The $\infty$-category $\mathcal{S}$ of spaces is the terminal $\infty$-topos: for every $\infty$-topos $\mathcal{X}$, there is an essentially unique geometric morphism $\mathcal{X} \to \mathcal{S}$.

$\mathcal{S}$ is the sheaf $\infty$-topos on the point (the trivial site). It satisfies descent because pullbacks of spaces preserve colimits.

For a topological space $X$, the $\infty$-category $\operatorname{Shv}(X)$ of sheaves of spaces on $X$ is an $\infty$-topos. Its objects are functors $\operatorname{Open}(X)^{\mathrm{op}} \to \mathcal{S}$ satisfying the sheaf condition (descent for open covers).

$\operatorname{Shv}(X)$ is a left exact localization of the presheaf category $\operatorname{Fun}(\operatorname{Open}(X)^{\mathrm{op}}, \mathcal{S})$.

When $X$ is a point, $\operatorname{Shv}(*) \simeq \mathcal{S}$.

For a scheme $X$, the etale $\infty$-topos $\operatorname{Shv}_{\mathrm{et}}(X)$ is the $\infty$-category of sheaves of spaces on the small etale site of $X$. This is an $\infty$-topos that contains the etale cohomology of $X$ as part of its structure.

The constant sheaf functor $\mathcal{S} \to \operatorname{Shv}_{\mathrm{et}}(X)$ and the global sections functor $\Gamma: \operatorname{Shv}_{\mathrm{et}}(X) \to \mathcal{S}$ form a geometric morphism.

For a site $(\mathcal{C}, \tau)$, the $\infty$-category $\operatorname{Shv}_\tau(\mathcal{C})$ of sheaves valued in spaces is an $\infty$-topos. Its objects are higher stacks: homotopy-coherent presheaves satisfying descent for the topology $\tau$.

When $\mathcal{C}$ is the category of schemes with the etale topology, $\operatorname{Shv}_{\mathrm{et}}(\mathbf{Sch})$ contains algebraic spaces, Deligne--Mumford stacks, and general higher stacks as objects.

If $\mathcal{X}$ is an $\infty$-topos and $X \in \mathcal{X}$ is an object, the overcategory $\mathcal{X}_{/X}$ is again an $\infty$-topos. This corresponds to the "base change" or "localization" at $X$.

For $\mathcal{X} = \operatorname{Shv}(B)$ and $X = f_*(1)$ for a continuous map $f: E \to B$, the slice $\mathcal{X}_{/X} \simeq \operatorname{Shv}(E)$.

In an $\infty$-topos, colimits are universal: for any map $f: X \to Y$ and any diagram $F: K \to \mathcal{X}_{/Y}$,

$f^*(\operatorname{colim} F) \simeq \operatorname{colim}(f^* \circ F)$

This means pullback preserves colimits. In particular, coproducts are disjoint ($X \times_{X \sqcup Y} Y \simeq \emptyset$) and pushouts are computed fiberwise.

This universality is the $\infty$-categorical analogue of the descent condition for sheaves.

In an $\infty$-topos, every groupoid object is effective. A groupoid object in $\mathcal{X}$ is a simplicial object $U_\bullet: \Delta^{\mathrm{op}} \to \mathcal{X}$ satisfying the Segal condition and completeness. Effectiveness means $U_\bullet$ is the Cech nerve of its colimit: $U_\bullet \simeq \check{C}(U_0 \to |U_\bullet|)$.

This is the $\infty$-categorical analogue of the fact that every equivalence relation in a Grothendieck topos is effective (has a quotient).

In an $\infty$-topos $\mathcal{X}$, the cohomology of an object $X$ with coefficients in an object $A$ is:

$H^n(X; A) = \pi_0 \operatorname{Map}_{\mathcal{X}}(X, K(A, n))$

where $K(A, n)$ is the Eilenberg--MacLane object. More generally, for any pointed connected object $B \in \mathcal{X}$, the nonabelian cohomology is $H^1(X; G) = \pi_0 \operatorname{Map}(X, BG)$ (classifying $G$-torsors).

The mapping space $\operatorname{Map}(X, A)$ computes the "derived global sections" of $A$ over $X$, with its full homotopical information.

An $\infty$-topos has homotopy dimension $\leq n$ if for every $n$-connected object $A$ (i.e., $\pi_k(A) = 0$ for $k \leq n$), the mapping space $\operatorname{Map}(1, A)$ is contractible (where $1$ is the terminal object).

For $\operatorname{Shv}(X)$ where $X$ is a topological space of covering dimension $\leq n$, the homotopy dimension is $\leq n$. This implies the vanishing of sheaf cohomology $H^k(X; \mathcal{F}) = 0$ for $k > n$ and locally constant sheaves $\mathcal{F}$.

A continuous map $g: X \to Y$ of topological spaces induces a geometric morphism $g: \operatorname{Shv}(X) \to \operatorname{Shv}(Y)$ with $g^*(\mathcal{F})(U) = \mathcal{F}(g^{-1}(U))$ (inverse image) and $g_*(\mathcal{G})(V) = \mathcal{G}(g^{-1}(V))$ (direct image).

A point of an $\infty$-topos $\mathcal{X}$ is a geometric morphism $p: \mathcal{S} \to \mathcal{X}$. The inverse image $p^*: \mathcal{X} \to \mathcal{S}$ computes the "stalk" of a sheaf at the point.

For $\operatorname{Shv}(X)$, points correspond to points of $X$. For the etale $\infty$-topos of a scheme, points correspond to geometric points.