\(\infty\)-Topoi¶
Job: generalize Grothendieck toposes to the \((\infty,1)\)-setting — categories of sheaves of spaces — with object classifiers, higher descent, and hypercompleteness as in Lurie’s HTT.
Core¶
An \(\infty\)-topos is an \(\infty\)-category that behaves like \(\mathcal{S}\)-valued sheaves on a site: presentable, satisfies higher Giraud axioms (colimits universal and effective, etc.), equivalently an accessible left exact localization of a presheaf \(\infty\)-category \(\mathcal{P}(C)=\mathrm{Fun}(C^{\mathrm{op}},\mathcal{S})\).
| Layer | Content |
|---|---|
| Definition | Giraud / left exact localization of \(\mathcal{P}(C)\) |
| Examples | \(\mathcal{S}\); \(\mathbf{Shv}(X)\); étale \(\infty\)-topos of a scheme |
| Classifiers | object classifier; subobject classifier for \(0\)-truncated |
| Hypercompleteness | Postnikov completeness; \(\infty\)-connective maps |
| Geometry | geometric morphisms; ringed \(\infty\)-topoi (SAG) |
Reading map¶
| Page | Focus |
|---|---|
| Foundations | HTT axioms; classifiers; hypercompleteness; sheaves of spaces |
Dependency order¶
- Category theory + topos theory (1-topos baseline).
- Higher categories (presentable \(\infty\)-cats, \(\mathcal{S}\)).
- Model categories optional (model toposes / local model structures).
- Applications: spectral AG, HoTT, DAG.
Interfaces¶
| Area | Contact |
|---|---|
| Topos theory | truncation \(h\mathcal{X}\) / \(0\)-truncated objects form a 1-topos |
| Higher categories | ambient language |
| Homotopy type theory | internal language of \(\infty\)-topoi (univalent universes) |
| Spectral AG | spectrally ringed \(\infty\)-topoi |
| Algebraic geometry | étale / fppf \(\infty\)-topoi of schemes |
| Model categories | presentations via local injective model structures |
Notation (paper-standard)¶
- \(\mathcal{X},\mathcal{Y}\) \(\infty\)-topoi; \(\mathcal{S}=\mathbf{Spaces}\); \(\mathcal{P}(C)\)
- \(f^*\dashv f_*\) geometric morphism; \(f_!\) essential
- \(\tau_{\le n}\), \(n\)-truncated; \(\mathcal{X}^\wedge\) hypercompletion
- \(\mathbf{Shv}(C,J)\) / \(\mathbf{Sh}(C;\mathcal{S})\) sheaves of spaces
See it¶
Descent that 1-toposes miss. For a hypercover \(U_\bullet\to X\), sheaves of sets only see \(0\)-truncated descent. Sheaves of spaces require $$ F(X)\;\simeq\;\lim_{\Delta} F(U_\bullet) $$ in \(\mathcal{S}\) — higher coherences for gluing. This is mandatory for stacks in groupoids, derived stacks, and spectral schemes: the ambient category of “spaces over the site” is an \(\infty\)-topos, not an elementary 1-topos.
Check¶
How does an ordinary Grothendieck topos sit inside an \(\infty\)-topos?
If \(\mathcal{X}\) is an \(\infty\)-topos, the full subcategory \(\tau_{\le 0}\mathcal{X}\) of \(0\)-truncated objects is a Grothendieck 1-topos. Conversely, sheaves of sets embed as discrete objects in sheaves of spaces.
Name one \(\infty\)-topos that is not hypercomplete.
Certain sheaf \(\infty\)-topoi on topological spaces / sites with bad hypercovers fail hypercompleteness (Dugger–Hollander–Isaksen; HTT examples): \(\infty\)-connected maps need not be equivalences. Hypercompletion \((-)^\wedge\) forces Postnikov completeness.
Pitfalls¶
- Presentability is essential: ad hoc “categories of stacks” may fail Giraud.
- Hypercomplete vs not: cohomology theories and Postnikov towers need the distinction.
- Object classifier size: universes / inaccessible cardinals as in HTT — not a small internal set theory issue to ignore.
- Geometric morphism vs logical: same hierarchy as 1-toposes; inverse image left exact.
Refs¶
- Lurie, Higher Topos Theory
- Rezk, Toposes and homotopy toposes (notes)
- Hoyois, Anel–Joyal, and HoTT book (internal language)
- Dugger–Hollander–Isaksen, hypercovers and simplicial presheaves
Next¶
Foundations · Higher categories · Topos theory · Homotopy type theory · Spectral AG · Derived AG · Model categories · Descent & Stacks · Shape & Cosmoi · Cohesive & Differential · Internal Language & HoTT