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Remark 6.5.3.14 in Lurie’s Higher Topos Theory says: “It follows that every ∞-topos can be obtained by starting with an ∞-category of presheaves P(C), selecting a collection of augmented simplicial objects U_•^+, and inverting the corresponding maps |U_•| → U_{−1}. The specification of the desired class of augmented simplicial objects can be viewed as a kind of “generalized topology” on C in which one specifies not only the covering sieves but also the collection of hypercoverings which are to become effective after localization. It seems plausible that this notion of topology can be described more directly in terms of the ∞-category C, but we will not pursue the matter further.”
It seems to me that a rather straightforward description of such “Grothendieck hypertopologies” can be pulled out of Lurie’s writeup quite easily, and I sketch one such construction below.
Is there anything like this in the literature? The construction below is nothing but a straightforward application of results from Higher Topos Theory, and I would be surprised if something like this hasn’t been invented before.
Here is the construction of a presentation:
Recall that an ∞-topos T is an accessible left exact localization L: PreSh(C)→T of the ∞-presheaf category PreSh(C) of some small ∞-category C. Recall (Proposition 5.5.4.15 in Higher Topos Theory) that accessible localizations bijectively correspond to strongly saturated classes of morphisms of small generation. Here strong saturation means closure under cobase changes, colimits in the category of morphisms, and the 2-out-of-3 property. Small generation means that there is a set of morphisms whose saturation coincides with the given class. Furthermore (Proposition 6.2.1.1 in Higher Topos Theory) recall that left exact accessible localizations bijectively correspond to strongly saturated classes of morphisms of small generation that are closed under base changes. (In both correspondences, to go one way, send a localization functor to the class of morphisms that it inverts; to go the other way, universally invert the given class of morphisms.)
Lemma. Given an accessible left exact localization L: PreSh(C)→T of the presheaf category PreSh(C) for some small ∞-category C, the strong saturation of the class of morphisms J with representable codomain inverted by L coincides with the class of all morphisms inverted by L.
Proof. Consider an arbitrary morphism f: A→B such that L(f) is invertible. Being an ∞-presheaf on a small category, B can be presented as a small ∞-colimit colim_{i∈I} X_i of representables X_i for some small indexing ∞-category I. The category of ∞-presheaves on C is an ∞-topos, hence (Theorem 6.1.3.9 in Higher Topos Theory) it satisfies descent, and the slice category T/B is equivalent to the ∞-limit of slice categories T/X_i. (To go one way, take base changes with respect to the morphisms X_i→B; to go the other way, take the ∞-colimit in the category of arrows.) Now apply this fact to the morphism f: A→B ∈ T/B. The resulting components g_i: Y_i→X_i are base changes of f, hence they are inverted by L. Thus we presented f as the ∞-colimit of morphisms g_i in the category of arrows. The latter operation is one of the operations from the definition of strong saturation.
Lemma. The class of morphisms J from the previous lemma satisfies the following properties reminiscent of covering sieves in an ordinary Grothendieck topology (see, e.g., Johnstone, Sketches of an Elephant, Definition C2.1.8): 1) It is closed under base changes along maps of representables; 2) The identity morphism of representables is always a covering ∞-sieve; 3) If F is a sieve on X such that the sieve ⋃_d {g: Y→X | g^*(F) covers Y} is a covering sieve of X, then F itself covers X.
Proof. 1) holds because the class of morphisms inverted by f is closed under base changes along arbitrary maps. 2) is trivial. 3) is an immediate consequence of descent for T (Theorem 6.1.3.9 in Higher Topos Theory).
Since J satisfies the ∞-categorical analog of the axioms of a Grothendieck topology, we could say that J is a Grothendieck ∞-topology on the small ∞-category C.
However, we can go further and give an even more concrete model for T in terms of hypercovers.
Using Lemma 5.5.8.13 in Higher Topos Theory, any morphism in J can be presented as a morphism F→X, where F is a simplicial object in ∞-presheaves on D and X is representable. Furthermore, as shown in the proof of Lemma 5.5.8.13, every simplicial level F_n of F splits as the coproduct of the nth latching object L_n(F→X) and a small coproduct of representables, and every map F_n→M_n(F→X) to the nth matching object is an effective ∞-epimorphism.
These two conditions (on latching and matching objects of F→X) can be seen as ∞-categorical analogs of the conditions in the definition of a split hypercover: the condition on matching objects is what gives us a hypercover, and the condition on latching objects is what makes it split.
In fact, one can further strictify the resulting presentation to obtain a purely 1-categorical model of T using simplicial presheaves and hypercovers.
To this end we present the small ∞-category C as a small relative category D. (Other models, such as small simplicial categories, would work equally well here.)
The ∞-category of ∞-presheaves on C can then be modeled as the left Bousfield localization of simplicial presheaves on D with respect to the Yoneda image of weak equivalences of D.
The morphism F→X can now be presented as a projective fibration of simplicial presheaves on D. The properties of F→X mentioned above (every simplicial level F_n splits as L_n(F→X) and a coproduct of representables) guarantee that F is cofibrant (in fact, cellular) simplicial presheaf that satisfies the splitness condition required of a split hypercover. The other condition on F→X (with matching objects) guarantees that F→X is a hypercover, in fact a split hypercover.
Thus we have shown that any ∞-topos T can be presented by the projective model structure on simplicial presheaves on a small relative category D, left Bousfield localized at the Yoneda images of weak equivalences in D, further left Bousfield localized at a (small) family of split hypercovers in D.
(In this statement the term “split hypercover” was used without the condition on matching objects; the latter would require us to introduce an ordinary Grothendieck topology first, which I don’t want to do.)
If I give you a collection of “split hypercovers” in D, can you tell me when it gives rise to a left-exact localization on -presheaves on ?
@CharlesRezk: After thinking about your question for a while, I think the answer is positive and one can indeed identify such conditions.
Below I use the word “hypersieve” to refer to a split hypercover that does not a priori belong to the Grothendieck hypertopology under consideration, and “hypercover” to refer to a hypersieve that does belong to the given Grothendieck topology.
The conditions one must impose on hypercovers are as follows (I silently identify objects with their representable presheaves):
1) The constant hypersieve on X is a hypercover of X.
2) Hypercovers are closed under base change.
3) The last property is also analogous to its cousin for ordinary Grothendieck topologies, but is more involved. Suppose H is a split hypercover, so that each simplicial level H_m is a coproduct of representables A_{m,i}; let’s call the latter “components” of H. Suppose furthermore that each component A_{m,i} itself has a hypercover G_{m,i} with components B_{m,i,n,j}. Define the diagonal hypersieve of (H,G) as the hypersieve diag(H,G) with components in degree m being B_{m,i,m,j}, where i and j are arbitrary. Then the diagonal hypersieve diag(H,G) is a hypercover of X.
To show that the localization associated to such a family of hypersieves is left exact we use the “associated hypersheaf” functor: if F is a presheaf, then the associated hypersheaf aF of F maps some representable X to the homotopy colimit over all hypercovers H of X of the descent object of F with respect to H (i.e., the obvious homotopy limit over Δ). Thus aF(X) = colim_{H→X} Hom(H,F) = colim_{H→X} lim_{m∈Δ} ∏_i F(A_{m,i}), where A denotes the components of H. To define aF on morphisms we take the relevant base change and use property 2).
Property 2) above ensures that hypercovers of a fixed representable X form a filtered ∞-category, which guarantees that the associated hypersheaf functor is left exact.
It remains to show that the associated hypersheaf functor is a localization functor. This reduces to showing that the canonical inclusion F→aF is a local map and aF is a hypersheaf, for any presheaf F.
The former property is formal and is proved in a largely analogous manner to part (a) of the proof of Proposition 6.2.2.7 in Lurie’s Higher Topos Theory.
The latter property can be established as follows. To show that aF is a hypersheaf, consider an arbitrary hypercover H→X with components A_{m,i}.
We have to show that aF(X)→Hom(H,aF)=lim_{m∈Δ} ∏_i aF(A_{m,i}) is an equivalence. Expanding the definition of aF on both sides, we have to show that colim_{H→X} Hom(H,F) → lim_m ∏_i colim_{G_{m,i}→A_{m,i}} Hom(G_{m,i},F) is an equivalence. Here G_{m,i} runs over all hypercovers of the representable A_{m,i}.
Now recall that filtered homotopy colimits of spaces distribute over small homotopy limits. (Distributivity here means something very different than commutativity, see the article distributivity of colimits over limits for more details.) In our case this allows us to rewrite the right side as colim_{∏_{m,i} {G_{m,i}→A_{m,i}}} lim_m ∏_i Hom(G_{m,i},F). Here the indexing category for the colimit is the product of all indexing categories for individual terms above, i.e., its objects are arbitrary families of hypercovers G_{m,i}→A_{m,i} for all m∈Δ and i.
Observe now that lim_m ∏_i Hom(G_{m,i},F) = lim_m ∏_i lim_n ∏_j F(B_{m,i,n,j}), where B denotes the components of G. Since Δ is cosifted, the diagonal Δ→Δ×Δ is final, so the above homotopy limit can be replaced by the homotopy limit lim_m ∏_{i,j} F(B_{m,i,m,j}), which is precisely the descent object for the diagonal hypercover diag G of G_{m,i}, as defined above. Altogether, we replaced the right side with colim_{∏_{m,i} {G_{m,i}→A_{m,i}}} Hom(diag G,F).
It is at this point that property 3) becomes crucial: the diagonal hypercover diag G is indeed a hypercover in our hypertopology by property 3), so a simple cofinality argument (i.e., refine covers using property 2) completes the proof.
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