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 $(C,J)$ a site with coverage $J$ and $D \to C$ any subcategory, the induced coverage $J_D$ on $D$ has as covering sieves the intersections of the covering sieves of $C$ with the morphisms in $D$.
Let $(C,J)$ be a site (possibly large). A subcategory $D \to C$ (not necessarily full) is called a dense sub-site with the induced coverage $J_D$ if
If $D$ is a full subcategory then the second condition is automatic.
The following theorem is known as the comparison lemma.
Let $(C,J)$ be a (possibly large) site with $C$ a locally small category and let $f : D \to C$ be a small dense sub-site. Then the pair of adjoint functors
with $f^*$ given by precomposition with $f$ and $f_*$ given by right Kan extension induces an equivalence of categories between the categories of sheaves
This appears as (Johnstone, theorem C2.2.3).
The nLab following Johnstone (2002, p.546) had initially the following form of condition 2 in definition :
2’. For every morphism $f : U \to V$ in $C$ with $V \in D$ there is a cover $S\in J(U)$ in $C$ generated by a family of morphisms $\{f_i : U_i \to U\}$ in $C$ such that the composites $f \circ f_i$ are in $D$.
But this is too weak to prove the comparison lemma as the following example shows:
Let $C$ be any groupoid, with the trivial topology (only maximal sieves cover), and let $D$ be the discrete category on the same objects. Then for any morphism $f:U\to V$, its inverse $f^{-1}:V\to U$ generates the maximal sieve on $U$, and the composite $f f^{-1} = 1_V$ is in $D$, so the conditions 1 and 2’ of the definition are satisfied. But the restriction $Set^{C^{op}} \to Set^{D^{op}}$ is not generally an equivalence.
See the dicussion here.
Let $X$ be a locale with frame $Op(X)$ regarded as a site with the canonical coverage ($\{U_i \to U\}$ covers if the join of the $U_i$ is $U$). Let $bOp(X)$ be a basis for the topology of $X$: a complete join-semilattice such that every object of $Op(X)$ is the join of objects of $bOp(X)$. Then $bOp(X)$ is a dense sub-site.
For $C = TopManifold$ the category of all topological manifolds equipped with the open cover coverage, the category CartSp${}_{top}$ is a dense sub-site: every topological manifold has an open cover by open balls homeomorphic to a Cartesian space.
For $C = PCompTopManifold$ the category of all paracompact topological manifolds equipped with the good open cover coverage, the category CartSp${}_{top}$ is a dense sub-site: every paracompact topological manifold has an good open cover by open balls homeomorphic to a Cartesian space.
Similarly for $C =$ Diff the category of paracompact smooth manifolds equipped with the good open cover coverage, the full subcategory CartSp${}_{smooth}$ is a dense sub-site: every such smooth manifold has a differentiably good open cover (see there): a good cover by open balls each of which are diffeomorphic to a Cartesian space.
Replacing sheaves by (∞,1)-sheaves of spaces produces a strictly stronger notion. See (∞,1)-comparison lemma for a sufficient criterion for a dense inclusion of (∞,1)-sites.
There is also the notion of dense subcategory, which is however only remotly related to the concept of a dense sub-site by both vaguely invoking the topological concept of a dense subspace.
The comparison lemma originates with the exposé III by Verdier in
A more general form is used to give a site characterization for étendue toposes in
A proof of the comparison lemma together with a nice list of examples is in
See also
Olivia Caramello, Denseness conditions, morphisms and equivalences of toposes , arXiv:1906.08737 (2019). (abstract)
Peter Johnstone, Sketches of an Elephant vol 2 , Oxford UP 2002. (Section C2.2, p.546)
Mike Shulman, Exact Completions and Small Sheaves , TAC 27 no.7 (2012) pp.97-173. (abstract; Section 11)
Last revised on May 31, 2022 at 12:02:55. See the history of this page for a list of all contributions to it.