1:11 AM
Anyway, I think there's a bijection between nullification functors (= slocalization at a set of maps with contractible codomain) on the $\infty$-category of spaces and modalities on the $\infty$-category of spaces. The bijection takes a nullification $P$ to the modality whose left class is those maps with $P$-acyclic fiber, and conversely simply takes a modality $(\mathcal E, \mathcal M)$ to localization at $\mathcal E$.
16 hours later…
5:11 PM
There's a theorem saying that if both $\mathcal{C}$ and $\mathcal{C}^{op}$ are (locally-) presentable 1-categories then $\mathcal{C}$ must be a lattice. I wonder if this is true for $\infty$-categories...
I believe it should fail. I would not consider it a completely fleshed out counterexample but here's my thought:
I think verdier duality can be stated as an equivalence $CoSh^{Sp}(\mathfrak{X}) := Fun^L(\mathfrak{X},Sp)^{op} \cong Sh^{Sp}(\mathfrak{X}) = Fun^{R}(\mathfrak{X}^{op},Sp)$ for $\mathfrak{X}$ the $\infty$-topos associated with a sufficiently nice locally compact hausdorff space. But regardless of that $Fun^{L}(\mathcal{C},\mathcal{D})$ when $\mathcal{C}$ and $\mathcal{D}$ are presentable is always presentable. I admit that I do not immediately have a reference for either statemen…
I think verdier duality can be stated as an equivalence $CoSh^{Sp}(\mathfrak{X}) := Fun^L(\mathfrak{X},Sp)^{op} \cong Sh^{Sp}(\mathfrak{X}) = Fun^{R}(\mathfrak{X}^{op},Sp)$ for $\mathfrak{X}$ the $\infty$-topos associated with a sufficiently nice locally compact hausdorff space. But regardless of that $Fun^{L}(\mathcal{C},\mathcal{D})$ when $\mathcal{C}$ and $\mathcal{D}$ are presentable is always presentable. I admit that I do not immediately have a reference for either statemen…
I just found a reference for this kind of Verdier duality statement! people.math.harvard.edu/~lurie/282ynotes/LectureXXI-Verdier.pdf
Still though, if the theorem I mentioned about 1-categories holds for $\infty$-categories it seems like no such statement can hold regardless of how nice $\mathfrak{X}$ is...
To see the relation between my formulations and lurie's: $Sh_{\mathcal{C}}(X) = Sh_{\mathcal{C}^{op}}(X)^{op} = Fun^{R}(\mathfrak{X}^{op},\mathcal{C}^{op})^{op} = Fun^{L}(\mathfrak{X},\mathcal{C})^{op}$
5:49 PM
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