For all definitions of $\infty$-categories I am aware of, an $(\infty,1)$-category is defined via reference to some shape, be it simplices in a form of a quasi-category or a cubical analogue of a Cisinski's model structure by Doherty–Kapulkin–Lindsey–Sattler in "CUBICAL MODELS OF (∞,1)-CATEGORIES". If I understand correctly even the definitions in terms of $\infty$-cosmoi and Simplicial Type Theory bake in simplices. In the case of $\infty$-cosmoi being defined as enriched in quasi-categories and quasi-category in turn defined in terms of simplices. In the case of Simplicial Type Theory, baking in theory of cubes at the cube layer. This brings me to my question: Is there a definition of $(\infty,1)$-categories that does not depend on a choice of a shape? Maybe abstracting the choice to being parametric on a test category.
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4$\begingroup$ All known definition involves some kind of concrete combinatorics and shape of the cells. I feel like there is more to say on this question, like try to argue that some shapes will automatically be present, but I'm not sure what to say exactly. Definitely, any kind of model categories will come with some notion of "cells". $\endgroup$– Simon HenryCommented Oct 10, 2023 at 1:28
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5$\begingroup$ I'm not sure what "shape-independent" means in general. For example, are topologically-enriched categories a "shape-independent" model? Or are we restricting attention to presheaf models? $\endgroup$– Tim CampionCommented Oct 10, 2023 at 2:37
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4$\begingroup$ Over here I argue that $\Delta \subseteq Cat_{(\infty,1)}$ can be characterized as the smallest dense full subcategory of $Cat_{(\infty,1)}$ which is closed under retracts. So somehow $\Delta$ is pretty inevitable. $\endgroup$– Tim CampionCommented Oct 10, 2023 at 2:53
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$\begingroup$ @TimCampion I think the point about Δ being the smallest dense full subcategory is what I was looking for. When I said shape-independent what I think I meant is some process only working with some definition of n-categories and taking a limit for n going to infinity, such that it coincides with the topological (shape-based) definitions. In other words the categories involved would be the (n+1)-categories of n-categories, instead of mentioning test categories. $\endgroup$– lemmanadeCommented Oct 10, 2023 at 3:30
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$\begingroup$ Looking around this seems to be similar to what nlab calls Trimble ∞-categories, but I am not sure how these relate to the simplicial definitions. I think my question can be rephrased as, is there a definition in terms of terminal coalgebra ala Trimble n-categories, such that its theory coincides with quasi-categories, or perhaps such a definition is known to be impossible. I have to give this a bit of thought. $\endgroup$– lemmanadeCommented Oct 10, 2023 at 3:34
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Yes. As shown in the paper The enriched Thomason model structure on 2-categories, the category of (strict) 2-categories can be equipped with a model structure that makes it Quillen equivalent to the Bergner model structure on the category of simplicial categories.
This proves that (strict) 2-categories form a model for (∞,1)-categories, and the definition of a strict 2-category makes no references to shapes likes simplices, cubes, or globes.
answered Oct 10, 2023 at 16:07
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4$\begingroup$ I don't think I agree with this. The notion of strict-2-category itself in a sense do not depends on shape (though this is debatable), but the Thomason model structure definitely does (it involves Kan subdivision functor in its definition). One can argue that the class of weak equivalences does not depends on this, so you can build the localization without using any kind of concrete shape - but as soon as you want to discuss the actual "higher structure" we are going to either need the model structure, or some model for $(\infty,1)$-category in which to take a Dwyer-Kan localizations. $\endgroup$ Commented Oct 10, 2023 at 16:19
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$\begingroup$ (of course, this is very subjective!) $\endgroup$ Commented Oct 10, 2023 at 16:20
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$\begingroup$ @SimonHenry: The relative category of strict 2-categories can be defined without any reference to shapes of any kind. Relative categories do form one legitimate setting in which you can discuss “higher structure”. They are provably equivalent to the other settings, and, of course, the proofs use shapes since all the other settings (such as simplicial categories, quasicategories, etc.) use shapes in their main definitions. $\endgroup$ Commented Oct 10, 2023 at 17:55
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$\begingroup$ @SimonHenry: That being said, the assertion that the model structure on strict 2-categories somehow requires shapes does not seem correct to me either: although the description in my paper is tailored to the theorem that I want to prove (which connects to simplicial categories), the generating cofibrations can be just easily defined without any reference to simplicial sets. Indeed, the “double subdivision” of Δ[n] can be described as the poset of increasing chains of subsets of [n]. This description makes no use of shapes. $\endgroup$ Commented Oct 10, 2023 at 17:58
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$\begingroup$ what I mean is that if you actually want to give a definition of what is "a $\infty$-category", and "a (weak) $\infty$-functor" and a natural transformation, you'll need to talk about fibrant and cofibrant object in your mode category, which will involves these shape dependant construction. And if you consider that the double subdivision isn't using some kind "shapes" then the theory of quasicategory isn't either: It just involves presheaves in finite non-empty ordinal after all. $\endgroup$ Commented Oct 10, 2023 at 18:26