“The six-functor formalism for rigid analytic motives” – Erratum

Lemma 2.1.4 in our article [2] is incorrect as we will demonstrate in the example below. (The argument given in [2] relied on the property that $F^*:\mathrm {Shv}_{\tau }(\mathcal {C}) \to \mathrm {Shv}_{\tau '}(\mathcal {C}')$ preserves $0$ -truncated objects, which is wrong in general.) The purpose of this erratum is to explain what additional assumptions have to be added in order to obtain the same conclusion. These additional assumptions, namely assumptions (3) and (4) below, are satisfied in each instance of the article where Lemma 2.1.4 was invoked. Hence the rest of the article remains unaffected.

We now state and prove a corrected version of Lemma 2.1.4. Similar results can be found in [4, Lemma C.3] and [6, Proposition 2.22].

Lemma 2.1.4. Consider two sites $(\mathcal {C},\tau )$ and $(\mathcal {C}',\tau ')$ , where $\mathcal {C}$ and $\mathcal {C}'$ are ordinary categories, and let $F:\mathcal {C} \to \mathcal {C}'$ be a functor. Assume the following conditions.

1. (1) The topologies $\tau $ and $\tau '$ are induced by pretopologies $\mathrm {Cov}_{\tau }$ and $\mathrm {Cov}_{\tau '}$ in the sense of [1, Exposé II, Définition 1.3].

2. (2) For $X\in \mathcal {C}$ , F takes a family in $\mathrm {Cov}_{\tau }(X)$ to a family in $\mathrm {Cov}_{\tau '}(F(X))$ . Moreover, if $a:U \to X$ is an arrow which is a member of a family belonging to $\mathrm {Cov}_{\tau }(X)$ and $b:V \to X$ a second arrow in $\mathcal {C}$ , we have $F(U\times _X V)\simeq F(U)\times _{F(X)}F(V)$ .

Then, the inverse image functors on presheaves induce by sheafification the following functors:

(⋆) $$ \begin{align} F^*:\mathrm{Shv}_{\tau}^{(\wedge)}(\mathcal{C})\to \mathrm{Shv}_{\tau'}^{(\wedge)}(\mathcal{C}') \qquad \text{and} \qquad F^*:\mathrm{Shv}_{\tau}^{(\wedge)}(\mathcal{C};\Lambda)\to \mathrm{Shv}_{\tau'}^{(\wedge)}(\mathcal{C}';\Lambda). \end{align} $$

Assume now, in addition, the following conditions.

1. (3) For $X\in \mathcal {C}$ , any family in $\mathrm {Cov}_{\tau '}(F(X))$ can be refined by the image by F of a family in $\mathrm {Cov}_{\tau }(X)$ .

2. (4) Every object $Y\in \mathcal {C}'$ admits a $\tau '$ -hypercover by objects lying in the essential image of F and, in the nonhypercomplete case, this hypercover can be chosen to be truncated.

3. (5) The functor F induces a fully faithful embedding $F^*:\mathrm {Shv}_{\tau }(\mathcal {C})_{\leq 0} \to \mathrm {Shv}_{\tau '}(\mathcal {C}')_{\leq 0}$ between the associated ordinary topoi.

Then the functors (⋆) are equivalences of $\infty $ -categories.

Proof. The case of (hyper)sheaves of $\Lambda $ -modules follows from the case of (hyper)sheaves of spaces using [2, Remark 2.3.3(2)]. Consider the functors

where $F^*$ is given by left Kan extension along F, $F^!$ by right Kan extension along F and $F_*$ by composition with F. Recall that $F_*$ is right adjoint to $F^*$ and $F^!$ is right adjoint to $F_*$ . We will prove the following assertions.

1. (A) Under the assumptions (1) and (2), $F^*$ takes $\tau $ -local equivalences to $\tau '$ -local equivalences (in both the hypercomplete and nonhypercomplete cases). Equivalently, $F_*$ takes $\tau '$ -(hyper)sheaves to $\tau $ -(hyper)sheaves.

2. (B) Under the assumptions (1) and (3), $F_*$ takes $\tau '$ -local equivalences to $\tau $ -local equivalences (in both the hypercomplete and nonhypercomplete cases). Equivalently, $F^!$ takes $\tau $ -(hyper)sheaves to $\tau '$ -(hyper)sheaves.

Assertion (A) is clear. Indeed, the assumptions (1) and (2), imply that $F^*$ takes $\tau $ -(hyper)covers to $\tau '$ -(hyper)covers. The argument for (B) is standard (see [6, Lemmas 2.13, 2.14, 2.18, 2.19]), but we give a proof for convenience. We first treat the nonhypercomplete case. In this case, the class of $\tau '$ -local equivalences is the smallest strongly saturated class (in the sense of [5, Definition 5.5.4.5]) containing the inclusions of $\tau '$ -covering sieves $R\hookrightarrow \mathrm {y}(Y)$ , for $Y\in \mathcal {C}'$ . Since $F_*$ is colimit-preserving, we only need to show that $F_*(R)\to F_*(\mathrm {y}(Y))$ is a $\tau $ -local equivalence. By the universality of colimits [5, Proposition 6.1.3.10], it is enough to show that for every $X\in \mathcal {C}$ and every morphism $u:\mathrm {y}(X) \to F_*(\mathrm {y}(Y))$ in $\mathcal {P}(\mathcal {C})$ , the monomorphism

$$ \begin{align*}P=\mathrm{y}(X)\times_{u,\,F_*(\mathrm{y}(Y))}F_*(R) \to \mathrm{y}(X)\end{align*} $$

is a $\tau $ -covering sieve. The morphism u corresponds by adjunction to a morphism $v:F(X) \to Y$ in $\mathcal {C}'$ , and it is easy to see that $P\hookrightarrow \mathrm {y}(X)$ is the inclusion of the sieve of X consisting of those morphisms $X'\to X$ in $\mathcal {C}$ such that $F(X')\to F(X)$ belongs to $\mathrm {y}(F(X))\times _{v,\,Y}R$ , which is a $\tau '$ -covering sieve of $F(X)$ . Assumption (3) implies that P is a $\tau $ -covering sieve of X as needed. Next, we explain how to deduce (B) in the hypercomplete case from the nonhypercomplete case. Let f be a $\tau '$ -local equivalence in $\mathcal {P}(\mathcal {C}')$ . Then, for every $n\in \mathbb {N}$ , $\tau _{\leq n}(f)$ is a truncated $\tau '$ -local equivalence, and $F_*\tau _{\leq n}(f)=\tau _{\leq n}F_*(f)$ is a truncated $\tau $ -local equivalence by (B) in the nonhypercomplete case. Since, in the hypercomplete case, $\tau $ -local equivalences are detected on the truncations, the result follows.

It is now easy to conclude. Assertion (A) implies the existence of the first functor in (⋆). To prove that this functor is fully faithful, we verify that the unit morphism $\mathrm {id} \to F_*F^*$ is an equivalence. (We stress that $F^*$ denotes the first functor in (⋆) and $F_*$ is the direct image functor on (hyper)sheaves.) Assertion (B) implies that $F_*$ has a right adjoint, which is the restriction of $F^!$ to (hyper)sheaves. In particular, $F_*$ is colimit-preserving, and it is enough to check that the unit morphism $\mathrm {id} \to F_*F^*$ is an equivalence on objects of the form $\mathrm {L}_{\tau }\mathrm {y}(X)$ , for $X\in \mathcal {C}$ . Now $\mathrm {L}_{\tau }\mathrm {y}(X)$ is $0$ -truncated and the same is true for $F^*\mathrm {L}_{\tau }\mathrm {y}(X)\simeq \mathrm {L}_{\tau }\mathrm {y}(F(X))$ . Since $F_*$ preserves $0$ -truncated objects (being exact), it follows that the unit morphism $\mathrm {L}_{\tau }\mathrm {y}(X)\to F_*F^*\mathrm {L}_{\tau }\mathrm {y}(X)$ identifies with the unit of the adjunction $(F^*,F_*)$ on the ordinary topoi associated to $(\mathcal {C},\tau )$ and $(\mathcal {C}',\tau ')$ . Thus we can now conclude using the assumption (5).

Finally, to finish the proof, it remains to see that $\mathrm {Shv}_{\tau '}(\mathcal {C}')$ can be generated under colimits by the image of the functor $F^*$ in (⋆). This follows immediately from the assumption (4).

Example 2.1.4 bis. ¹ We now remark that without the additional assumption (4), which was missing in the original Lemma 2.1.4, the last statement is incorrect. For this, consider the poset J built as in [3, Example A.9]. (It is called $\mathcal {C}$ in loc. cit.) The objects are pairs $(i,j)$ of nonnegative integers with $|i-j|\leq 1$ and an arrow $(i,j)\to (i',j')$ exists if $i\geq i'$ and $j\geq j'$ . One can define a topology on J for which a family $\{(i_\alpha ,j_\alpha )\to (i,j)\}_\alpha $ is covering if $\min (i_\alpha )=i$ and $\min (j_\alpha )=j$ . We also consider the subposet $I\subset J$ spanned by the pairs $(i,j)$ with $|i-j|=1$ . Observe that the topology induced on this subposet is the trivial one. In particular, it is clearly induced by a pretopology. The same is true in the case of J since the latter admits finite limits. This shows that the inclusion $F\colon I\hookrightarrow J$ satisfies assumptions (1)–(3) of the lemma above. By the comparison lemma [1, Exposé III, Théorème 4.1], the functor F induces an equivalence on the associated ordinary topoi, thus assumption (5) is also satisfied. On the other hand, $\mathrm {Shv}(I)=\mathrm {PSh}(I)$ is hypercomplete while $\mathrm {Shv}(J)$ is not by [3, Example A.9]. Hence these two $\infty $ -topoi cannot be equivalent.