(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
The notion of $(\infty,1)$-category of $(\infty,1)$-sheaves is the generalization of the notion of category of sheaves from category theory to the higher category theory of (∞,1)-categories.
An $(\infty,1)$-category of $(\infty,1)$-sheaves is a reflective sub-(∞,1)-category
of an (∞,1)-category of (∞,1)-presheaves such that the following equivalent conditions hold
$L$ is a topological localization;
there is the structure of an (∞,1)-site on $C$ such that the objects of $Sh(C)$ are precisely those (∞,1)-presheaves $A$ that are local objects with respect to the covering monomorphisms $p : U \to j(c)$ in $PSh(C)$ in that
is an (∞,1)-equivalence in ∞Grpd.
This is HTT, def. 6.2.2.6.
An $(\infty,1)$-category of $(\infty,1)$-sheaves is an (∞,1)-topos.
Equivalence (1) is the descent condition and the presheaves satisfying it are the (∞,1)-sheaves .
Typically $U$ here is the Cech nerve
of a covering family $\{U_i \to c\}$ (where the colimit is the (∞,1)-categorical colimit or homotopy colimit) so that the above descent condition becomes
Sometimes (∞,1)-sheaves are called ∞-stacks, though sometimes the latter term is reserved for hypercomplete $(\infty,1)$-sheaves and at other times again it refers to (∞,2)-sheaves.
The (n,1)-categorical counting is:
$(2,1)$-sheaf = stack = 1-truncated $(\infty,1)$-sheaf
$(3,1)$-sheaf = 2-stack = 2-truncated $(\infty,1)$-sheaf
etc.
$(\infty,1)$-sheaf = ∞-stack (or = hypercomplete $(\infty,1)$-sheaf).
We reproduce the proof that the two characterization in def. 1 above are indeed equivalent.
For $C$ an (∞,1)-site, the full sub-(∞,1)-category of $PSh(C)$ on local objects with respect to the covering monomorphisms in $PSh(C)$ is indeed a topological localization, and hence $Sh(C)$ is indeed an exact reflective sub-(∞,1)-category of $PSh(C)$ and hence an (∞,1)-topos.
This is HTT, Prop. 6.2.2.7
We must prove that the (∞,1)-sheafification functor $L \colon PSh(C)\to Sh(C)$ preserves finite (∞,1)-limits. To do so we give an explicit construction of $L$. Given a presheaf $F\in PSh(C)$, define a new presheaf $F^+$ by the formula
where the colimit is taken over all covering sieves $U$ of $c$; this is called the plus construction. It defines a functor $PSh(C)\to PSh(C)$ and there is an obvious morphism $F\to F^+$ natural in $F$.
It is clear that the construction $F\mapsto F^+$ preserves finite (∞,1)-limits, since filtered (∞,1)-colimits do, and it is easy to see that the map $F\to F^+$ becomes an equivalence in $Sh(C)$. Given an ordinal $\alpha$, let $F^{(\alpha)}$ be the $\alpha$-iteration of the plus construction applied to the presheaf $F$. Then the functor $F\mapsto F^{(\alpha)}$ preserves finite limits and the canonical map $F\to F^{(\alpha)}$ becomes an equivalence in $Sh(C)$. In particular, if $F^{(\alpha)}$ is a sheaf, then $F^{(\alpha)}\simeq L(F)$. Thus, it suffices to show that there exists an ordinal $\alpha$ such that, for every $F\in PSh(C)$, $F^{(\alpha)}$ is a sheaf.
Fix $c\in C$ and a covering sieve $U$ of $C$. Given a presheaf $G\in PSh(C/c)$, we define an auxiliary presheaf $Match(U,G)\in PSh(C/c)$ by the formula
Restriction maps induce a morphism $\theta_G: G\to Match(U,G)$. Since we clearly have $G(u)\stackrel{\sim}{\to} Match(U,G)(u)$ for $u\in U$, the functor $Match(U,-)$ is idempotent in the sense that $Match(U,\theta_G)$ and $\theta_{Match(U,G)}$ are (equivalent) equivalences.
By definition, $F\in PSh(C)$ is a sheaf if and only if $F(c)\stackrel{\sim}{\to} Match(U,F|_{C/c})(c)$ for every $c\in C$ and every covering sieve $U$ of $c$. Our goal is therefore to find an ordinal $\alpha$ (depending only on the (∞,1)-site $C$) such that, for every $F\in PSh(C)$, the map
is an equivalence.
The morphism $G\to G^+$ in $PSh(C/c)$ factors as
Applying $Match(U,-)$ to this factorization, we get a commutative diagram
in which the map $\theta_{Match(U,G)}$ is an equivalence since $Match(U,-)$ is idempotent. By cofinality, the colimit of the maps $\theta_{G^{(n)}}$ as $n\to\infty$ is an equivalence. Applying this to $G=F|_{C/c}$, we get
This almost means that $F^{(\omega)}$ is a sheaf. The problem is that the filtered colimit on the right-hand side need not commute with the limit appearing in the definition of $Match(U,-)$, that is, the canonical map
need not be an equivalence. To solve this problem, we choose a cardinal $\kappa$ such that for every $c\in C$ and every covering sieve $U$ of $c$, the functor $Match(U,(-)|_{C/c})(c):Psh(C)\to \infty Grpd$ preserves $\kappa$-filtered colimits. This is possible because $C$ is small and each of these functors, being the composition of the restriction functor $PSh(C)\to PSh(U)$ and the limit functor $PSh(U)\to\infty Grpd$, has a left adjoint (∞,1)-functor and is therefore accessible (see HTT Prop. 5.4.7.7). Then the above map with $\omega$ replaced by $\kappa$ is an equivalence. For every ordinal $\alpha\lt\kappa$, applying the above to $F^{(\alpha)}$ shows that
Since $\kappa$ is a limit ordinal, we deduce that $F^{(\kappa)}$ is a sheaf by cofinality.
And conversely:
(equivalence of site structures and categories of sheaves)
For $C$ a small (∞,1)-category, there is a bijective correspondence between structure of an (∞,1)-site on $C$ and equivalence classes of topological localizations of $PSh(C)$.
This is HTT, prop. 6.2.2.9.
For $C$ a small (∞,1)-site and $Sh(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh(C)$ the corressponding reflective inclusion of (∞,1)-sheaves into (∞,1)-presheaves on $C$ we have that the image under $L$ of a sub-$(\infty,1)$-functor $p : U \to j(c)$ of a representable $j(c)$ is covering precisely if $L(p)$ is an equivalence.
This is HTT, lemma 6.2.2.8.
Since $Sh(C)$ is the reflectuive localization of $PSh(C)$ at covering monomorphisms, it is clear that if $p : U \to j(c)$ is covering, then $L(p)$ is an equivalence.
To see the converse, form the 0-truncation of $L i$ and conclude as for ordinary sheaves on the homotopy catgegory of $C$.
…
We have seen in (…) that for every structure of an $(\infty,1)$-site on $C$ we obtain a topological localization of the presheaf category, and that this is an injective map from site structures to equivalence classes of sheaf categories. It remains to show that it is also a surjective map, i.e. that every topological localization of $PSh(C)$ comes from the structure of an (∞,1)-site on $C$.
So consider $S \subset Mor(PSh(C))$ a strongly saturated class of morphisms which s topological (closed under pullbacks). Write $S_0 \subset S$ for the subcalss of those that are monomorphisms of the form $U \to j(c)$.
Observe that then $S$ is indeed generated by (is the smallest strongly saturated class containing) $S_0$: since by the co-Yoneda lemma every object $X \in PSh(C)$ is a colimit $x \simeq {\lim_\to}_k j(\Xi_k)$ over representables. It follows that every monomorphism $f : Y \to X$ is a colimit (in $Func(\Delta[1],PSh(C))$) of those of the form $U \to j(c)$: for consider the pullback diagram
where the equivalence is due to the fact that we have universal colimits in $PSh(C)$. This realizes $f$ as a colimit over morphisms of the form $f^* j(\Xi_k) \to j(\Xi_k)$ that are each a pullback of a monomorphism. Since monomorphisms are stable under pullback (see monomorphism in an (∞,1)-category for details), all these component morphisms are themselves monomorphisms.
So every monomorphism in $S$ is generated from $S_0$, but by the assumption that $S$ is topological, it is itself entirely generated from monomorphisms, hence is generated from $S_0$.
So far this establishes that evry topological localization of $PSh(C)$ is a localization at a collection of sieves/ subfunctors $U \to j(c)$ of representables. It remains to show that this collection of subfunctors is indeed an Grothendieck topology and hence exhibits on $C$ the structure of an (∞,1)-site. We check the necessary three axioms:
equivalences cover – The equivalences $j(c) \stackrel{\simeq}{\to} j(c)$ belong to $S$ and are monomorphisms, hence belong to $S_0$.
pullback of a cover is covering - Since monomorphisms are stable under pullback, we haave for every $p : U \to j(c)$ in $S$ and every $j(f) : j(d) \to j(c)$ that also the pullback $f^* p$
is a monomorphism and in $S$, hence in $S_0$.
if restriction of a sieve to a cover is covering, then the sieve is covering – Let $p : U \to j(c)$ be an arbitrary monomorphism and $f : X \to j(d)$ in $S_0$. Write $X \simeq {\lim_\to}_k \Xi_k$ and consider the pullback
where again we made use of the universal colimits in $PSh(C)$. Now notice that
$f$ is in $S$ by assumption;
$p^* f$ is by pullback stability of $S$;
each of the $f_k p$ is in $S$ by assumption, hence ${\lim_k f_k^* p}$ is by the fact that $S$ is strongly saturated.
so by commutativity $p \circ p^*f$ is in $S$.
finally by 2-out-of-3 this means that $p$ is in $S$.
We discuss how $(\infty,1)$-sheaves over a paracompact topological space are equivalent to topological spaces over $X$. This the analogue of the 1-categorical statement that sheaves on $X$ are equivalent to etale spaces over $X$: an etale space over $X$ is one whose fibers are discrete topological space, hence 0-truncated spaces. Then n-category analogy has homotopy n-types as fibers.
For $Y \to X$ a morphism in Top, and $U \in Op(X)$ an open subset of $X$, write
for the simplicial set (in fact a Kan complex) of continuous maps
form $U$ times the topological $k$-simplex $\Delta^k$ into $Y$, that are sections of $Y \to X$.
This is a relative version of the singular simplicial complex functor.
Let $(X, \mathcal{B})$ be a topological space equipped with a base for the topology $\mathcal{B}$.
There is a model category structure on the over category $Top/X$ with weak equivalences and fibration precisely those morphisms $Y \to Z$ over $X$ such that for each $U \in \mathcal{B}$ the induced morphism $Sing_X(Y,U) \to Sing_X(Z,U)$ is a weak equivalence or fibration, respectively, in the standar model structure on simplicial sets.
This is HTT, prop 7.1.2.1.
Write $(Top/X)^\circ$ for the (∞,1)-category presented by this model structure.
Let $X$ be a paracompact topological space and write as usual $Sh_{(\infty,1)}(X) := Sh_{(\infty,1)}(Op(X))$ for the $(\infty,1)$-category of $(\infty,1)$-sheaves on the category of open subsets of $X$; equipped with the canonical structure of a site.
Let $\mathcal{B}$ be the set of $F_\sigma$-open subsets of $X$. This are those open subsets that are countable unions of closed subsets, equivalently the 0-sets of continuous functions $X \to [0,1]$.
Let $Top/X^\circ$ be the corresponding $(\infty,1)$-categoty according to the above proposition. Then $Sing_X(-,-)$ constitutes an equivalence of (∞,1)-categories
This is HTT, corollary 7.1.4.4.
The (∞,1)-toposes that are $(\infty,1)$-categories of sheaves, i.e. that arise by topological localization from an (∞,1)-category of (∞,1)-presheaves, enjoy a number of special properties over other classes of $(\infty,1)$-toposes, such as notably hypercomplete (∞,1)-toposes.
The following lists these properties. (HTT, section 6.5.4.)
The construction of (∞,1)-sheaf (∞,1)-toposes on a given locale is singled out over the construction of other kinds of $(\infty,1)$-toposes (such as hypercomplete (∞,1)-toposes) by the following universal property:
forming $(\infty,1)$-sheaves is, roughly, right adjoint to the functor $\tau_{\leq -1}$ that sends each $(\infty,1)$-topos to its underlying locale of subobjects of the terminal object.
See HTT, item 1) of section 6.5.4.
For $X,Y$ two $(\infty,1)$-toposes, write $Geom(X,Y) \subset Func(X,Y)$ for the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors on those that are geometric morphisms.
For $C$ an small (n,1)-category with finite (∞,1)-limits and equipped with the structure of an (∞,1)-site and for $Y$ an (∞,1)-topos, the truncation functor
is an equivalence (of (∞,1)-categories).
This is HTT, lemma 6.4.5.6.
See also n-localic (∞,1)-topos.
Let $X$ be a coherent topological space and let $Op(X)$ be its category of open subsets with the standard structure of an (∞,1)-site.
Then $Sh_{(\infty,1)}(X) := Sh_{(\infty,1)}(Op(X))$ is compactly generated in that it is generated by filtered colimits of compact objects.
Moreover, the compact objects of $Sh_{(\infty,1)}(X)$ are those that are stalkwise compact objects in ∞Grpd and locally constant along a suitable stratification of $X$.
This is HTT, prop. 6.5.4.4.
This statement is false for the hypercompletion of $Sh_{(\infty,1)}(X)$, in general.
For $X$ a topological space, let
be the global sections terminal geometric morphism.
For $A \in \infty Grpd$, the (nonabelian) cohomology of $X$ with coefficients in $A$ is usually defined in ∞Grpd as
where $Sing X$ is the fundamental ∞-groupoid of $X$. On the other hand, if we send $A$ into $Sh_{(\infty,1)}(X)$ via $LConst$, the there is the intrinsic cohomology of the $(\infty,1)$-topos $Sh_{(\infty,1)}(X)$
Noticing that $X$ is in fact the terminal object of $Sh_{(\infty,1)}(X)$ and that $Sh_{(\infty,1)}(X)(X,-)$ is in fact that global sections functor, this is equivalently
If $X$ is a paracompact space, then these two definitins of nonabelian cohomology of $X$ with constant coefficients $A \in \infty Grpd$ agree:
This is HTT, theorem 7.1.0.1.
The topological localizations of an (∞,1)-category of (∞,1)-presheaves are presented by the left Bousfield localization of the global model structure on simplicial presheaves at the set of Cech covers.
The hypercomplete $(\infty,1)$-sheaf toposes are presented by the local Joyal-Jardine model structure on simplicial presheaves.
Detailed discussion of this model category presentation is at
The study of simplicial presheaves apparently goes back to
which considers locally Kan simplicial presheaves as a category of fibrant objects.
This was later conceived in terms of a model structure on simplicial presheaves and on simplicial sheaves by Joyal and Jardine. Toën summarizes the situation and emphasizes the interpretation in terms of ∞-stacks living in $(\infty,1)$-categories for instance in
B. Toën, Higher and derived stacks: a global overview (arXiv) .
This concerns mostly hypercomplete $(\infty,1)$-sheaves, though.
The full picture in terms of Grothendieck-(∞,1)-toposes of (∞,1)-sheaves is the topic of
Jacob Lurie, Higher Topos Theory .
localization $(\infty,1)$-functors ($(\infty,1)$-sheafification for the present purpose) are discussed in section 5.2.7;
local objects ($(\infty,1)$-sheaves for the present purpose) and local isomorphisms are discussed in section 5.5.4;
the definition of $(\infty,1)$-topoi of $(\infty,1)$-sheaves is then definition 6.1.0.4 in section 6.1;
the characterization of $(\infty,1)$-sheaves in terms of descent is in section 6.1.3
the relation between the Brown–Joyal–Jardine model and the general story is discussed at length in section 6.5.4
An overview is in