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1. • CommentRowNumber1.
   • CommentAuthorJohn T
   • CommentTimeJun 3rd 2025
   • (edited Jun 3rd 2025)
   • PermaLink
   Author: John T
   Format: MarkdownItexHello! I am rather new here, so apologies if I get the format wrong at all. I am following advice to cross-post [the following question](https://categorytheory.zulipchat.com/#narrow/channel/229199-learning.3A-questions/topic/Modal.20HoTT.3A.20More.20structured.20analog.20of.20pointing/near/521761374) from the category theory Zulip, in particular to see whether Urs Schreiber might be interested: ###The question### Quite some time ago, I was reading about negative moments vaguely around [here](https://ncatlab.org/nlab/show/Science+of+Logic#NegativeMoment) in nLab. It struck me that the negative of a monad couldn't necessarily be evaluated at the initial object, as the construction requires $\bigcirc X$ to be pointed (which we might get by simply taking $X$ itself to be pointed). Well, pointing is a morphism from $*$ to $X$, and, having just read the nearby section on $\emptyset\dashv*$ being the "initial opposition," I started wondering about morphisms like ʃ$X\to X$. I decided that I would want to work in the category of diagrams of the form [Diagram link](https://q.uiver.app/#q=WzAsNSxbMCwwLCJcXGJpZ2NpcmNcXGJpZ2NpcmMgWCJdLFswLDIsIlxcYmlnY2lyYyBYIl0sWzIsMCwiXFxiaWdjaXJjIFgiXSxbMiwyLCJYIl0sWzQsMiwiXFxiaWdjaXJjIFgiXSxbMCwxLCJcXGJpZ2NpcmMgcyJdLFswLDIsIlxcbXUiLDJdLFsxLDMsInMiXSxbMiwzLCJzIiwyXSxbMiw0LCJcXG9wZXJhdG9ybmFtZXtJZH1fe1xcYmlnY2lyYyBYfSIsMl0sWzMsNCwiXFxldGEiLDJdXQ==) ![Diagram that shows a morphism from circle X to X obeying laws similar to those for an algebra](https://categorytheory.zulipchat.com/user_uploads/thumbnail/21317/jErcxjzC8vLFj_pM2z4JJXGn/image.png/840x560.webp) which by chasing split monos can be simplified to the rules that $\bigcirc s=\mu$ and that $s$ is a right-inverse for $\eta$. I couldn't find a name for this (it's sort of similar to an algebra but with the inverse being on the wrong side), so I'll call it a $\bigcirc$-store here. A $*$-store is a pointed object, a ʃ-store is an object with something like a canonical embedding of its fundamental infinity-groupoid into itself, et cetera. The $\bigcirc$-stores form a category in the obvious way, with morphisms being $f:X\to Y$ such that everything commutes after $\bigcirc^i f$ is applied to the corners of the storage diagram. My question is when the category of $\bigcirc$-stores (say, in a topos or (inf,1)-topos) has pullbacks (which would allow us to look at "relatively negative" moments by pulling back along the unit map and its composition with the storage map). It certainly should if $\bigcirc$ preserves pullbacks, but I think the category of ʃ-stores might have pullbacks as well, despite ʃ itself not preserving them? There's certainly a unique way we might hope to lift storage maps on three objects sitting in a cospan to one on the pullback, per this diagram: [Diagram link](https://q.uiver.app/#q=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) ![A cube where the three front vertical edges are storage diagrams arranged in a cospan, with the bottom rear corner being a pullback U, p1, p2 of the lower-face cospan on objects X, Y, Z, and the top rear corner being circle U with morphisms circle p1 and circle p2 to circle X and circle Y. These compose with the storage maps to make a commutative diagram, so there should be a unique induced morphism from circle U into the pullback U of the bottom face.](https://categorytheory.zulipchat.com/user_uploads/thumbnail/21317/tGkuXvkJTzhb1rDE1FWgGQlv/image.png/840x560.webp) The question is really whether this induced map is a right inverse for the unit map, and I've worked through a couple examples without feeling like I've learned much. Assuming the pullbacks do exist, I would think that it would be interesting to pull back unit maps for various monads along (said unit composed with) storage maps, to look at what structure the monad is collapsing in regions local to more generic "small" subobjects besides just points. The cofiber of a counit could also be generalized to give some relative negation there (and of course we could dually talk about co-stores and the like), but I don't think that taking a relative negation of a comonad against a monad requires the store structure in order to be defined. Negating flat along shape could still be interesting (I'd think it would handle objects with distinct components of differing dimension "better" somehow), but I'll admit I'm not particularly familiar with this area, hence my asking about pullbacks in [Zulip channel "learning: questions"] rather than asserting something in a results channel. Learning resources would be welcome as well as answers or hints or whatnot. Thanks! ###Additional thoughts### Apologies again if I've messed up the post at all or if I'm missing anything simple regarding the problem or misunderstanding something about your work. Ultimately my hope with this was to prove that, if C is some sufficiently nice category with a particular nice monad M and D is the category of M-stores in C, negative moments relative to M could be defined as pullbacks in D, and would have some nice structure such as the relative negation of a monad on C w.r.t. M being comonadic on D, in a manner generalizing properties of standard negative moments in particular. Perhaps there would even be something like a relative differential cohomology hexagon? But I really don't yet know enough to be speculating. I hope this was interesting!

   Hello! I am rather new here, so apologies if I get the format wrong at all. I am following advice to cross-post the following question from the category theory Zulip, in particular to see whether Urs Schreiber might be interested:

   The question

   Quite some time ago, I was reading about negative moments vaguely around here in nLab. It struck me that the negative of a monad couldn’t necessarily be evaluated at the initial object, as the construction requires $\bigcirc X$ to be pointed (which we might get by simply taking $X$ itself to be pointed). Well, pointing is a morphism from $*$ to $X$, and, having just read the nearby section on $\emptyset\dashv*$ being the “initial opposition,” I started wondering about morphisms like ʃ$X\to X$. I decided that I would want to work in the category of diagrams of the form Diagram link

   which by chasing split monos can be simplified to the rules that $\bigcirc s=\mu$ and that $s$ is a right-inverse for $\eta$. I couldn’t find a name for this (it’s sort of similar to an algebra but with the inverse being on the wrong side), so I’ll call it a $\bigcirc$-store here. A $*$-store is a pointed object, a ʃ-store is an object with something like a canonical embedding of its fundamental infinity-groupoid into itself, et cetera. The $\bigcirc$-stores form a category in the obvious way, with morphisms being $f:X\to Y$ such that everything commutes after $\bigcirc^i f$ is applied to the corners of the storage diagram.

   My question is when the category of $\bigcirc$-stores (say, in a topos or (inf,1)-topos) has pullbacks (which would allow us to look at “relatively negative” moments by pulling back along the unit map and its composition with the storage map). It certainly should if $\bigcirc$ preserves pullbacks, but I think the category of ʃ-stores might have pullbacks as well, despite ʃ itself not preserving them? There’s certainly a unique way we might hope to lift storage maps on three objects sitting in a cospan to one on the pullback, per this diagram: Diagram link

   The question is really whether this induced map is a right inverse for the unit map, and I’ve worked through a couple examples without feeling like I’ve learned much.

   Assuming the pullbacks do exist, I would think that it would be interesting to pull back unit maps for various monads along (said unit composed with) storage maps, to look at what structure the monad is collapsing in regions local to more generic “small” subobjects besides just points. The cofiber of a counit could also be generalized to give some relative negation there (and of course we could dually talk about co-stores and the like), but I don’t think that taking a relative negation of a comonad against a monad requires the store structure in order to be defined. Negating flat along shape could still be interesting (I’d think it would handle objects with distinct components of differing dimension “better” somehow), but I’ll admit I’m not particularly familiar with this area, hence my asking about pullbacks in [Zulip channel “learning: questions”] rather than asserting something in a results channel. Learning resources would be welcome as well as answers or hints or whatnot. Thanks!

   Additional thoughts

   Apologies again if I’ve messed up the post at all or if I’m missing anything simple regarding the problem or misunderstanding something about your work. Ultimately my hope with this was to prove that, if C is some sufficiently nice category with a particular nice monad M and D is the category of M-stores in C, negative moments relative to M could be defined as pullbacks in D, and would have some nice structure such as the relative negation of a monad on C w.r.t. M being comonadic on D, in a manner generalizing properties of standard negative moments in particular. Perhaps there would even be something like a relative differential cohomology hexagon? But I really don’t yet know enough to be speculating.

   I hope this was interesting!

2. • CommentRowNumber2.
   • CommentAuthorJohn T
   • CommentTimeJun 3rd 2025
   • PermaLink
   Author: John T
   Format: MarkdownItex(I just noticed- the map from $\bigcirc Z$ to $Z$ in the cube diagram should be labeled $t$, not $s$, my bad)

   (I just noticed- the map from $\bigcirc Z$ to $Z$ in the cube diagram should be labeled $t$, not $s$, my bad)

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 5th 2025
   • (edited Jun 5th 2025)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThis looks interesting --- though it's also a little artificial, isn't it, which may not be unrelated to your question: If the construction were a standard one of categorical algebra, then there would be a good chance that existence of limits like pullbacks would follow from general facts. But if the motivation is (?) mostly to bring pointed objects into the picture, based on the initial opposition, then I'd point to the simple but neat observation ([here](https://ncatlab.org/schreiber/show/Modern+Physics+formalized+in+Modal+Homotopy+Type+Theory#AspectsOfTheInitialOpposition)) that the negative aspect of the pure non-being of $X$ is $$ \overline{\varnothing} X \;\equiv\; cofib(\varnothing \to X) \;\simeq\; X \sqcup \ast \,. $$ Now, this operation $\overline{\varnothing}$ --- the determinate negation of pure nothingness --- is [[maybe monad|itself a monad]]. While not idempotent, we can still regard its modules as its modal objects (as such I like to call them the *modales*) and these $\overline{\varnothing}$-modales [are](https://ncatlab.org/nlab/show/maybe+monad#EMCategoryAndRelationToPointedObjects) exactly the pointed objects! So if the goal is to see how pointed objects would arise in the *Prozeß*, then $\overline{\varnothing}$ suggests itself as the answer. (It's also curious to note that pointedness together with the smash product that comes with it is the first inkling of linearity. The objects that are not only $\overline{\varnothing}$-modal but also infinitesimal modal are like linear spaces.)

   This looks interesting — though it’s also a little artificial, isn’t it, which may not be unrelated to your question: If the construction were a standard one of categorical algebra, then there would be a good chance that existence of limits like pullbacks would follow from general facts.

   But if the motivation is (?) mostly to bring pointed objects into the picture, based on the initial opposition, then I’d point to the simple but neat observation (here) that the negative aspect of the pure non-being of $X$ is

   $$\overline{\varnothing} X \;\equiv\; cofib(\varnothing \to X) \;\simeq\; X \sqcup \ast \,.$$

   Now, this operation $\overline{\varnothing}$ — the determinate negation of pure nothingness — is itself a monad. While not idempotent, we can still regard its modules as its modal objects (as such I like to call them the modales) and these $\overline{\varnothing}$-modales are exactly the pointed objects!

   So if the goal is to see how pointed objects would arise in the Prozeß, then $\overline{\varnothing}$ suggests itself as the answer.

   (It’s also curious to note that pointedness together with the smash product that comes with it is the first inkling of linearity. The objects that are not only $\overline{\varnothing}$-modal but also infinitesimal modal are like linear spaces.)

4. • CommentRowNumber4.
   • CommentAuthorJohn T
   • CommentTimeJun 5th 2025
   • PermaLink
   Author: John T
   Format: MarkdownItexThank you for your response! Has more been written on the linearity point? That sounds particularly interesting. Regarding the motivation here- My *initial* motivation was pretty much just to mess around and see whether I could learn something, as this is interesting but dissimilar from what I normally spend my time on, and I could use a better intuition for how things work if I'd like to get much more out of it than what I have thus far. If I had to justify it *now* though: my current mental model of what the negative of an idempotent monad "does" is that it's *sort of* like localization at a point, extracting the geometric structure in the original object that gets reduced to a plain point upon applying the monad's unit map. Of course, even if we assume that the point in $\bigcirc X$ that gets placed into the pullback square really comes from a point in $X$, just how "local" this is would depend on which monad is getting applied. It could range from the application of no change at all (negative of * is Id) to picking out the entire connected component that the point is in (negative of shape?) to actual properly local structure for monads that "leave points where they are" and only contract away local structure. Now the motivation is that, if it's interesting to examine the sort of structure that's "local" to a point subobject, it might be interesting to try "localizing" at more complicated (but still somehow "skeletal") subobjects in the same way. We wouldn't expect to get anything extra if the monad we're looking at just contracts these subobjects to points anyway, but perhaps matching a monad that "leaves underlying points where they are" with a shape-store that points each component independently and doesn't reduce away homotopy structure would give us additional interesting information. If the monad we're negating normally removed/trivialized just enough information/structure to make tangent spaces trivial, then perhaps looking locally at what gets contracted along the nontrivial point-to-self paths in a shape-store would tell us something about holonomy along those paths in the original space? Though, now that I've written that, we might want to try specifically for information that doesn't depend on the particular $\bigcirc$-store we chose to work with, and I'm not sure that that's built in to this proposal. As for why $\bigcirc$-stores specifically might still be an interesting starting point for this, the original motivation remains the same (and that's at least part of why the Zulip question is phrased the way it is), but the in-retrospect justification I'd give now is that they just seemed to me like the "natural" way to get a subobject that represented the minimal amount of information *preserved* by a given idempotent monad. The specific storage rules were really motivated by trying to match pointing in the case of * and to make it "play nicely" with the monad in general, though (aside from a similarity to how algebras for a monad work) I've forgotten the exact chain of reasoning that led me to believe that this was the correct way to do it. Again, I'm not really native to this area of math, so to speak, so it's very possible that there's some less-"artificial" thing that would work better, or that "stores" aren't the most natural thing to combine with this localization/ relative negation concept generally. In any case, I'm eager to learn. :)

   Thank you for your response! Has more been written on the linearity point? That sounds particularly interesting. Regarding the motivation here-

   My initial motivation was pretty much just to mess around and see whether I could learn something, as this is interesting but dissimilar from what I normally spend my time on, and I could use a better intuition for how things work if I’d like to get much more out of it than what I have thus far.

   If I had to justify it now though: my current mental model of what the negative of an idempotent monad “does” is that it’s sort of like localization at a point, extracting the geometric structure in the original object that gets reduced to a plain point upon applying the monad’s unit map. Of course, even if we assume that the point in $\bigcirc X$ that gets placed into the pullback square really comes from a point in $X$, just how “local” this is would depend on which monad is getting applied. It could range from the application of no change at all (negative of * is Id) to picking out the entire connected component that the point is in (negative of shape?) to actual properly local structure for monads that “leave points where they are” and only contract away local structure. Now the motivation is that, if it’s interesting to examine the sort of structure that’s “local” to a point subobject, it might be interesting to try “localizing” at more complicated (but still somehow “skeletal”) subobjects in the same way. We wouldn’t expect to get anything extra if the monad we’re looking at just contracts these subobjects to points anyway, but perhaps matching a monad that “leaves underlying points where they are” with a shape-store that points each component independently and doesn’t reduce away homotopy structure would give us additional interesting information. If the monad we’re negating normally removed/trivialized just enough information/structure to make tangent spaces trivial, then perhaps looking locally at what gets contracted along the nontrivial point-to-self paths in a shape-store would tell us something about holonomy along those paths in the original space? Though, now that I’ve written that, we might want to try specifically for information that doesn’t depend on the particular $\bigcirc$-store we chose to work with, and I’m not sure that that’s built in to this proposal.

   As for why $\bigcirc$-stores specifically might still be an interesting starting point for this, the original motivation remains the same (and that’s at least part of why the Zulip question is phrased the way it is), but the in-retrospect justification I’d give now is that they just seemed to me like the “natural” way to get a subobject that represented the minimal amount of information preserved by a given idempotent monad. The specific storage rules were really motivated by trying to match pointing in the case of * and to make it “play nicely” with the monad in general, though (aside from a similarity to how algebras for a monad work) I’ve forgotten the exact chain of reasoning that led me to believe that this was the correct way to do it. Again, I’m not really native to this area of math, so to speak, so it’s very possible that there’s some less-“artificial” thing that would work better, or that “stores” aren’t the most natural thing to combine with this localization/ relative negation concept generally. In any case, I’m eager to learn. :)

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 6th 2025
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItex> Has more been written on the linearity point? There's a little more at [Science of Logic](https://ncatlab.org/nlab/show/Science+of+Logic#quantum_mechanics_quantumchemism). It gets mentioned from time to time, as [here](https://nforum.ncatlab.org/discussion/6538/modern-physics-in-modal-homotopy-theory/?Focus=121152#Comment_121152).

   Has more been written on the linearity point?

   There’s a little more at Science of Logic.

   It gets mentioned from time to time, as here.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 6th 2025
   • (edited Jun 6th 2025)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSo the basic idea is that in the progression of cohesive modalities ([[cohesion - table|here]]) there appears the [[reduction modality]] "$\Re$" which has the interpretation that $\Re X$ is what remains of $X$ if any "infinitesimal extension" is removed. Conversely, this means that the $\Re$-antimodal objects, $\Re X \,\simeq\, \ast$, are *nothing but* an infinitesimal thickening of the point, hence are [[infinitesimal halos]]. In standard models, the maps between such halos are order-$n$ polynomial if the halos are order-$n$ infinitesimal, hence are linear for first-order infinitesimals. The same antimodal objects are obtained for the dual [[infinitesimal shape modality]] $\Im$. (For reference I could point to section 6.5.2 of dcct ([pp. 727](https://ncatlab.org/schreiber/files/dcct170811.pdf#page=753)) but I am not suggesting that you should spend time looking at that.) $\,$ Instead, back to the idea of "$\bigcirc$-Stores": Maybe you could illustrate an intended example application, beyond the case $\bigcirc \equiv \ast$?

   So the basic idea is that in the progression of cohesive modalities (here) there appears the reduction modality “$\Re$” which has the interpretation that $\Re X$ is what remains of $X$ if any “infinitesimal extension” is removed.

   Conversely, this means that the $\Re$-antimodal objects, $\Re X \,\simeq\, \ast$, are nothing but an infinitesimal thickening of the point, hence are infinitesimal halos. In standard models, the maps between such halos are order-$n$ polynomial if the halos are order-$n$ infinitesimal, hence are linear for first-order infinitesimals.

   The same antimodal objects are obtained for the dual infinitesimal shape modality $\Im$.

   (For reference I could point to section 6.5.2 of dcct (pp. 727) but I am not suggesting that you should spend time looking at that.)

   $\,$

   Instead, back to the idea of “$\bigcirc$-Stores”:

   Maybe you could illustrate an intended example application, beyond the case $\bigcirc \equiv \ast$?

7. • CommentRowNumber7.
   • CommentAuthorJohn T
   • CommentTimeJun 6th 2025
   • (edited Jun 6th 2025)
   • PermaLink
   Author: John T
   Format: MarkdownItexThank you both for the references! Examples aren't applications, but if you don't mind I'd like to think out loud for a minute about what it is that the different kinds of (co)stores do before trying to *assert* that what they do is useful. If I appear to have anything wrong in my picture of how this modal cohesion stuff works I'm more than happy to take corrections, but you can also just skip to the end for a more direct answer. ###The tangent### I think I have a decent conceptual picture of levels 0 and 1 of your singular cohesion tower, but I'm not yet particularly confident with the kinds of structure that are getting removed past that point. If we throw out $*$, that leaves the $\sharp$ and ʃ stores and the $\flat$ and $\emptyset$ costores for examples. As a costore requires a one-sided inverse to the counit, the only $\emptyset$-costore would be the trivial one on $\emptyset$ itself. So a $\emptyset$-costore on $X$ would witness that $X$ is empty. A $\flat$-costore on $X$ would witness that $X$ has no nontrivial paths, as $X$ can be mapped to $\flat X$ continuously in a manner that preserves "where points are." I'm still a bit fuzzy on how the higher levels of structure work formally, but I don't *think* this would further require that $X$ have no infinitesimal structure (my hesitation mostly coming from the ʃ=$\operatorname{loc}_{\mathbb{R}}$ axiom). Such structure would simply map to zero along the costore map. If I had to guess, I would say that in general the existence of a $\square$-costore structure on $X$ would witness that $X$ has none of the minimal type of structure that $\square$ removes (no points or "being," no paths, no purely infinitesimal extensions, no supergeometric odd part?), without going quite so far as to assert that $X$ is a modal object for $\square$, lacking higher-level structure than what $\square$ removes. A costore for the identity comonad would just be the identity map on an object $X$. We have already addressed $\bigcirc=*$. A ʃ-store on $X$ would embed the fundamental infinity-groupoid of $X$ back into $X$. This would entail pointing each connected component of $X$. Assuming I've worked this out correctly, the storage axioms assert that this embedding is "full and faithful" on (higher) paths- if a path or homotopy exists in ʃ$X$, it must map to a path in $X$, such that the ʃ unit map sends it back to the original path or homotopy. As for what it means for higher structure on $X$, I'm unsure. A $\sharp$-store would embed $\sharp X$ into $X$, but I'm unsure how $\sharp$ acts on geometric structure at a higher level than points and paths, so I'm not quite sure how to continue past that. A store for the identity monad would of course just be the identity map on an object $X$. In every case the storage map is an embedding of $\bigcirc X$ back into $X$, with the storage conditions forcing the embedding to faithfully preserve any structure of $X$ that $\bigcirc$ doesn't trivialize. ###Answering the question### As for applications- based on the examples of (co)stores I have access to, it seems like they serve as a way to talk about the kind of structure that $\bigcirc$ or $\square$ collapses, without the collateral damage to *higher*-level structure that comes with directly applying the (co)monad itself (lower-level structure being left intact either way). All modal types are stores in the trivial way (the storage map is just be the identity, as the unit acts as the identity on these objects), but stores are allowed to carry extra higher structure that modal types cannot (or, at least, that they cannot have in a *nontrivial* way). If we run with the analogy between stores and algebras (where the definitions are similar except that stores invert the unit map on the right and algebras on the left), perhaps stores might be able to generalize or expand on some of the things modal types (being in the image of the associated monad) are useful for while allowing for the possibility of extra higher structure, in vaguely a similar way to how algebras generalize and expand on many of the things that free algebras (being in the image of the associated monad) are useful for while allowing additional relations. I really don't know enough to be *claiming* that that's even loosely representative of reality, but I can at least say that I'd learn something interesting if I could see where the analogy failed, or learn something else if it carried through. The more specific, less nebulous example I have of something stores might be useful for is that relative negation thing, trying to extract the information about an object $X$ that's collapsed by some monad $\Diamond$ while retaining $X$'s structure up to the level collapsed by $\bigcirc$. For example, perhaps (if my picture of infinitesimals as related to (co)tangent spaces is right) the negation of ℜ relative to a ʃ-store would retain information about the infinitesimal neighborhoods of the points in each component of $X$, and at the level of paths would also say something about holonomy in the different components of $X$ (I've managed to convince myself that this would be informative again- if it does say anything about holonomy then the base point X shouldn't really matter, as we're retaining *all* paths from the base point to itself, not just some of them, meaning we can transport information from point A to point B by conjugating by a path between them, as normal).

   Thank you both for the references!

   Examples aren’t applications, but if you don’t mind I’d like to think out loud for a minute about what it is that the different kinds of (co)stores do before trying to assert that what they do is useful. If I appear to have anything wrong in my picture of how this modal cohesion stuff works I’m more than happy to take corrections, but you can also just skip to the end for a more direct answer.

   The tangent

   I think I have a decent conceptual picture of levels 0 and 1 of your singular cohesion tower, but I’m not yet particularly confident with the kinds of structure that are getting removed past that point. If we throw out $*$, that leaves the $\sharp$ and ʃ stores and the $\flat$ and $\emptyset$ costores for examples.

   As a costore requires a one-sided inverse to the counit, the only $\emptyset$-costore would be the trivial one on $\emptyset$ itself. So a $\emptyset$-costore on $X$ would witness that $X$ is empty.

   A $\flat$-costore on $X$ would witness that $X$ has no nontrivial paths, as $X$ can be mapped to $\flat X$ continuously in a manner that preserves “where points are.” I’m still a bit fuzzy on how the higher levels of structure work formally, but I don’t think this would further require that $X$ have no infinitesimal structure (my hesitation mostly coming from the ʃ=$\operatorname{loc}_{\mathbb{R}}$ axiom). Such structure would simply map to zero along the costore map.

   If I had to guess, I would say that in general the existence of a $\square$-costore structure on $X$ would witness that $X$ has none of the minimal type of structure that $\square$ removes (no points or “being,” no paths, no purely infinitesimal extensions, no supergeometric odd part?), without going quite so far as to assert that $X$ is a modal object for $\square$, lacking higher-level structure than what $\square$ removes.

   A costore for the identity comonad would just be the identity map on an object $X$.

   We have already addressed $\bigcirc=*$.

   A ʃ-store on $X$ would embed the fundamental infinity-groupoid of $X$ back into $X$. This would entail pointing each connected component of $X$. Assuming I’ve worked this out correctly, the storage axioms assert that this embedding is “full and faithful” on (higher) paths- if a path or homotopy exists in ʃ$X$, it must map to a path in $X$, such that the ʃ unit map sends it back to the original path or homotopy. As for what it means for higher structure on $X$, I’m unsure.

   A $\sharp$-store would embed $\sharp X$ into $X$, but I’m unsure how $\sharp$ acts on geometric structure at a higher level than points and paths, so I’m not quite sure how to continue past that.

   A store for the identity monad would of course just be the identity map on an object $X$.

   In every case the storage map is an embedding of $\bigcirc X$ back into $X$, with the storage conditions forcing the embedding to faithfully preserve any structure of $X$ that $\bigcirc$ doesn’t trivialize.

   Answering the question

   As for applications- based on the examples of (co)stores I have access to, it seems like they serve as a way to talk about the kind of structure that $\bigcirc$ or $\square$ collapses, without the collateral damage to higher-level structure that comes with directly applying the (co)monad itself (lower-level structure being left intact either way). All modal types are stores in the trivial way (the storage map is just be the identity, as the unit acts as the identity on these objects), but stores are allowed to carry extra higher structure that modal types cannot (or, at least, that they cannot have in a nontrivial way). If we run with the analogy between stores and algebras (where the definitions are similar except that stores invert the unit map on the right and algebras on the left), perhaps stores might be able to generalize or expand on some of the things modal types (being in the image of the associated monad) are useful for while allowing for the possibility of extra higher structure, in vaguely a similar way to how algebras generalize and expand on many of the things that free algebras (being in the image of the associated monad) are useful for while allowing additional relations. I really don’t know enough to be claiming that that’s even loosely representative of reality, but I can at least say that I’d learn something interesting if I could see where the analogy failed, or learn something else if it carried through.

   The more specific, less nebulous example I have of something stores might be useful for is that relative negation thing, trying to extract the information about an object $X$ that’s collapsed by some monad $\Diamond$ while retaining $X$’s structure up to the level collapsed by $\bigcirc$. For example, perhaps (if my picture of infinitesimals as related to (co)tangent spaces is right) the negation of ℜ relative to a ʃ-store would retain information about the infinitesimal neighborhoods of the points in each component of $X$, and at the level of paths would also say something about holonomy in the different components of $X$ (I’ve managed to convince myself that this would be informative again- if it does say anything about holonomy then the base point X shouldn’t really matter, as we’re retaining all paths from the base point to itself, not just some of them, meaning we can transport information from point A to point B by conjugating by a path between them, as normal).