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Firstly a small disclaimer. I am not an expert in the theory of higher sheaves and their presentation in the model categories, so please feel free to correct all inaccuracies in the question itself!

Assume we have two presheaves on the $1$-site of topological spaces (equipped with the structure of the model category) $\mathsf{F}, \mathsf{G}: \mathsf{Top} \to \mathsf{C}$ which take values in an ambient model category $\mathsf{C}$, which is homotopy-complete model cateogory. Further assume that they both satisfy Čech homotopy descent condition, namely that there are weak equivalences $\mathsf{F}(X) \tilde{\rightarrow} \mathsf{holim}(\mathsf{F}(\mathsf{Č}(\mathfrak{U} \to X)_\bullet))$ and $\mathsf{G}(X) \tilde{\rightarrow} \mathsf{holim}(\mathsf{G}(\mathsf{Č}(\mathfrak{U} \to X)_\bullet))$, where $\mathsf{Č}(\mathfrak{U} \to X)_\bullet$ is the Čech nerve of any covering $\mathfrak{U} \to X$. To say it differently, they are supposed to be sheaves in some sense convenient for the homotopy theory. A good reference for this might be https://arxiv.org/abs/math/0205027 from Dugger, Hollander and Isaksen, where they consider $\mathsf{C} = \mathsf{sSet}$. However, my favourite category $\mathsf{C}$ is $\mathsf{DGCA}$, the category of differential graded commutative algebras with its model structure.

Question: What is a suitable notion of a natural transformation between $\mathsf{F}$ and $\mathsf{G}$? Can we get any component $\alpha_X: \mathsf{F}(X) \to \mathsf{G}(X)$ of this natural transformation from series of components $\{\alpha_{\mathsf{Č}(\mathfrak{U} \to X)_k}: \mathsf{F}(\mathsf{Č}(\mathfrak{U} \to X)_k) \to \mathsf{G}(\mathsf{Č}(\mathfrak{U} \to X)_k) \}_{k \in \mathbb{N}}$? If so, what are coherence relations between these components?

It tempts me to think about higher category theory generalisation of a natural transformation between $n$-functors, but I can not find any reference for it neither being able to find an answer by myself.

Nary
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  • After some time of further thinking, I got the feeling that maye I had made the situation overcomplicated. Even though $\mathsf{F}$ and $\mathsf{G}$ present higher sheaves/stacks, they are still ordinary functors and therefore their natural transformations have to be ordinary ones. If it is true, it is quite a mystery for me how is it possible to retell the higher sheaf story in the language of ordinary functors. And there is still another question, if the homotopy-limit functor is full on hom-sets between presheaves satisfying homotopy sheaf condition. – Nary Jul 22 '21 at 09:46
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    You're right, as categories of sheaves are often constructed as subcategories of presheaves, so maps between them are just natural transformations. To discuss higher coherences, you can either look at the model structures used in the DHI paper you referenced or similar places (see work of Joyal and Jardine) where the higher coherence is in some sense ``strictified'', or you can use the infinity-categorical language found in the work of Lurie for example (Higher Topos Theory or section 1.3.1 of Spectral Algebraic Geometry). Could you also expand on your second question in the comment, please? – Jack Davies Aug 02 '21 at 13:00
  • Thank you for these references! Just to make things clear, in the setting of model categories (when functors $\mathsf{F}, \mathsf{G}$ land in model category $\mathsf{C}$), the only coherence condition is the ordinary naturality condition on natural transformation between these 1-functors, agree? As to your question - assume we have any natural transformation $\alpha: \mathsf{F} \to \mathsf{G}$, I would like to know "how well" can I approximate its arbitrary $X$-evaluation $\alpha_X$ (as a concrete morphism in $\mathsf{C}$) by the set ${ \alpha_{Č( \mathfrak{U}\to X )_k} }_k$. – Nary Aug 02 '21 at 13:28
  • It is in the complete analogy with a requirement on presheaves to be homotopy sheaves, but now I focus not on objects in this presheaf category but on morphisms between them. And when I see $\mathsf{holim}$ as a functor, this question can be phrased as if this functor is full (at least between presheaves of my interest - $\mathsf{F}$ and $\mathsf{G}$. Or more generally, what we can say about the acting of $\mathsf{holim}$ on hom-sets? – Nary Aug 02 '21 at 13:41
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    Given a cover $U\to X$ in our site (so in topological space in your question), then for any presheaf $F$ there is a natural map $h_F\colon F(X)\to lim F(U^\bullet)$, where $U^\bullet$ is the \v{C}ech nerve of the cover. For a sheaf $F$, this map is an isomorphism, of course. The fact this map is natural (from the universal property of a limit!) means that for any map of presheaves $f\colon F\to G$, we have the commutative represented by the equation $lim f(U^\bullet)\circ h_F= h_G\circ f(X)$. – Jack Davies Aug 02 '21 at 18:44
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    This says that the limit of $f(U^\bullet)$ ``converges'' to you $f(X)$ up to natural isomorphism. In the $\infty$-categorical setting, all the isomorphisms above are (weak) equivalences, and the obvious diagram commutes up to homotopy (but this data is given by the map $f$, as it is a natural transformation in this setting). In the model categorical setting you need to be careful with how you model these homotopy limits (sometimes called totalisations when they are of this shape), but in the end you will still say the homotopy limit of $f(U^\bullet)$ is naturally w.e. to $f(X)$. – Jack Davies Aug 02 '21 at 18:48
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    I would maybe need more elaboration on homotopy limits (rather than ordinary ones), but I believe that this argumentation gets through. Thanks again! – Nary Aug 03 '21 at 08:20

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