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Higher Topos Theory · Chapter 3 · Section 3.4

Hypercompletion

3.4. Hypercompletion

Finite truncations need not detect all morphisms in a topos. A morphism may be \(n\)-connected for every finite \(n\) without being an isomorphism, so that it is invisible at every finite stage of the Postnikov hierarchy. Hypercompletion is the localization which forces all such morphisms to become isomorphisms. We now construct this localization and show that it is left exact, hence again a topos.

Definition 3.36.

A morphism \(f\colon X \to Y\) in \(T\) is called \(\infty\)-connected if it is \(n\)-connected for every \(n\). It is called \(\infty\)-truncated (or hypercomplete) if it is right orthogonal to the class of \(\infty\)-connected maps.

Definition 3.37.

Let \(T_{\leq \infty}\) denote the full subcategory of \(\infty\)-truncated objects in \(T\). An alternative notation for this subcategory is \(T^{\hyp}\). We say that \(T\) is hypercomplete if every object is \(\infty\)-truncated.

Remark 3.38.

The term “hypercomplete” is historical but slightly misleading: hypercompleteness is a separation condition rather than a completeness condition. “Postnikov separated” might have been a better term. In [Anel et al. 2025], the authors use the term “hyperreduced” instead.

Proposition 3.39.

Let \(T\) be a topos.

  1. The \(\infty\)-connected maps form a strongly saturated class of small generation, stable under base change and diagonals.

  2. The inclusion \(T_{\leq \infty} \hookrightarrow T\) admits a left adjoint \(\tau_{\infty}\colon T \to T_{\leq \infty}\).

  3. The pair (\(\infty\)-connected, \(\infty\)-truncated) is a factorization system on \(T\).

Proof
(1) The \(n\)-connected maps form a saturated class (i.e. closed under pushouts and closed under colimits in \(\Ar(T)\)), by the factorization system from Proposition 3.18. In particular, the \(\infty\)-connected maps form a saturated class.For the 2-out-of-3 property, it remains to check left cancellation. If \(gf\) and \(g\) are \(\infty\)-connected, then for every \(n\) the map \(gf\) is \(n\)-connected and \(g\) is \((n+1)\)-connected, so Corollary 3.33 shows that \(f\) is \(n\)-connected. Thus \(f\) is \(\infty\)-connected. Closure under base change follows from part (2) of Proposition 3.18. Closure under diagonals follows from Theorem 3.22: if \(f\) is \(\infty\)-connected, then for every \(n\) it is \((n+1)\)-connected, and hence \(\Delta_f\) is \(n\)-connected.Small generation is the additional accessibility input. The full subcategory of \(\Ar(T)\) spanned by the \(\infty\)-connected morphisms is accessible, and therefore the resulting strongly saturated class is of small generation. We refer to [Lurie 2009, Proposition 6.5.2.8] for this argument.(2) The \(\infty\)-truncated objects are precisely the local objects with respect to this strongly saturated class of small generation. The existence and accessibility of the reflector \(\tau_{\infty}\) therefore follow from Proposition A.15, or equivalently from [Lurie 2009, Proposition 5.5.4.15].For (3), it remains to show that any morphism \(f\colon X \to Y\) can be factored into an \(\infty\)-connected morphism followed by an \(\infty\)-truncated one. The same argument as in part (1) of Proposition 3.18 shows that the map \(X \to \tau_{\infty}(X/Y)\) provides such a factorization.

Corollary 3.40.

The left adjoint \(\tau_{\infty}\colon T \to T_{\leq \infty}\) is left exact. In particular, \(T_{\leq \infty}\) is again a topos.

Proof
We first explain why the \(\infty\)-connected maps are closed under finite limits in \(\Ar(T)\). They contain the terminal object. For pullbacks, consider a morphism between cospans whose three component maps are \(\infty\)-connected. The induced map between their pullbacks factors as
\[X\times_ZY \longrightarrow X\times_{Z'}Y \longrightarrow X'\times_{Z'}Y \longrightarrow X'\times_{Z'}Y'.\]
The last two maps are base changes of \(X\to X'\) and \(Y\to Y'\), while the first is a base change of the diagonal of \(Z\to Z'\). Each factor is therefore \(\infty\)-connected by Proposition 3.39, and so is their composite. This is the same three-factor argument used in the proof of Lemma 4.15.For a finite diagram \(\{X_i\}_{i \in I}\) in \(T\), it follows that the map \(\lim_i X_i \to \lim_i \tau_{\infty}X_i\) is \(\infty\)-connected. Since \(\infty\)-truncated objects are local objects for a class of morphisms, they are closed under limits, so the target is \(\infty\)-truncated. The map therefore exhibits the target as the hypercompletion of \(\lim_iX_i\), giving an isomorphism
\[\tau_{\infty}(\lim_i X_i) \iso \lim_i \tau_{\infty}X_i.\]

References

  1. Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal. Left-exact localizations of $\infty$-topoi III: The acyclic product. 2025.
  2. Jacob Lurie. Higher topos theory. Ann. Math. Stud. 170, Princeton, NJ: Princeton University Press. 2009.