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John T (Jun 02 2025 at 14:27):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 to be pointed (which we might get by simply taking itself to be pointed). Well, pointing is a morphism from to , and, having just read the nearby section on being the "initial opposition," I started wondering about morphisms like ʃ . I decided that I would want to work in the category of diagrams of the form
Diagram link
image.png
which by chasing split monos can be simplified to the rules that and that is a right-inverse for . 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 -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 -stores form a category in the obvious way, with morphisms being such that everything commutes after is applied to the corners of the storage diagram.
My question is when the category of -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 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
image.png
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 one 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 #**learning: questions> rather than asserting something in a results channel. Learning resources would be welcome as well as answers or hints or whatnot. Thanks!
John T (Jun 02 2025 at 14:29):I suppose I should also be asking how to properly format math here...
Morgan Rogers (he/him) (Jun 02 2025 at 14:35):For inline, you need to make sure there aren't typos (you have \dsahv in the first one) and you have to have a space or newline immediately before (that's what broke the second one). For tikz, taking a screenshot of the diagram in quiver (or a compiled pdf of the tex if you didn't build the diagram in quiver) is recommended. Tikzcd isn't natively supported afaik
David Corfield (Jun 03 2025 at 11:10):I've only ever heard Urs Schreiber talk about such matters, so you might want to post this question on the nForum.
John T (Jun 03 2025 at 20:50):David Corfield said:
I've only ever heard Urs Schreiber talk about such matters, so you might want to post this question on the nForum.
Will do, thanks!
John T (Jun 03 2025 at 21:38):It's now cross-posted here.