Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
(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 be a locally small (∞,1)-category, a small (∞,1)-category, and a fully faithful functor. Let and be quasi-topologies on and , respectively. Suppose that:
a. Every -sieve is generated by a cover such that:
the fiber products exist and are preserved by ;
is a -cover.
b. For every and every -sieve , is a -sieve in .
c. Every admits a -cover such that the fiber products exist and belong to the essential image of .
Then the adjunction restricts to an equivalence of ∞-categories .
Here, a quasi-topology is a collection of sieves closed under pullback and is the coarsest topology containing a quasi-topology . The stability under pullback ensures that .
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 -sieve , is an equivalence.
b. For every and every -sieve , is a -sieve in .
c. For every , its image in belongs to the smallest subcategory generated by the image of 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 4, 2019 at 03:18:39. See the history of this page for a list of all contributions to it.