Cohomology and homotopy
In higher category theory
Notions of subcategory
A dense sub-site is a subcategory of a site such that a natural functor between the corresponding categories of sheaves is an equivalence of categories.
For a site with coverage and any subcategory, the induced coverage on has as covering sieves the intersections of the covering sieves of with the morphisms in .
Let be a site (possibly large). A subcategory (not necessarily full) is called a dense sub-site with the induced coverage if
every object has a covering in with all in ;
for every morphism in with there is a covering family such that the composites are in .
The following theorem is known as the comparison lemma.
Let be a (possibly large) site with a locally small category and let be a small dense sub-site. Then the pair of adjoint functors
with given by precomposition with and given by right Kan extension induces an equivalence of categories between the categories of sheaves
This appears as (Johnstone, theorm C2.2.3).