization systems of geometric morphisms.
• but left-generated factorization systems also have a logical interpretation: the finitely presented left maps from which the system is generated code for a class of function symbols (for they are dual to some provably functional formulas in the syntactic category of your theory) whose pushouts under arbitrary objects are to be seen as their interpretation in those objects ; on the other side, right maps are those that reflects the objects constructed from those function symbols. For instance, localizations create new invertibles and conservative morphisms reflect those latter.
  However those explanations make sense in the context of a spectral construction, where not all objects of your ambient category are to be seen as points for your geometry - only a distinguished class satisfying some condition relating them to your right class - e.g. : local rings.

Axel Osmond (Nov 05 2022 at 23:09):

@Tim Campion but if I understand well, the view point your are describing is quite different as you want to see morphisms in the factorization system to express data between points. So let us describe in detail, from a somewhat ionadic point of view, what can of topological situation this could express.

Morphisms between (generalized) points of a topos should be seen somewhat encoding specialization order; then a factorization system would distinguish two kinds of specializations between points, obeying dual cancellations properties and stability under (co)limits. For a topos $\mathcal{E} \simeq Sh(\mathcal{C}, J)$, basic compact open for the topology on the points of $\mathcal{E}$ - which correspond to objects of the site $\mathcal{C}$- induce discrete opfibrations of points through the evaluation functor $\mathcal{C} \rightarrow \mathcal{S}^{pt(\mathcal{E})}$, where the lifting property of opfibration tells you that a witness that a point $X_1$ lies in an basic open $c$ (that is, an element $a$ of $X_1(c)$) is sent along a specialization map $f :X_1 \rightarrow X_2$ to a witness that its codomain $X_2$ is also in this open $c$ ("open are upclosed" for the specialization).

But now if you have a distinguished class of maps $\Lambda$ in the site (as in the case of a left generated system), this can be seen as a distinguished class of inclusions between basic compact opens of your basis, stipulating that an compact open is "$\Lambda$-sufficiently" included inside of another. From a logical point of view, seeing $\mathcal{C}$ as the syntactic site of some geometric theory $\mathbb{T}$ and hence $\Lambda$ as a class of functional symbols, this notion for a basic open to being "sufficiently" inside another expresses that you only need operations coded by $\Lambda$ to produce witnesses that a point $X$ lies inside a basic open $U$ sufficiently containing another one $V$ from a witness that $X$ was in $V$: in other words, this is inclusion that can be "tested" at the level of points in term of $\Lambda$ operations.

Axel Osmond (Nov 06 2022 at 01:00):

But now what is the meaning of left and right maps of the factorization system induced by $\Lambda$ on points - at least to set-valued points ?

For left maps, as they are filtered colimits of (duals of) $\Lambda$ maps, i would say that a left map $l : X_1 \rightarrow X_2$ is a specialization that uses only $\Lambda$-operations to transfer witnesses : for any witness $a$ that $X_1$ lies in $U$, there is a $\Lambda$ map $n : V \rightarrow U$ such that the witness $l(a)$ that $X_2$ is also in $U$ comes from some witness that it lies in $V$ through the $\Lambda$ operation coded by $n$. So indeed, it could be somewhat seen as a notion of "closeness" or "reachability" through $\Lambda$ between points in the sense that on can glide from a point to another related throuhg a left map using only $\Lambda$ operations on witnesses, in a "$\Lambda$-continuous way".

Dually, a right map between points $r : X_1 \rightarrow X_2$ is a natural transformation in $LexSite[\mathcal{C}, \mathcal{S}]$ whose naturality square at a $\Lambda$-map $n : U \rightarrow V$ of $\mathcal{C}$ is a pullback exhibiting $X_1(U_1) \simeq X_1(V) \times_{X_2(V)} X_2(U)$. In some sense witnesses are reflected along right maps : for $U$ $\Lambda$-sufficiently inside $V$, if a witness that $X_1$ is in $V$ is transported along $r$ to something that can be produced through a $\Lambda$ operation from a witness that $X_2$ is in $U$, then you could already extract a witness that $X_1$ was in $U$ along the same operation. I am not sure right now how to interpret it more spatially in terms of reachability, but this looks somewhat like a inaccessibility condition: you cannot enter through $r$ into an open sufficiently inside another one you started in, unless you are actually already inside. So in some sense, right maps are somewhat "slow paths" that cannot get you "on time" from outside to inside.

I dont know whether this fits the kind of interpretation of factorization systems you need. I will try to figure out what more we can say from this viewpoint. Concerning your question relative to the link with the theory of spectra, i would say this is a very different approach as spectra dont consider ambient object where the factorization system lives as points, rather as spaces, and the left and right maps as kind of inclusions, while here we really see the whole category as a single space and the morphisms as some topological data on the points.

Tim Campion (Nov 06 2022 at 15:58):

Thanks for the detailed analysis, @Axel Osmond ! I like the idea of the left class encoding some kind of "specific notion of reachability".

Example 1: I'm envisioning now a space whose points are points on the Earth, and whose morphisms are not just paths, but itineraries from one point to another; then maybe we have a "factorization system" (maybe more like a prefactorization system) where the left half only includes itineraries by foot and/or car; itineraries which involve a plane or train are excluded from the left class. So we've constrained the operations available to us to reach the destination.

Example 2: Or maybe we have a space where the points are people, and the paths are ways to send a message from one to another. Maybe there's a factorization system where the left half only allows electronic communication. Maybe the right half then gives some notion of message-passing where electronic communication counts as "free", so that only the parts requiring a courier sort of count as "costly" in the messaging system... that's not quite consistent with the initial picture, but maybe there's a way to turn this into a sensible picture...

Tim Campion (Nov 06 2022 at 16:06):

Example 3: It's probably also worth mentioning that the (connected, truncated) factorization systems you bring up are very much on my mind here. There's an analogy (used in Goodwillie calculus for instance) which treats the $\infty$-category of $Spaces$ as a "directed space", with points being the spaces, and paths being the maps of spaces. An $n$-connected morphism $Y \to X$ is thought of as exhibiting $Y$ as being close to $X$ -- the bigger $n$, the closer it is, and more specifically it's thought of as exponential in $n$. That is, an $n$-connected morphism $Y \to X$ exhibits $Y$ as "lying in a ball of radius $R^{-n}$ centered at $X$" where $R > 1$ is some fixed "scale factor".

Vista: (So when I think about factorization systems as encoding some notion of closeness, one thing I eventually hope to be able to touch on is how different factorization systems can fit together to provide something like the structure of a metric space, or a uniform space -- giving different relative notions of "closeness" which fit together systematically.)

Tim Campion (Nov 06 2022 at 16:09):

Wild speculation: (It occurs to me now that if this perspective is really to be taken seriously, one might even dream of doing something like the following: consider all the different commonly-used topologies in algebraic geometry -- there's a whole zoo of them, and they generally are each related to some sort of factorization system. One might wildly speculate that these different topologies / factorization systems fit together in some sort of meta structure, just as the different entourages in a uniform space fit together into a uniform structure.)

Tim Campion (Nov 06 2022 at 16:16):

Clarification: More to the point, I suppose another thing to say is that for the kind of geometric picture I have in mind, it's maybe not essential to be thinking about factorization systems per se. Really it looks to me like a story of classes of morphisms with various closure properties (focusing mostly on "left classes") -- you could imagine varying the closure properties and still being able to interpret the resulting notions in some kind of "geometric" way as encoding different kinds of "reachability" notions.

Tim Campion (Nov 06 2022 at 16:22):

Technical point: Specifically in the essentially algebraic setting, I still don't know how to think about this phenomenon where a finitely-presented model of the theory plays two roles: It's both a "point", as a model of the theory, and also a "compact open", as an object of the site presenting the theory. Should I be thinking in analogy to the little Zariski site of a ring $R$, where the points $P$-- and especially the non-closed points -- play a similar sort of dual role, where you often think of them in terms of their closures $\bar P$, which are more like "predicates" than points? (Hmm... but the closure of a point is generally not open... though $\bar P$ does correspond to an object (and hence a "compact open") of the big Zariski site of $R$... which, unlike the little Zariski spectrum, actually classifies an essentially algebraic theory... )