nLab (∞,1)-comparison lemma

Idea

The (∞,1)-comparison lemma says that, under certain conditions, a functor between (∞,1)-sites induces an equivalence between the categories of (∞,1)-sheaves on the sites.

In this paper, Lemma C.3, Hoyois proves the following comparison lemma.

Lemma

Let $D$ be a locally small (∞,1)-category, $C$ a small (∞,1)-category, and $u : C \to D$ a fully faithful functor. Let $\tau$ and $\rho$ be quasi-topologies on $C$ and $D$, respectively. Suppose that:

a. Every $\tau$-sieve is generated by a cover $\{U_i \to X\}$ such that:

1. the fiber products $U_{i_0} \times_X \cdots\times_X U_{i_n}$ exist and are preserved by $u$;

2. $\{u(U_i) \to u(X)\}$ is a $\bar\rho$-cover.

b. For every $X \in C$ and every $\rho$-sieve $R \hookrightarrow u(X)$, $u^*(R) \hookrightarrow X$ is a $\bar\tau$-sieve in $C$.

c. Every $X \in D$ admits a $\bar\rho$-cover $\{U_i \to X\}$ such that the fiber products $U_{i_0} \times_X \cdots \times_X U_{i_n}$ exist and belong to the essential image of $u$.

Then the adjunction $u^* \dashv u_*$ restricts to an equivalence of ∞-categories $Shv_\rho(D) \simeq Shv_\tau(C)$.

Here, a quasi-topology is a collection of sieves closed under pullback and $\bar\tau$ is the coarsest topology containing a quasi-topology $\tau$. The stability under pullback ensures that $Shv_\tau(C)=Shv_{\bar\tau}(C)$.

Generalization

It seems difficult to find a useful generalization not assuming the existence of some pullbacks. For the conclusion of the lemma, the following conditions (b is unchanged) are both necessary and sufficient:

a. For every $\tau$-sieve $U \hookrightarrow X$, $a_\rho u_!(U \to X)$ is an equivalence.

b. For every $X \in C$ and every $\rho$-sieve $R \hookrightarrow u(X)$, $u^*(R) \hookrightarrow X$ is a $\bar\tau$-sieve in $C$.

c. For every $X \in D$, its image in $Shv_\rho(D)$ belongs to the smallest subcategory generated by the image of $C$ under colimits.

We can take these conditions a and b to define, respectively, the notions of cover-preserving functor (continuous functor) and comorphism of sites (cocontinuous functor) for (∞,1)-sites.

Related concepts

• comparison lemma

• dense subsite

References

• Marc Hoyois, A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula. arXiv:1309.6147