Orlov's theorem states that any exact fully faithful functor between $D^b Coh$ of smooth projective varieties can be realised as an integral transform, at least if you allow arbitrary objects of $D^b(X × Y)$ as integral kernels. That certainly applies to your case, so yes, the inverse functor is an integral transform. An inverse functor is in particular right adjoint, and a right adjoint to an integral transform can be realised as an integral transform by the dual sheaf, considered as an object of $D^b(M × X)$ and dualized in that category.

The problem is not the $(\infty,1)$-categories themselves but functors between them. Even if you rectify the categories you must still consider the functors in a weak sense, with all possible homotopy coherent data. I think this is explained well in the introduction of Lurie's Higher Topos Theory. By the way, the homotopy hypothesis per se isn't about globular sets, it is about an existence of some notion of $(\infty,1)$-category where the subcategory of groupoids would be equivalent to homotopy types. Globular groupoids is just one possible model.

Nonconnective rings also tend to be huge, since any negative degree element multiplied by a positive opposite degree element gives an element in $\pi_0$. Thus unless such multiplications lose a lot of information $\pi_0$ will likely be non-Noetherian and intractable. You would also need to know all homotopy groups at once, which is problematic. In the same vein, a connective ring can be constructed inductively from f.g. discrete by iterated finite extensions, which makes it in principle possible to compute anything. With nonconn. rings there appears no way to treat their homotopy inductively.

I suppose it's mostly (1). As an example, note that in "Survey of ell.coh." Lurie requires a representing topos with a sheaf of nonconnective rings, but its existence is proved via restricting to the connective case, lifting it from classical moduli space and localizing. One could list many ways why nonconnective rings are problematic on their own: there is no functor relating it to discrete rings, constructing Postnikov systems is problematic, spectral sequences often don't converge, category of modules is contrintuitive etc. The most tractable case thus is a localization of connective ring.

The line $y \in \mathbb Z[x]$ in op's question makes me think he's asking not about points on curves, but about the roots in the polynomial ring.

Dimension 4 is required because any proof rests either on the Poincare duality (in compact case) or on the intersection form on 2-cohomology in general. Note that you also need orientability. I didn't read the paper you cite, but perhaps this paper will help.

n-groupoid should mean an $(n,0)$-category, or an $(\infty, 0)$-category which represents it. Homotopy n-types themselves form an $(\infty, 1)$-subcategory of the $(\infty,1 )$-category of spaces, since there are no non-invertible $n>2$-morphisms between them.

Assuming that you formalize Whitehead's programme as an existence of fully faithful embedding of HoTop into some algebraic 1-category (i.e. algebras over some monad on $Set$), it can be proved impossible. See this paper of Peter Freyd. This means that basically any algebraic formalism that you use must be at least as complicated as spaces themselves, so defining n-groupoids as n-types is more or less required. The best you can do is choose some more tractable presentation of spaces, like simplicial sets or cubical sets or something.

In fact, if you just consider presheaves of sets on a discrete category $C$, then the described construction is tautological: you get again $[C^{op}, Set]$, which is purely discrete and 1-categorical. I'm not sure how one can formulate a theorem for a general $C$ that would nicely interpolate between a purely homotopy-theoretic and a purely discrete world.

If I'd take a guess I'd say that for moderately good $C$'s the corresponding theory would be the subcategory of spaces generated by spaces of the form $P\otimes N(C/\cdot)$, where $N$ is the nerve, $P: [C^{op}, Set]$, $\otimes$ means the homotopy coend (which can be described just as a homotopy colimit over a different category) and $C/\cdot$ is the covariant functor of overcategory. The problems is that 1). I see no reason why all maps between spaces would be generated this way, in fact often they won't; 2). objects of $C$ may have equivalent nerves, but not connected by a chain of equivs.

Since $C/X$ always has a final object $X \xrightarrow{1} X$, its nerve is contractible. As described, all objects of $C$ will be weakly equivalent. In the definition of a test category one considers presheaves $[C^{op}, Set]$ and overcategories only of the form $C/X$. I assume you're asking what is the homotopy theory of presheaves on $C$ for a general $C$ via the above construction.

@SteveHuntsman Yes, it is cited from a book of Gelfand, Kapranov and Zelevinski on hyperdeterminants and resultants, but the book only mentions it in passing. I couldn't find any references for the algebra itself.

@SteveHuntsman Thank you for the references, I saw one of them. Both describe constructions very different from the one that I describe above. Most usual definitions of ternary associativity have two multiplications on the left and on the right, while in my case the numbers are different. The bracket I describe isn't a hyperdeterminant, it is a ternary product, and a hyperdeterminant is a high degree polynomial which classifies its degeneracy.

(cont...) A fibre functor should map this 2-category of representations to the 2-category of $\mathbb C$-abelian categories. Invertible representations defined above will map to invertible categories, which for $R = \mathbb C$ are all isomorphic to $Vect_{\mathbb C}$, an isomorphism defined by a line over $\mathbb C$. Thus a fiber functor gives a point on the double dual curve, i.e. on $E$. An automorphism of fiber functor also corresponds to a point on double dual, thus we reconstruct $E$ tannakially. The details are complicated, you should post a question if you are interested.

@მამუკაჯიბლაძე Conceptually, you should consider the 2-category of representations of $E$ on (sufficiently good) abelian categories over $R$ (here $R$ is the base ring which I will assume $=\mathbb C$ for fun). For example, a representation of $E$ on $Vect_{\mathbb C}$ is the same as a representation of cogroup $QCoh(E)$ on $Vect_{\mathbb C}$ ($QCoh(E)$ has obvious multiplication and inherits comultiplication from $E$), which is the same as a line bundle on $E$. Thus linear 2-representations of $E$ correspond to points on the dual elliptic curve.

In general we need to develop the full $G$-equivariant homotopy theory, with $G$-spaces, $G$-spectra etc, and it can be quite nonobvious what should constitute an equivariant analogue of some classic cohomology theory: for K-theory we have to work with equivariant bundles, for elliptic cohomology the construction is quite intricate, and for an arbitrary cohomology theory there is no reason to suspect equivariant analogue. However, for ordinary cohomology the situation is much simpler: the homotopy quotient construction reduces the equivariant theory to a nonequivariant one uniquely.