(Infinity,1)-Topos

Let $\mathcal{E}$ be a presentable $(\infty,1)$-category.

Definition. An $(\infty,1)$-topos is a presentable $(\infty,1)$-category $\mathcal{E}$ satisfying the following $\infty$-Giraud axioms:

1. Colimits in $\mathcal{E}$ are universal: for every morphism $f: X \to Y$, the pullback functor $f^*: \mathcal{E}_{/Y} \to \mathcal{E}_{/X}$ preserves colimits.
2. Coproducts in $\mathcal{E}$ are disjoint: for any coproduct $X \amalg Y$, the pullback $X \times_{X \amalg Y} Y$ is initial.
3. Every groupoid object in $\mathcal{E}$ is effective: it is the Cech nerve of its own colimit.

Equivalently, $\mathcal{E}$ is an $(\infty,1)$-topos if and only if it arises as an accessible left-exact localization of a presheaf $(\infty,1)$-category $\mathrm{PSh}(\mathcal{C}) = \mathrm{Fun}(\mathcal{C}^{\mathrm{op}}, \mathcal{S})$ for some small $(\infty,1)$-category $\mathcal{C}$.

Proposition. Every $(\infty,1)$-topos $\mathcal{E}$ is equivalent to the $(\infty,1)$-category $\mathrm{Sh}(\mathcal{C}, \tau)$ of sheaves of spaces on some site $(\mathcal{C}, \tau)$.

Proof sketch. By the equivalence above, $\mathcal{E}$ is a left-exact localization of $\mathrm{PSh}(\mathcal{C})$, and the localization is determined by a Grothendieck topology $\tau$ on $\mathcal{C}$.

Proposition. The full subcategory $\tau_{\leq 0}\mathcal{E}$ of $0$-truncated objects in an $(\infty,1)$-topos $\mathcal{E}$ is a $1$-topos (Grothendieck topos).

Proposition. Geometric morphisms between $(\infty,1)$-topoi are adjunctions $f^* \dashv f_*$ where $f^*$ preserves finite $\infty$-limits. The $(\infty,1)$-category $\mathbf{RTop}$ of $(\infty,1)$-topoi and geometric morphisms admits all small limits.

Proposition. Every $(\infty,1)$-topos $\mathcal{E}$ has an internal logic that is constructive and higher-dimensional: the law of excluded middle ($p \vee \neg p = \top$) holds iff $\mathcal{E}$ is Boolean (i.e., every subobject is complemented), propositions are $(-1)$-truncated objects, sets are $0$-truncated objects, and general objects are homotopy types of arbitrary truncation level. Identity types carry homotopical content.

Examples.

• Spaces: The $(\infty,1)$-category $\mathcal{S}$ of spaces is the terminal $(\infty,1)$-topos: for every $(\infty,1)$-topos $\mathcal{E}$, there is a unique geometric morphism $\mathcal{E} \to \mathcal{S}$.
• Sheaves on a space: For a topological space $X$, the $(\infty,1)$-category $\mathrm{Sh}(X)$ of sheaves of spaces on $X$ is an $(\infty,1)$-topos. Its $0$-truncated part recovers the ordinary topos $\mathrm{Sh}(X, \mathbf{Set})$.
• Etale sheaves: For a scheme $X$, the $(\infty,1)$-category of hypercomplete etale sheaves of spaces on $X$ is an $(\infty,1)$-topos, providing the natural setting for etale cohomology with higher categorical coefficients.
• Cohesive topoi: A cohesive $(\infty,1)$-topos is one equipped with an adjoint quadruple $(\Pi \dashv \mathrm{Disc} \dashv \Gamma \dashv \mathrm{coDisc})$ over $\mathcal{S}$, encoding geometric structure via the shape, flat, and sharp modalities.