equivalences in/of $(\infty,1)$-categories
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
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.
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:
the fiber products $U_{i_0} \times_X \cdots\times_X U_{i_n}$ exist and are preserved by $u$;
$\{u(U_i) \to u(X)\}$ 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 $Shv_\tau(C)=Shv_{\bar\tau}(C)$.
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.
Last revised on July 3, 2019 at 23:18:39. See the history of this page for a list of all contributions to it.