I don't know what Toën was talking about, but I suspect that it was about finiteness conditions for Artin stacks: the problem is that the usual finiteness conditions we look at for schemes (like the notion of constructibility for l-adic sheaves) do not extend to stacks in a straightforward way, which gives some trouble if one wants to count points
(i.e. to define things like Euler characteristics). Some notions of finiteness are developed to define Grothendieck rings of Artin stacks (e.g. in Toën's paper arXiv:0509098 and in Ekedahl's paper arXiv:0903.3143), which can be realized by our favourite cohomologies (l-adic, Hodge, etc), but the link with a good notion of finiteness for categories of coefficients over Artin stacks (l-adic sheaves, variation of mixed Hodge structures) does not seem to be fully understood yet, at least conceptually (and by myself).
As for finiteness conditions for sheaves (in some homotopical context), the kind of properties we might want to look at are of the following shape.
Consider a variety of you favourite kind X, and a derived category D(X) of sheaves over
some site S associated to X (e.g. open, or étale, or smooth subvarieties over X etc.).
For instance, D(X) might be the homotopy category of the model category of simplicial sheaves, or the derived category of sheaves of R-modules.
Important finiteness properties can be expressed by saying that for any U in the site S,
we have
(1) hocolimᵢ RΓ(U,Fᵢ)= RΓ(U,hocolimᵢ Fᵢ)
where {Fᵢ} is a filtered diagram of coefficients. If you are in such a context, then
you can look at the compact objects in D(X), i.e. the objects A of D(X) such that
(2) hocolimᵢ RHom(A,Fᵢ)= RHom(A,hocolimᵢ Fᵢ)
for any filtered diagram {Fᵢ}. In good situations, condition (1) will imply that the category of compact objects will coincide with constructible objects (i.e. the smallest subcategory of D(X) stable under finite homotopy colimits (finite meaning: indexed by finite posets) which contains the representable objects).
Sufficient conditions to get (1) are the following:
a) For simplicial sheaves (as well as sheaves of spectra or R-modules...), a sufficient condition is that the topology on S is defined by a cd-structure in the sense of Voevodsky (see arXiv:0805.4578). These include the Zariski topology, the Nisnevich topology, as well as the cdh topology (the latter being generated by Nisnevich coverings as well as by blow-ups in a suitable sense), at least if we work with noetherian schemes of finite dimension. Note also that topologies associated to cd structures define what Morel and Voevodsky call a site of finite type (in the language of Lurie, this means that, for such sites, the notion of descent is the same as the notion of hyperdescent: descent for infinity-stacks over S can be tested only using truncated hypercoverings; this is the issue discussed by David Ben Zvi above).
In practice, the existence of a cd structure allows you to express (hyper)descent using only Mayer-Vietoris-like long exact sequences (the case of Zariski topology was discovered in the 70's by Brown and Gersten, and they used it to prove Zariski descent for algebraic K-theory).
b) For complexes of sheaves of R-modules, a sufficient set of conditions are
i) the site S is coherent (in the sense of SGA4).
ii) any object of the site S is of finite cohomological dimension (with coefficients in R).
The idea to prove (1) under assumption b) is that one proves it first when all the Fᵢ's are concentrated in degree 0 (this is done in SGA4 under assumption b)i)).
This implies the result when the Fᵢ's are uniformly bounded. Then, one uses the fact, that, under condition b)ii), the Leray spectral sequence converges strongly, even for unbounded complexes (this done at the begining of the paper of Suslin and Voevodsky "Bloch-Kato conjecture and motivic cohomology with finite coefficients").
This works for instance for étale sheaves of R-modules, where R=Z/n, with n prime to the residual characteristics. Note moreover that, in the derived category of R-modules, the compact objects (i.e. the complexes A satisfying (2)) are precisely the perfect complexes.
The fact that the six Grothendieck operations preserves constructibility can then be translated into the finiteness of cohomology groups (note however that the notion of constructiblity is more complex then this in general: if we work with l-adic sheaves
(with Ekedahl's construction, for instance), then the notion of constructiblity does not agree with compactness anymore). However, condition (1) is preserved after taking the Verdier quotient of D(X) by any thick subcategory T obtained as the smallest thick subcategory which is closed under small sums and which contains a given small set of compact objects of D(X) (this is Thomason's theorem). This is how such nice properties survive in the context of homotopy theory of schemes for instance.
Note also that, in a stable (triangulated) context, condition (2) for A implies that we have the same property, but without requiring the diagrams {Fᵢ} to be filtering.
For your second question, the extension of a cohomology theory to simplicial varieties is automatic (whenever the cohomology is given by a complex of presheaves), at least if we have enough room to take homotopy limits, which is usually the case (and not difficult to force if necessary). The only trouble is that you might lose the finiteness conditions, unless you prove that your favorite simplicial object A satisfies (2). The fact that Hironaka's resolution of singularities gives the good construction (i.e. gives nice objects for open and/or singular varieties) can be expained by finiteness properties related to descent by blow-ups (i.e. cdh descent), but the arguments needed for this use strongly that we work in a stable context (I don't know any argument like this for simplicial sheaves). The fuzzy idea is that if a cohomology theory satisfies Nisnevich descent and homotopy invariance, then it satisfies cdh descent (there is a nice very general proof of this in Voevodsky's paper arXiv:0805.4576 (thm 4.2, where you will see we need to be able to desuspend)); then, thanks to Hironaka, locally for the cdh topology, any scheme is the complement of a strict normal crossing divisor in a projective and smooth variety. As cdh topology has nice finiteness properties (namely a)), and as
any k-scheme of finite type is coherent in the cdh topos, this explains, roughly, why we get nice extensions of our cohomology theories (as far as you had a good knowledge of smooth and projective varieties). If we work with rational coefficients, the same principle applies for schemes over an excellent noetherian scheme S of dimension lesser or equal to 2, using de Jong's results instead of Hironaka's, and replacing the cdh topology by the h topology (the latter being obtained from the cdh topology by adding finite surjective morphisms): it is then sufficient to have a good control of proper regular S-schemes.
In the context of Goodwillie's paper, he's got an explicit natural transformation $f:holim_I(X)\to holim_J(X|_J)$, where $X:I\to Top$ is a functor to spaces, and $J\subset I$ is a subcategory of $I$. With the construction of holim he's using, this map is always a fibration.
What if you tried to use a different construction of holim? Then maybe you get a map $f'$ which is not a fibration anymore. In that case, you could still have taken the homotopy fiber of $f'$, and this would be a notion which is invariant under weak equivalence. That is, you could (functorially) replace $f'$ with a fibration via the path construction, and take the fiber of that.
Of course, the homotopy fiber is exactly the thing he wants here. In fact, he's manufactured the situation exactly so that the homotopy fiber he wants is just the fiber of this map.
(It's worthwhile to note that in his setting, the category $I$ (which is a cube) has an initial object $\varnothing$. This means that the evident map $holim_I(X)\to X(\varnothing)$ is a weak equivalence. In other words, $holim_I(X)$ is really just $X$ evaluated at $\varnothing$, but modified so that it maps to (and fibers over) $holim_J X|_J$.)
Best Answer
Actually, for simplicial sheaves, and to be more accurate infinity sheaves of infinity groupoids, you do not need hypercovers. Your "naive" idea about multiple intersections (actually fibered products) is correct. If you instead use hypercovers, you get the notion of a *hyper*sheaf. Both infinity sheaves and hypersheaves form an infinity topos, and the infinty topos hypersheaves is a left exact localization of the infinity topos of infinity sheaves. They COULD be the same, but this depends on your Grothendieck site. They are the same if and only if the infinity topos of infinity sheaves satisfies Whitehead's theorem internally with respect to internal homotopy sheaves; for any infinity sheaf $X$ over $\mathcal{C}$, one has sheaves $\pi_n(X) \in Sh_{\infty}\left(C\right)/X$ (it actually is a discrete object thereof) analogous to homotopy groups of spaces. The infinity topos of infinity sheaves is hypercomplete if and only if any morphism $g:X \to Y$ which induces an isomorphism on $\pi_n$ for all $n$ must be an equivalence. If any arbitrary infinity topos is not hypercomplete, we may hypercomplete it by localizing with respect to those morphisms which induces isomorphisms on all homotopy sheaves, and this is again an infinity topos, called its hypercompletion. So the more precise statement is that for an infinity topos of infinity sheaves over a site $\mathcal{C}$, its hypercompletion can be identified with the infinity topos of hypersheaves.
One should note that an infinity topos is defined to be a left exact (accessible) localization of an infinity category of infinity presheaves on some infinity category, and that this DOESN'T imply that you are the infinity topos of infinity sheaves on some site, in the infinite case; the infinity topos of hypersheaves over some site need not be equivalent to ordinary infinity sheaves over any site.
Now to answer the question in the comments below:
Why do we need the full simplicial Cech diagram for infinity sheaves?
Well, let go back to ordinary sheaves of sets first. A Grothendieck topology should actually be defined in terms of covering sieves not in terms of covering families. What most people call a Grothendieck topology is really a basis for a Grothendieck topology. A sieve on an object $C$ of $\mathcal{C}$ is a subobject $S$ of the representable presheaf $y\left(C\right)$ in the category $Set^{\mathcal{C}^{op}}$. If we are given a basis for a Grothendieck topology, and a cover $\mathcal{U}:=\left(f:U_\alpha \to C\right)$ of $C,$ then we can consider the sieve $S_\mathcal{U}$ to be the subobject which assigns an object $D$ the morphisms $D \to C$ which factor through some $f_\alpha$. A sieve $R$ is a called a covering sieve if there is some cover $V$ such that $S_V \subset R$. Now a presheaf $F$ is said to be a sheaf if for any covering sieve $R \subset y(C),$ the induced map $$F(C) \cong Hom(y(C),F) \to Hom(R,F)$$ is an isomorphism. One can check that this is equivalent to requiring this condition on sieve of the form $S_\mathcal{U}$ for some cover. But, $S_\mathcal{U}$ can be described as the colimit of the diagram $$\coprod\limits_{\alpha,\beta}{y\left(U_\alpha \times_C U_\beta\right)}\rightrightarrows \coprod\limits_{\alpha}{y\left(U_\alpha\right)}$$ in presheaves (you can check this easily by computing this "object-wise"). But this means that $$Hom(S_\mathcal{U},F) \cong \varprojlim \left(\prod \limits_{\alpha}{Hom(y\left(U_\alpha\right),F)} \rightrightarrows \prod\limits_{\alpha,\beta}{Hom(y\left(U_\alpha \times_C U_\beta\right),F)}\right).$$ After Yoneda, this recovers the ordinary definition of sheaf with covers.
But now one can simply say that an infinity sheaf is an infinity presheaf $F$ of infinity groupoids such that for any covering sieve $R \subset y(C),$ the induced map $$F(C) \simeq Hom(y(C),F) \to Hom(R,F)$$ is an equivalence of infinity groupoids (i.e. a weak homotopy equivalence of simplicial sets). Again, it suffices to check this condition for sieves of the form $S_\mathcal{U}$. Now consider the infinity colimit (i.e. homotopy colimit) of the Cech diagram coming from $\mathcal{U}$. Call it $S'$. We can compute its 0-truncation $S'_0$, and since this functor is left-adjoint to the inclusion of ordinary presheaves, the unit of the adjunction furnishes us with a map $l:S' \to S'_0$. But since it is a left-adjoint, it preserves colimits, so it is equal to the colimit of the same diagram in ordinary presheaves. But, the "2-stage" diagram is cofinal in this case, so we recover $S_\mathcal{U}$. But, the homotopy fibers of $l:S' \to S'_0=S_\mathcal{U}$ are discrete, so this means $l$ is both 0-connected and 0-truncated, hence an equivalence. But this means that $S_\mathcal{U}$ is the homotopy colimit of the whole simplicial diagram, when we consider it as an object in infinity presheaves. This in terms tells us that $Hom(S_\mathcal{U},F)$ is a homotopy limit of a cosimplicial diagram, and envoking the infinity Yoneda lemma, we get that this diagram is equivalent to the one involving applying $F$ to iterative fibered products.