nLab
(infinity,1)-category of (infinity,1)-sheaves

Idea

The notion of $(\mathrm{\infty },1)$-category of $(\mathrm{\infty },1)$-sheaves is the generalization of the notion of category of sheaves from category theory to the higher category theory of (∞,1)-categories.

Definition

Definition

An $(\mathrm{\infty },1)$-category of $(\mathrm{\infty },1)$-sheaves is a reflective sub-(∞,1)-category

$$\mathrm{Sh}(C)\stackrel{\stackrel{L}{\leftarrow }}{\hookrightarrow }\mathrm{PSh}(C)$$Sh(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh(C)

of an (∞,1)-category of (∞,1)-presheaves such that

• $L$ is a topological localization;

• equivalently: there is the structure of an (∞,1)-site on $C$ such that the objects of $\mathrm{Sh}(C)$ are precisely those (∞,1)-presheaves $A$ that are local objects with respect to the covering monomorphisms $p:U\to j(c)$ in $\mathrm{PSh}(C)$ in that

  $$A(c)\simeq \mathrm{PSh}(j(c),A)\stackrel{\mathrm{PSh}(p,A)}{\to }\mathrm{PSh}(U,A)$$A(c) \simeq PSh(j(c),A) \stackrel{PSh(p,A)}{\to} PSh(U,A)

  is an (∞,1)-equivalence in ∞Grpd.

  This is the descent condition and the presheaves satisfying it are the (∞,1)-sheaves .

  Typically $U$ here is the Cech nerve

  $$C(\{U_i\})=\underset{{\to }_{[n]}}{\lim }{U}_{i_0,\cdots U_{i_n}}$$C(\{U_i\}) = \lim_{\to_{[n]}} U_{i_0, \cdots U_{i_n}}

  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

  $$A(c)\simeq \mathrm{PSh}(\underrightarrow{\lim }{U}_{\cdots },A)\simeq \underleftarrow{\lim }A(U_{\cdots })=\underleftarrow{\lim }(\cdots \overrightarrow{\overrightarrow{\to }}\prod _{i,j}A(U_i){\times }_{A(c)}A(U_j)\overrightarrow{\to }\prod _{i}A(U_i)).$$A(c) \simeq PSh(\lim_\to U_\cdots, A) \simeq \lim_{\leftarrow} A(U_\cdots) = \lim_{\leftarrow} \left( \cdots \stackrel{\to}{\stackrel{\to}{\to}} \prod_{i,j} A(U_i) \times_{A(c)} A(U_j) \stackrel{\to}{\to}\prod_i A(U_i) \right) \,.

This is HTT, def. 6.2.2.6.

An $(\mathrm{\infty },1)$-category of $(\mathrm{\infty },1)$-sheaves is an (∞,1)-topos.

Terminology

Sometimes (∞,1)-sheaves are called ∞-stacks, though sometimes the latter term is reserved for hypercomplete $(\mathrm{\infty },1)$-sheaves.

The (n,1)-categorical counting is

• sheaf = 0-stack = 0-truncated $(\mathrm{\infty },1)$-sheaf

• $(2,1)$-sheaf = stack = 1-truncated $(\mathrm{\infty },1)$-sheaf

• $(3,1)$-sheaf = 2-stack = 2-truncated $(\mathrm{\infty },1)$-sheaf

• etc.

• $(\mathrm{\infty },1)$-sheaf = ∞-stack (or = hypercomplete $(\mathrm{\infty },1)$-sheaf).

Properties

Localizations and Grothendieck topology

We reproduce the proof that the two definitions above really are equivalent.

Proposition

For $C$ an (∞,1)-site, the full sub-(∞,1)-category of $\mathrm{PSh}(C)$ on local objects with respect to the covering monomorphisms in $\mathrm{PSh}(C)$ is indeed a topological localization. and hence $\mathrm{Sh}(C)$ is indeed an exact reflective sub-(∞,1)-category of $\mathrm{PSh}(C)$ and hence an (∞,1)-topos.

This is HTT, lemma 6.2.2.7

Proof

Write $S_0$ for the class of covering monomorphisms $U\to j(C)$ in $\mathrm{PSh}(C)$ and write $S$ for the strongly saturated class that it generates.

Since $\mathrm{PSh}(C)$ has all limits, by the characterization of exact localizations it is sufficient to show that $S$ is closed under pullback.

By the very definition of (∞,1)-site we have that the covering monomorphisms are stable under pullback along morphisms $j(f):j(d)\to j(c)$ in the image of the (∞,1)-Yoneda embedding $j:C\to \mathrm{PSh}(C)$. So the task is to show that this statement lifts to pullbacks along arbitrary morphisms $X\to j(c)$ in $\mathrm{PSh}(C)$.

For that, let $S\prime \subset \mathrm{Mor}(\mathrm{PSh}(C))$ be the collection of morphisms whose pullback lands in $S$. As in the discussion of exact localizations one finds from the fact that $\mathrm{PSh}(C)$ has universal colimits that $S\prime$ itself is strongly saturated. Since we need to show that $S\subset S\prime$ and since $S$ is the smallest strongly saturated class containing $S_0$, it is sufficient to show that $S_0\subset S\prime$, i.e. that the pullback of a covering monomorphisms lands in the strong saturation of covering monomorphisms.

This follows by realizing every morphism $X\to j(c)$ as a colimit of representables and then using the stability of covering monomorphisms under pullback along representables to conclude that the pullback of $U\to j(c)$ to $X$ is a colimit of covering monomorphism, hence is in the strongly saturated class generated by them.

We spell this out in more detail.

We want to show that for a pullback diagram

$$\begin{array}{ccc}{f}^{*}U=p^{*}X& \to & U\\ \downarrow & & {\downarrow }^p\\ X& \stackrel{f}{\to }& j(c)\end{array}$$\array{ f^* U = p^* X &\to& U \\ \downarrow && \downarrow^p \\ X &\stackrel{f}{\to}& j(c) }

with $U\to j(c)$ a covering monomorphism, i.e. in $S_0$, $g^{*}U\to X$ is in $S$.

Using the equivalence of presheaves on overcategories

$$\mathrm{Sh}(C{)}_{/j(c)}\simeq \mathrm{PSh}(C_{/c})$$Sh(C)_{/j(c)} \simeq PSh(C_{/c})

and the co-Yoneda lemma in $\mathrm{PSh}(C_c)$ we have that every $X\to j(c)$ may be written as a colimit over a diagram

$$\Xi :K\to C_c\stackrel{j}{\to }\mathrm{PSh}(C_c)\simeq \mathrm{PSh}(C{)}_{j(c)}$$\Xi : K \to C_{c} \stackrel{j}{\to} PSh(C_{c}) \simeq PSh(C)_{j(c)}

that factors through the Yoneda embedding.

$$X\simeq \underrightarrow{\lim }\Xi .$$X \simeq \lim_\to \Xi \,.

Since colimits in over-categories $C_{/c}$ are computed as colimits in $C$ under the projection $C_{/c}\to C$, we may write out pullback diagram equivalently as

$$\begin{array}{ccc}{p}^{*}{\underrightarrow{\lim }}_k{\Xi }_k& \to & U\\ \downarrow & & \downarrow \\ {\underrightarrow{\lim }}_k{\Xi }_k& \stackrel{f}{\to }& j(c)\end{array}\simeq \begin{array}{ccc}{\underrightarrow{\lim }}_k{p}^{*}{\Xi }_k& \to & U\\ \downarrow & & \downarrow \\ {\underrightarrow{\lim }}_k{\Xi }_k& \stackrel{f}{\to }& j(c)\end{array}$$\array{ p^* {\lim_\to}_k \Xi_k &\to& U \\ \downarrow && \downarrow \\ {\lim_\to}_k \Xi_k &\stackrel{f}{\to}& j(c) } \;\;\;\;\;\; \simeq \;\;\;\;\;\; \array{ {\lim_\to}_k p^* \Xi_k &\to& U \\ \downarrow && \downarrow \\ {\lim_\to}_k \Xi_k &\stackrel{f}{\to}& j(c) }

in $\mathrm{PSh}(C)$. Since $\mathrm{PSh}(C)$ has universal colimits the morphism on the left is equivalent to ${{\lim }_{\to }}_k{p}^{*}{\Xi }_k\to {{\lim }_{\to }}_k{\Xi }_k$, as indicated above. But, as mentioned before, each component $p^{*}{\Xi }_k\to {\Xi }_k$ is in $S_0$, so this morphism is a colimit in the functor category $\mathrm{Func}(\Delta [1],\mathrm{PSh}(C))$ over objects in $S_0$. By the definition of strong saturation, this is in $S$.

And conversely:

Proposition

(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 $\mathrm{PSh}(C)$.

This is HTT, prop. 6.2.2.9.

Lemma

For $C$ a small (∞,1)-site and $\mathrm{Sh}(C)\stackrel{\stackrel{L}{\leftarrow }}{\hookrightarrow }\mathrm{PSh}(C)$ the corressponding reflective inclusion of (∞,1)-sheaves into (∞,1)-presheaves on $C$ we have that the image under $L$ of a sub-$(\mathrm{\infty },1)$-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.

Proof of the Lemma

Since $\mathrm{Sh}(C)$ is the reflectuive localization of $\mathrm{PSh}(C)$ 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 $Li$ and conclude as for ordinary sheaves on the homotopy catgegory of $C$.

…

Proof of the Proposition

We have seen in (…) that for every structure of an $(\mathrm{\infty },1)$-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 $\mathrm{PSh}(C)$ comes from the structure of an (∞,1)-site on $C$.

So consider $S\subset \mathrm{Mor}(\mathrm{PSh}(C))$ 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 $X\in \mathrm{PSh}(C)$ 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 $\mathrm{Func}(\Delta [1],\mathrm{PSh}(C))$) of those of the form $U\to j(c)$: for consider the pullback diagram

$$\begin{array}{ccc}{f}^{*}({\underrightarrow{\lim }}_k{\Xi }_k)& \to & Y\\ {\downarrow }^{\simeq f}& & {\downarrow }^f\\ {\underrightarrow{\lim }}_k{\Xi }_k& \stackrel{\simeq }{\to }& X\end{array}\simeq \begin{array}{ccc}({\underrightarrow{\lim }}_k{f}^{*}{\Xi }_k)& \to & Y\\ {\downarrow }^{\simeq f}& & {\downarrow }^f\\ {\underrightarrow{\lim }}_k{\Xi }_k& \stackrel{\simeq }{\to }& X\end{array}$$\array{ f^* ({\lim_\to}_k \Xi_k) &\to& Y \\ \downarrow^{\mathrlap{\simeq f}} && \downarrow^{\mathrlap{f}} \\ {\lim_\to}_k \Xi_k &\stackrel{\simeq}{\to}& X } \;\;\;\;\; \simeq \;\;\;\;\; \array{ ({\lim_\to}_k f^* \Xi_k) &\to& Y \\ \downarrow^{\mathrlap{\simeq f}} && \downarrow^{\mathrlap{f}} \\ {\lim_\to}_k \Xi_k &\stackrel{\simeq}{\to}& X }

where the equivalence is due to the fact that we have universal colimits in $\mathrm{PSh}(C)$. 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 $\mathrm{PSh}(C)$ 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:

1. equivalences cover – The equivalences $j(c)\stackrel{\simeq }{\to }j(c)$ belong to $S$ and are monomorphisms, hence belong to $S_0$.

2. 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 $f^{*}p$

   $$\begin{array}{ccc}{f}^{*}U& \to & U\\ {\downarrow }^{f^{*}p}& & {\downarrow }^p\\ j(d)& \stackrel{f}{\to }& j(c)\end{array}$$\array{ f^* U &\to& U \\ \downarrow^{\mathrlap{f^* p}} && \downarrow^{\mathrlap{p}} \\ j(d) &\stackrel{f}{\to}& j(c) }

   is a monomorphism and in $S$, hence in $S_0$.

3. 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

   $$\begin{array}{ccc}{\underrightarrow{\lim }}_k{p}^{*}{\Xi }_k& \stackrel{p^{*}f}{\to }& U\\ {\downarrow }^{{\underrightarrow{\lim }}_k{f}_k^{*}p}& & {\downarrow }^p\\ {\underrightarrow{\lim }}_k{\Xi }_k& \stackrel{f}{\to }& j(c)\end{array},$$\array{ {\lim_\to}_k p^* \Xi_k &\stackrel{p^* f}{\to}& U \\ \downarrow^{{\lim_\to}_k f_k^* p} && \downarrow^{\mathrlap{p}} \\ {\lim_\to}_k \Xi_k &\stackrel{f}{\to}& j(c) } \,,

   where again we made use of the universal colimits in $\mathrm{PSh}(C)$. Now notice that

   1. $f$ is in $S$ by assumption;

   2. $p^{*}f$ is by pullback stability of $S$;

   3. 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.

   4. so by commutativity $p\circ p^{*}f$ is in $S$.

   5. finally by 2-out-of-3 this means that $p$ is in $S$.

Over paracompact topological spaces

We discuss how $(\mathrm{\infty },1)$-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.

Definition

For $Y\to X$ a morphism in Top, and $U\in \mathrm{Op}(X)$ an open subset of $X$, write

$${\mathrm{Sing}}_X(Y,U):={\mathrm{Hom}}_X(U\times {\Delta }^{•},X)$$Sing_X(Y,U) := Hom_X(U \times \Delta^\bullet, X)

for the simplicial set (in fact a Kan complex) of continuous maps

$$\begin{array}{ccccc}U\times {\Delta }^k& & \to & & Y\\ & \searrow & & \swarrow \\ & & X\end{array}$$\array{ U \times \Delta^k && \to && Y \\ & \searrow && \swarrow \\ && X }

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.

Proposition

Let $(X,ℬ)$ be a topological space equipped with a base for the topology $ℬ$.

There is a model category structure on the over category $\mathrm{Top}/X$ with weak equivalences and fibration precisely those morphisms $Y\to Z$ over $X$ such that for each $U\in ℬ$ the induced morphism ${\mathrm{Sing}}_X(Y,U)\to {\mathrm{Sing}}_X(Z,U)$ is a weak equivalence or fibration, respectively, in the standar model structure on simplicial sets.

This is HTT, prop 7.1.2.1.

Write $(\mathrm{Top}/X{)}^{\circ }$ for the (∞,1)-category presented by this model structure.

Proposition

Let $X$ be a paracompact topological space and write as usual ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(X):={\mathrm{Sh}}_{(\mathrm{\infty },1)}(\mathrm{Op}(X))$ for the $(\mathrm{\infty },1)$-category of $(\mathrm{\infty },1)$-sheaves on the category of open subsets of $X$; equipped with the canonical structure of a site.

Let $ℬ$ 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 $\mathrm{Top}/X^{\circ }$ be the corresponding $(\mathrm{\infty },1)$-categoty according to the above proposition. Then ${\mathrm{Sing}}_X(-,-)$ constitutes an equivalence of (∞,1)-categories

$$\mathrm{Top}/X^{\circ }\simeq {\mathrm{Sh}}_{(\mathrm{\infty },1)}(X).$$Top/X^\circ \simeq Sh_{(\infty,1)}(X) \,.

This is HTT, corollary 7.1.4.4.

Difference to more general $(\mathrm{\infty },1)$-toposes

The (∞,1)-toposes that are $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-toposes, such as notably hypercomplete (∞,1)-toposes.

The following lists these properties. (HTT, section 6.5.4.)

Universal property

The construction of (∞,1)-sheaf (∞,1)-toposes on a given locale is singled out over the construction of other kinds of $(\mathrm{\infty },1)$-toposes (such as hypercomplete (∞,1)-toposes) by the following universal property:

forming $(\mathrm{\infty },1)$-sheaves is, roughly, right adjoint to the functor ${\tau }_{\le -1}$ that sends each $(\mathrm{\infty },1)$-topos to its underlying locale of subobjects of the terminal object.

See HTT, item 1) of section 6.5.4.

For $X,Y$ two $(\mathrm{\infty },1)$-toposes, write $\mathrm{Geom}(X,Y)\subset \mathrm{Func}(X,Y)$ for the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors on those that are geometric morphisms.

Lemma

For $C$ an small (n,1)-category equipped with the structure of an (∞,1)-site and for $Y$ an (∞,1)-topos, the truncation functor

$${\tau }_{\le n-1}:\mathrm{Geom}(Y,\mathrm{Sh}(C))\to \mathrm{Geom}({\tau }_{\le n-1}Y,{\tau }_{\le n-1}\mathrm{Sh}(C))$$\tau_{\leq n-1} : Geom(Y, Sh(C)) \to Geom(\tau_{\leq n-1} Y, \tau_{\leq n-1} Sh(C))

is an equivalence (of (∞,1)-categories).

This is HTT, lemma 6.4.5.6.

See also n-localic (∞,1)-topos.

Compact generation

Proposition

Let $X$ be a coherent topological space and let $\mathrm{Op}(X)$ be its category of open subsets with the standard structure of an (∞,1)-site.

Then ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(X):={\mathrm{Sh}}_{(\mathrm{\infty },1)}(\mathrm{Op}(X))$ is compactly generated in that it is generated by filtered colimits of compact objects.

Moreover, the compact objects of ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(X)$ 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 ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(X)$, in general.

Nonabelian cohomology

For $X$ a topological space, let

$$(\mathrm{LConst}⊣\Gamma ):{\mathrm{Sh}}_{(\mathrm{\infty },1)}(X)\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{\Gamma }{\to }}$$(LConst \dashv \Gamma) : Sh_{(\infty,1)}(X) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}

be the global sections terminal geometric morphism.

For $A\in \mathrm{\infty }\mathrm{Grpd}$, the (nonabelian) cohomology of $X$ with coefficients in $A$ is usually defined in ∞Grpd as

$$H(X,A):={\pi }_0\mathrm{Func}(\mathrm{Sing}X,A),$$H(X,A) := \pi_0 Func(Sing X, A) \,,

where $\mathrm{Sing}X$ is the fundamental ∞-groupoid of $X$. On the other hand, if we send $A$ into ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(X)$ via $\mathrm{LConst}$, the there is the intrinsic cohomology of the $(\mathrm{\infty },1)$-topos ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(X)$

$$H\prime (X,A):={\pi }_0{\mathrm{Sh}}_{(\mathrm{\infty },1)}(X)(X,\mathrm{LConst}A).$$H'(X,A) := \pi_0 Sh_{(\infty,1)}(X)(X, LConst A) \,.

Noticing that $X$ is in fact the terminal object of ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(X)$ and that ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(X)(X,-)$ is in fact that global sections functor, this is equivalently

$$\cdots \simeq {\pi }_0\Gamma \mathrm{LConst}A.$$\cdots \simeq \pi_0 \Gamma LConst A \,.

Theorem

If $X$ is a paracompact space, then these two definitins of nonabelian cohomology of $X$ with constant coefficients $A\in \mathrm{\infty }\mathrm{Grpd}$ agree:

$$H(X,A):={\pi }_0\mathrm{\infty }\mathrm{Grpd}(\mathrm{Sing}X,A)\simeq {\mathrm{Sh}}_{(\mathrm{\infty },1)}(X)(X,\mathrm{LConst}A).$$H(X,A) := \pi_0 \infty Grpd(Sing X,A) \simeq Sh_{(\infty,1)}(X)(X,LConst A) \,.

This is HTT, theorem 7.1.0.1.

Models

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 $(\mathrm{\infty },1)$-sheaf toposes are presented by the local Joyal-Jardine model structure on simplicial presheaves.

Detailed discussion of this model category presentation is at

• models for (infinity,1)-category of (infinity,1)-sheaves .

References

The study of simplicial presheaves apparently goes back to

• K. Brown, Abstract Homotopy Theory and Generalized Sheaf Cohomology

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. Toë summarizes the situation and emphasizes the interpretation in terms of ∞-stacks living in $(\mathrm{\infty },1)$-categories for instance in

B. Toën, Higher and derived stacks: a global overview (arXiv) .

This concerns mostly hypercomplete $(\mathrm{\infty },1)$-sheaves, though.

The full picture in terms of Grothendieck-(∞,1)-toposes of (∞,1)-sheaves is the topic of

• Jacob Lurie, Higher Topos Theory .

  • localization $(\mathrm{\infty },1)$-functors ($(\mathrm{\infty },1)$-sheafification for the present purpose) are discussed in section 5.2.7;

  • local objects ($(\mathrm{\infty },1)$-sheaves for the present purpose) and local isomorphisms are discussed in section 5.5.4;

  • the definition of $(\mathrm{\infty },1)$-topoi of $(\mathrm{\infty },1)$-sheaves is then definition 6.1.0.4 in section 6.1;

  • the characterization of $(\mathrm{\infty },1)$-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