I am studying homotopical cosheaves, and I came up with the following "conjecture".
We can see an "additive" precosheaf in chain complexes (such that corestrictions do not commute on the nose) as a functor of infinity operads $\mathcal{F}: N(Open_M^{\otimes}) \to Chain(k)^{\oplus}$. Here $Open_M$ is the operad such that $Mul(U_1, \ldots, U_n; M)$ is $\{*\}$ if $U_i$ are disjoint and contained in $U$, and empty otherwise. $Chain(k)$ are the chain complexes over $k$, with symmetric monoidal structure given by $\oplus$. Additive precosheaf means that sends disjoint unions to direct sums.
An homotopical cosheaf is classically defined as follows. For every cover $\mathfrak{U}$ of $U \in Open_M$, define the $\mathfrak{U}$ cech complex of some precosehaf $\mathcal{F}$ as
$$ \check{C}(\mathfrak{U}, \mathcal{F})_{\bullet} = Tot_{i,j}( \bigoplus_{U_{\alpha_0}, \ldots, U_{\alpha_i} \in \mathfrak{U} } \mathcal{F}(U_{\alpha_0} \cap \ldots \cap U_{\alpha_i} )_j ) = hocolim_{i \in \Delta}( \bigoplus_{U_{\alpha_0}, \ldots, U_{\alpha_i} \in \mathfrak{U} } \mathcal{F}(U_{\alpha_0} \cap \ldots \cap U_{\alpha_i} ) )$$
I was thinking if one could improve the last equality to a more "global" hocolimit, by the intuition that direct sum is a colimit and that the "top" ranges over $U \in \mathfrak{U}$. Formally, I ask whether if (eventually under some mild assumptions on $\mathfrak{U}$, like being closed for intersections) the cech complex is computing the homotopy colimit of $\mathcal{F} | N(\mathfrak{U})$.
My aim, to be honest, is the following. I would like to have a formulation of homootpy cosheaf in terms of kan extensions, which is natural to ask. In particular, I showed that if the previous fact is true (or probably some slight variants), then
$\mathcal{F}$ is a homotopy cosheaf iff
for every cover $\mathfrak{U}$, $\mathcal{F}$ is a left Kan extension of $\mathcal{F}| \mathfrak{U} : N(\mathfrak{U})^{\otimes} \to N(Open_M^{\otimes}) \to Chain(k)$.
This formulation has various advantages; one being that the formulation is codomain independent, and the other is that the extension from a basis lemma seems to be easy to show, also in an infinity category setting.
I would be happy also if you have some suggestions on my final aim, that uses another approach from the one I formulated.
