$\infty$-Topoi: Foundations¶

Job: define $\infty$-topoi following Lurie HTT, record object classifiers and hypercompleteness, and relate them to sheaves of spaces on sites.

Core¶

Definitions (equivalent packages)¶

An $\infty$-category $\mathcal{X}$ is an $\infty$-topos if it satisfies any of the following equivalent conditions (HTT):

Giraud theorem (∞). $\mathcal{X}$ is presentable; colimits in $\mathcal{X}$ are universal (stable under pullback); coproducts are disjoint; groupoid objects are effective (every groupoid is the Čech nerve of its colimit map); and $\mathcal{X}$ has a small set of generators.
Left exact localization. There is a small $\infty$-category $C$ and an accessible left exact localization functor $\mathcal{P}(C)\to\mathcal{X}$ (reflective localization whose reflector preserves finite limits).
Sheaves of spaces. $\mathcal{X}\simeq\mathbf{Shv}(C,J)$ for some site $(C,J)$ with values in $\mathcal{S}$ (up to hypercompletion subtleties of the topology).

1-topos (Giraud)	$\infty$-upgrade
Set-valued sheaves	space-valued sheaves
Effective eq.\ relations	effective groupoid objects
$\Omega$ subobject classifier	object classifier / universes
Finite limits + power objects	presentable + left exact localization

Examples. $\mathcal{S}$ itself; $\mathcal{P}(C)$ presheaves; $\mathbf{Shv}(X)$ on a topological space / locale; étale $\infty$-topos of a scheme; slice $\mathcal{X}_{/X}$; products of $\infty$-topoi.

Sheaves of spaces¶

For a Grothendieck topology $J$ on $C$, a functor $F:C^{\mathrm{op}}\to\mathcal{S}$ is a sheaf if for every covering sieve / hypercover (depending on formulation), $$ F(c)\;\xrightarrow{\sim}\;\lim_{c'\to c\in S} F(c'). $$ Sheafification is left exact accessible localization of $\mathcal{P}(C)$. Using only Čech covers vs full hypercovers yields possibly different localizations — the gap is measured by hypercompleteness.

Model presentation. Injective/local projective model structures on simplicial presheaves (Jardine, …) present $\mathbf{Shv}(C;\mathcal{S})$; fibrant objects are local Kan complexes satisfying descent.

Truncation and the underlying 1-topos¶

The truncation functors $\tau_{\le n}:\mathcal{X}\to\mathcal{X}$ exist; $n$-truncated objects are those with $(n+1)$-connective diagonal mapping spaces.

$0$-truncated objects $\tau_{\le 0}\mathcal{X}$ form a Grothendieck 1-topos (ordinary sheaves of sets when $\mathcal{X}=\mathbf{Shv}(C)$).
$n$-topoi interpolate (Rezk, Lurie): $(n,1)$-categorical sheaves.

Homotopy groups / sheaves of $\pi_n$ for pointed objects internalize classical algebraic topology inside $\mathcal{X}$.

Object classifiers¶

An $\infty$-topos has object classifiers: for sufficiently small morphisms (relatively $\kappa$-compact), there is a universal family $\mathcal{U}_\kappa\to\mathcal{U}_\kappa$ such that maps $X\to\mathcal{U}_\kappa$ classify bundles $E\to X$ with $\kappa$-compact fibers, up to equivalence.

Classifier	Classifies
Subobject classifier $\Omega$	monomorphisms (0-truncated $(-1)$-truncated maps)
Object classifier $\mathcal{U}$	maps with small fibers (univalent universe)
$n$-truncated classifiers	truncated bundles

This is the $\infty$-categorical home of univalent universes in HoTT: internal language of $\infty$-topoi supports homotopy type theory with univalence (Shulman, …).

Hypercompleteness¶

A morphism $f$ in $\mathcal{X}$ is $\infty$-connected if $\tau_{\le n}f$ is an equivalence for all $n$ (all homotopy sheaves vanish). $\mathcal{X}$ is hypercomplete if every $\infty$-connected morphism is an equivalence.

Fact	Content
Hypercompletion	$\mathcal{X}\to\mathcal{X}^\wedge$ left exact localization inverting $\infty$-connected maps
Postnikov	hypercomplete $\Leftrightarrow$ every object is limit of its Postnikov tower
Whiteheads	hypercomplete $\Rightarrow$ Whitehead theorem: $\pi_*$-iso $\Rightarrow$ eq
Failure	some $\mathbf{Shv}(X)$ for non-hypercomplete topological sites

When it matters. Descent along hypercovers, comparison of cohomology theories, and identification of mapping spaces in geometry (derived stacks, spectral schemes) often implicitly pass to hypercomplete sheaves. HTT carefully distinguishes $\mathbf{Shv}$ vs $\mathbf{Shv}^\wedge$.

Geometric morphisms¶

A geometric morphism $f:\mathcal{Y}\to\mathcal{X}$ is an adjunction $f^*\dashv f_*$ with $f^*$ left exact. Same logic as 1-toposes:

inverse image $f^*$ preserves finite limits and all colimits (as left adjoint);
points $f:\mathcal{S}\to\mathcal{X}$ give stalk functors;
morphisms of sites induce geometric morphisms of sheaf $\infty$-topoi.

Essential geometric morphisms: $f^*$ has further left adjoint $f_!$. Étale geometric morphisms: slices $\mathcal{X}_{/U}\to\mathcal{X}$.

Relation to higher geometry¶

Structure	Role of $\infty$-topos
Derived / spectral stacks	sheaves of spaces on SCR / $\mathrm{CAlg}$
Ringed $\infty$-topos	structure sheaf in $\mathrm{CAlg}(\mathcal{X})$ — SAG
Motivic spaces	$\mathbb{A}^1$-localization of Nisnevich sheaves of spaces
Internal logic	HoTT; synthetic homotopy theory

See it¶

Étale $\infty$-topos of a scheme $X$. Objects are sheaves of spaces on the étale site $X_{\mathrm{\acute{e}t}}$. Discrete objects recover the classical étale topos. Higher homotopy encodes stacks: a sheaf of groupoids satisfying descent is a $1$-truncated object — Deligne–Mumford stacks embed into the étale $\infty$-topos. Derived enhancements enlarge the site (SCR-étale) but still produce $\infty$-topoi as ambient categories of sheaves.

Non-hypercomplete warning. There exist spaces / sites where a map induces isomorphisms on all $\pi_n$ sheaves yet fails to be an equivalence in $\mathbf{Shv}$ — Postnikov towers do not converge. Working with $\mathbf{Shv}^\wedge$ restores Whitehead’s theorem at the cost of changing colimits slightly. Spectral algebraic geometers typically use hypercomplete sheaves of spectra when stating descent.

Check¶

Why does left exactness of the reflector matter in $\mathcal{P}(C)\to\mathcal{X}$?

Left exact localization preserves finite limits, so the localization still has universal colimits interacting well with pullbacks — the higher Giraud exactness. Arbitrary accessible localizations give presentable $\infty$-categories that need not be topoi (e.g.\ stable localizations produce spectra-like categories, not topoi).

Prove sketch: $\mathcal{S}$ is an $\infty$-topos.

$\mathcal{S}\simeq\mathcal{P}(*)$; identity is left exact localization. Alternatively Giraud: presentable; colimits universal (spaces); groupoids effective (homotopy hypothesis / nerves); generated by a point.

Pitfalls¶

Site choice: same $\infty$-topos may arise from many sites; morphisms of sites $\neq$ all geometric morphisms.
$n$-truncated reflection changes mapping spaces; limits of truncated objects need not compute limits in $\mathcal{X}$.
Sheaves of spectra form a stable $\infty$-category, not an $\infty$-topos — topos $=$ spaces-like, not spectra-like.
Size: object classifiers require universe stratification; “the” universe is always relative to a regular cardinal.

Refs¶

Lurie, Higher Topos Theory (esp.\ §§6–7)
Rezk, Toposes and homotopy toposes
Anel–Joyal, Topo-logie; Hoyois notes on $\infty$-topoi
Univalent Foundations, Homotopy Type Theory book (internal language)

Next¶

∞-topoi index · Higher categories · Topos theory · Homotopy type theory · Spectral AG · Derived AG · Model categories · Algebraic geometry

\(\infty\)-Topoi: Foundations¶