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I’ve been looking at logic in infinity groupoids, and realized one can do a lot working exclusively in the homotopy category by introducing a new quantifier for “there exists unique up to contractible choice” (teuucc).
For example, if I’ve not made an error, one can define limits in infinity categories entirely by working in the category Ho(catinf) (with hom-sets []), in an analogous way to the Q-sequences from Categories, Allegories, namely
A functor in $[J, C]$ has a limit iff:
where the teuucc quantifier would pick out the subset of $[(1 \amalg 1) \star J, C]$ of points for which $Map([1] \star J, C) \to Map((1 \amalg 1) \star J, C)$ would have contractible fibers.
Is there any existing work that approaches infinity category theory in a method like this? E.g. by pushing down to the homotopy category and augmenting the predicate logic on hom-sets to better encode homotopical information?
That’s what derivators are: the systems of homotopy categories of all $\infty$-categories of small diagrams in a given $\infty$-category.
It will take some work to see how quantifier approach would fit in, but it’s an interesting idea to try to apply them to semiderivators rather than Ho(Catinf), thanks for the suggestion!
But I see now I misread your question:
I thought $[J,C]$ was notation for the homotopy category of the $\infty$-functor $\infty$-category from the small diagram 1-category $J$ to the $\infty$-category $C$. The derivator-incarnation of the $\infty$-category $C$ is the system of all these homotopy categories $Ho(Func_\infty(J,C))$ as $J$ varies over small 1-categories.
But I see now you want $[J,C]$ to denote just the set of connected components of that homotopy category. That information is contained in the derivator of $C$, clearly, but your point seems to be that it’s actually sufficient to retain just this 0-truncated form of the derivator of $C$?
Yes – well, more precisely, if you also remember the relation “the connected component of $x$ is equivalent to the connected component of $f(x)$”, when you have a morphism $f$ of infinity-groupoids, which you use to define the more refined version of unique existence on the truncation.
My motivation, incidentally, is that a lot of Higher Topos Theory involves constructing a category (usually a functor category), and then restricting to a full subcategory of objects satisfying some proposition, which seemed like a rather pleasing way to argue where you don’t have to deal with the higher structure because the machinery takes care of it for you, and I also really liked the Q-sequences approach out of Categories, Allegories for defining properties of diagrams in a category, and I wanted to mimic that. (i.e. the approach of quantifying over the existence of extensions along a sequence of diagram categories)
I switched to the set of connected components of the core because I was having technical difficulties figuring out to do with full subcategories of the $Ho(Fun_\infty(J,C))$, and the path forward there seemed more clear after going down to sets. Maybe I’m losing something, but it still seems promising enough. The test for myself when poking at it was to be able to express the fact that, if limits exist, they are functoral, and I think I’ve nearly settled that, so I’m thinking of looking at this in earnest and am now curious to look for existing similar work.
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