This is related to a previous post, but a bit softer and should probably stand on its own.
In Appendix A of "Higher Algebra", Lurie shows that for a reasonably good topological space, there is a Riemann-Hilbert correspondence for locally constant $\infty$-sheaves on it ($\mathcal{S}$ is the $\infty$-category of spaces): $$ \operatorname{Sh}_{\infty}^{lc}(X) \simeq \operatorname{Fun}(\operatorname{Sing}(X), \mathcal{S})=\operatorname{Fun}(\underset{\operatorname{Sing}(X)}{\operatorname{colim}}*, \mathcal{S}) = \underset{\operatorname{Sing}(X)}{\operatorname{lim}}\mathcal{S}$$ A similar exodromy result is proven a bit later, for constructible sheaves on a good stratified space $X \rightarrow A$ - this is the first $\simeq$ below. I am guessing for the following manipulations that every $(\infty,1)$-category should probably be the "$(\infty,2)$-colimit" over its objects, whatever that is: $$ \operatorname{Sh}_{\infty}^{cb}(X\rightarrow A) \simeq \operatorname{Fun}(\operatorname{Sing}^A(X), \mathcal{S})=\operatorname{Fun}(\underset{\operatorname{Sing}^A(X)}{\operatorname{(\infty,2)-colim}} \, *, \mathcal{S}) = \underset{\operatorname{Sing}^A(X)}{\operatorname{(\infty,2)-lim}}\,\mathcal{S}$$
The colimit statement about $\mathbb{E}_M$ at the end of Remark 5.4.5.2, that I was asking about in my previous post, looks exacly like this - note that algebras over $\mathbb{E}_M$ are the same thing as locally constant factorization algebras, and given by $$ \operatorname{Alg}_{\mathbb{E}_M}(\mathcal{C}) = \operatorname{Fun}(\mathbb{E}_M^\otimes,\mathcal{C}^\otimes) =\operatorname{Fun}(\underset{\operatorname{Sing}(X)}{\operatorname{colim}} \, \mathbb{E}_n^\otimes, \mathcal{C}^\otimes) = \underset{\operatorname{Sing}(X)}{\operatorname{lim}}\,\operatorname{Alg}_{\mathbb{E}_n}(\mathcal{C})$$ where $\operatorname{Fun}$ now denotes the mapping $(\infty,1)$-category in the $(\infty,2)$-category of operads $\operatorname{Op}_\infty$, i.e. the first equality shouls hold by definition of it. I would suspect that something similar also holds for constructible factorization algebras over a good stratified space $X \rightarrow A$, as defined in https://arxiv.org/abs/1409.0848: $$ \{ \text{constr. fact. algebras over $X$ in } \mathcal{C} \} = \operatorname{Fun}(\mathbb{E}_X^\otimes,\mathcal{C}^\otimes) =\operatorname{Fun}(\underset{\operatorname{Sing}^A(X)}{\operatorname{(\infty,2)-colim}} \, \mathbb{E}_B^\otimes, \mathcal{S}) = \underset{\operatorname{Sing}^A(X)}{\operatorname{(\infty,2)-lim}}\,\operatorname{Alg}_{\mathbb{E}_B}(\mathcal{C})$$
Here, we identify the exit path category $\operatorname{Sing}^A(X)$ with $\mathcal{B}sc_{/X}$ (basic stratied spaces of the form $\mathbb{R}^i \times C(Z)$ with $Z$ compact, over $X$), as Lurie does in the manifold case in the mentioned remark - the statement still holds for stratied spaces. $B$ should then denote the respective basic spaces.
I hope this rambling makes any sense, I am aware that there are probably many mistakes in by ideas. I want to ask the following:
- Is /should an $(\infty,2)$-limit indexed by a Kan-complex just be the same thing as an $(\infty,1)$-limit? This would make the statements above fit together even better, note that the exit-path category is not a Kan complex, while $\operatorname{Sing}(X)$ is.
- Do you know of any statements that resemble my last claim? Maybe some weaker versions of it, like a recollement result for stratified factorization algebras (these often appear in the physics literature, but only in individual cases).
- Do you think this Riemann-Hilbert-correspondence view on Lurie's Remark is a good point of view/ fruitful/ does what I say even make sense? I've tried to apply it to simple cases like $M=S^1$, but I generally have no good intuition for such (co)limits.
