A morphism of sites is, unsurprisingly, the appropriate sort of morphism between sites. It is defined exactly so as to induce a geometric morphism between toposes of sheaves (or, more generally, exact completions).
Let and be sites.
A functor is a morphism of sites if
is covering-flat, and
preserves covering families, i.e. for every covering of an object , the family is a covering of .
If has finite limits and all covering families in are strong epic, then covering-flatness of is equivalent to preserving finite limits, i.e. being a left exact functor, or equivalently to being a representably flat functor. Thus, frequently in the literature one finds a definition of a morphism of sites as being representably flat and preserving covering families.
For general and , however, being representably flat implies being covering-flat, but not conversely. Thus, the above definition of morphism of sites is more general than the common one. There are few practical examples where the distinction matters, but our definition has better formal properties (see below).
If and are frames regarded as sites via their canonical coverages, then a morphism of sites is equivalently a frame homomorphism, a function preserving finite meets and arbitrary joins.
(slice sites)
For a site and , the comma category inherits a topology from , such that the forgetful functor constitutes a morphism of sites. This is also called the big site of . There are natural operations for restriction and extension of sheaves from a sub-site to and back.
For instance, if is a topological space and is an open subset, then regarded as a topological space in its own right has corresponding to it the site .
For and regular categories equipped with their regular coverages, a morphism of sites is the same as a regular functor, i.e. a functor preserving finite limits and covers.
More generally, if and are ∞-ary regular categories with their -canonical topologies, then a morphism of sites is the same as a -ary regular functor (preserving finite limits and -ary effective-epic families).
For any site with finite limits, there is canonically a morphism of sites to its tangent category. See there for details.
We discuss how morphisms of sites induce geometric morphisms of the corresponding sheaf toposes, and conversely. The reader might want to first have a look at the discussion of geometric morphisms – between presheaf toposes.
Let be a morphism of sites, with and small. Then precomposition with defines a functor between categories of presheaves .
There is a geometric morphism between the categories of sheaves
where is the restriction of to sheaves.
For the classical definition of morphisms of sites, using representably-flat functors, this appears for instance as (Johnstone, lemma C2.3.3, cor. C2.3.4). We give the proof in this special case; for the general case see (Shulman).
By the assumption that preserves covers we have that the restriction of to indeed factors through .
Because for a cover in and a sheaf on , we have that (assuming here for simplicity that has finite limits)
where we used the Yoneda lemma, the fact that the hom functor sends colimits in the first argument to limits, and the assumption that preserves the pullbacks involved.
Also preserves all limits, because for presheaves these are computed objectwise. And since the inclusion is right adjoint (to sheafification) we have that
preserves all limits. Therefore by the adjoint functor theorem it has a left adjoint. Explicitly, this is the composite of the left adjoint to and to sheaf inclusion. The first is left Kan extension along and the second is sheafification on , so the left adjoint is the composite
Here the first morphism preserves all limits, the last one all finite limits. Hence the composite preserves all finite limits if the left Kan extension does. This is the case if is a flat functor.
(Because the left Kan extension is given by the colimit over the comma category which is a filtered category if is flat, and filtered colimits are precisely those that commute with finite limits. For more details on this argument see the discussion at Geometric morphisms between presheaf toposes.)
This is a “contravariant” construction in that a morphism of sites going one way gives a geometric morphisms of toposes going the opposite way. Another condition, the covering lifting property gives a covariant assignment.
Conversely, any geometric morphism which restricts and corestricts to a functor between sites of definition is induced by a morphism between those sites:
Let be a small site and let be a small-generated site. Then a geometric morphism
is induced by a morphism of sites precisely if the inverse image functor respects the Yoneda embeddings, i.e. there is a functor making the following diagram commute:
In the special case when and have finite limits and is subcanonical, so that morphisms of sites can be defined using representably flat functors, this appears as (Johnstone, lemma C2.3.8). We give the proof in this case; for the general case see (Shulman, Prop. 11.14). Note that the general case would not be true for the classical definition of “morphism of sites”.
It suffices to show that given , the factorization is, if it exists, necessarily a morphism of sites: because since is left adjoint and thus preserves all colimits and every object in is a colimit of representables, is fixed by the factorization. By uniqueness of adjoint functors this means then that together with its right adjoint it is the geometric morphism induced from the morphism of sites, by prop. .
So we show that is necessarily a morphisms of sites:
since the Yoneda embedding and sheafification as well as inverse images preserve finite limits, so does and hence preserves finite limits, hence is a flat functor;
preserves coverings (maps them to epimorphisms in ) and since is assumed to be subcanonical it follows from this prop. that also reflects covers. Therefore preserves covers.
Let be a small site and let be any sheaf topos. Then we have an equivalence of categories
between the geometric morphisms from to and the morphisms of sites from to the big site for the canonical coverage on .
This appears as (Johnstone, cor. C2.3.9).
Since for the canonical coverage the Yoneda embedding is the identity, this follows directly from prop. .
Corollary leads to the notion of classifying toposes. See there for more details.
If and are ∞-ary sites, then a functor is a morphism of sites if and only if it preserves finite local -prelimits.
See (Shulman, Prop. 4.8)
Kashiwara-Schapira, Categories and Sheaves, section 17.2 (in terms of local isomorphisms).
Saunders MacLaneIeke Moerdijk, Sheaves in Geometry and Logic, section VII. 10 of (in terms of covering sieves).
Peter Johnstone, Sketches of an Elephant, section C2.3.
Michael Shulman, “Exact completions and small sheaves”. Theory and Applications of Categories, Vol. 27, 2012, No. 7, pp 97-173. Free online
Last revised on May 14, 2021 at 19:33:21. See the history of this page for a list of all contributions to it.