nLab dense sub-site

Idea

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.

Definition

Definition

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 $D$.

Definition

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

1. every object $U \in C$ has a covering sieve generated by maps $U_i \to U$ with $U_i \in D$.

2. for every morphism $f : U \to V$ in $C$ with $U, V \in D$ there is a covering sieve $\{f_i : U_i \to U\}$ of $U$ in $D$ such that the composites $f \circ f_i$ are in $D$.

Remark

If $D$ is a full subcategory then the second condition is automatic.

The following theorem is known as the comparison lemma.

Theorem

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

$$(f^* \dashv f_*) : PSh(D) \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} PSh(C)$$

with $f^*$ given by precomposition with $f$ and $f_*$ given by right Kan extension induces an equivalence of categories between the categories of sheaves

$$(f_* \dashv f^*) : Sh_{J_D}(D) \underoverset {\underset{f_*}{\to}}{\overset{f^*} {\leftarrow}} {\simeq} Sh_J{C} \,.$$

This appears as (Johnstone, theorem C2.2.3).

Problems with another definition

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.

Examples

• 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 $X$ a locally contractible space, $Op(X)$ its category of open subsets and $cOp(X)$ the full subcategory of contractible open subsets, we have that $cOp(X)$ 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.

Warning

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.

Remark

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.

Related entries

• dense

• comparison lemma

• (∞,1)-comparison lemma

• covering lifting property

References

The comparison lemma originates with the exposé III by Verdier in

• M. Artin, A. Grothendieck, J. L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA4), LNM 269 Springer Heidelberg 1972. (p.288)

A more general form is used to give a site characterization for étendue toposes in

• A. Kock, I. Moerdijk, Presentations of Etendues , Cah. Top. Géom. Diff. Cat. XXXII 2 (1991) pp.145-164. (numdam, pp.151f)

A proof of the comparison lemma together with a nice list of examples is in

• S. Mac Lane, I. Moerdijk, Sheaves in Geometry and Logic , Springer Heidelberg 1994. (Appendix 4, pp.590ff)

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)