Could not include topos theory - contents
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 $C$ and $D$ be sites.
A functor $f:C\to D$ is a morphism of sites if
$f$ is covering-flat, and
$f$ preserves covering families, i.e. for every covering $\{p_i : U_i \to U\}$ of an object $U \in C$, the family $\{f(p_i) : f(U_i) \to f(U)\}$ is a covering of $f(U) \in D$.
If $C$ has finite limits and all covering families in $D$ are strong epic, then covering-flatness of $f$ is equivalent to $f$ 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 $C$ and $D$, 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 $A$ and $B$ are frames regarded as sites via their canonical coverages, then a morphism of sites $A \to B$ is equivalently a frame homomorphism, a function preserving finite meets and arbitrary joins.
(slice sites)
For $C$ a site and $U \in C$, the comma category $(C / U)$ inherits a topology from $C$, such that the forgetful functor $(C/U) \to C$ constitutes a morphism of sites. This is also called the big site of $U$. There are natural operations for restriction and extension of sheaves from a sub-site $U$ to $X$ and back.
For instance, if $X$ is a topological space and $U \in Op(X)$ is an open subset, then $U$ regarded as a topological space in its own right has corresponding to it the site $Op(U) = Op(X) \downarrow U$.
For $C$ and $D$ 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 $C$ and $D$ are κ-ary regular categories with their $\kappa$-canonical topologies, then a morphism of sites is the same as a $\kappa$-ary regular functor (preserving finite limits and $\kappa$-ary effective-epic families).
For $C$ 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 $f : (\mathcal{C},J) \to (\mathcal{D},K)$ be a morphism of sites, with $\mathcal{C}$ and $\mathcal{D}$ small. Then precomposition with $f$ defines a functor between categories of presheaves $(-)\circ f : PSh(\mathcal{D}) \to PSh(\mathcal{C})$.
There is a geometric morphism between the categories of sheaves
where $f_*$ is the restriction of $(-)\circ f$ to sheaves.
For the classical definition of morphisms of sites, using representably-flat functors, this appears for instance as (Johnstone, lemma C2.2.3, cor. C2.2.4). We give the proof in this special case; for the general case see (Shulman).
By the assumption that $f$ preserves covers we have that the restriction of $(-)\circ f$ to $Sh_K(\mathcal{D})$ indeed factors through $Sh(\mathcal{C}) \hookrightarrow PSh(\mathcal{C})$.
Because for $\{U_i \to U\}$ a cover in $\mathcal{C}$ and $F$ a sheaf on $\mathcal{D}$, we have that (assuming here for simplicity that $\mathcal{C}$ has finite limits)
where we used the Yoneda lemma, the fact that the hom functor $PSh(-,-)$ sends colimits in the first argument to limits, and the assumption that $f$ preserves the pullbacks involved.
Also $(-)\circ f$ preserves all limits, because for presheaves these are computed objectwise. And since the inclusion $Sh_K(\mathcal{D}) \to PSh(\mathcal{D})$ 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 $(-)\circ f$ and to sheaf inclusion. The first is left Kan extension $Lan_f$ along $f$ and the second is sheafification $L_J$ on $(\mathcal{C},J)$, 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 $Lan_f$ does. This is the case if $f$ is a flat functor.
(Because the left Kan extension is given by the colimit $Lan_f X : d \mapsto {\underset{\to}{\lim}}((f^{op}/d) \to {\mathcal{C}}^{op} \stackrel{X}{\to} Set)$ over the comma category $f^{op}/d$ which is a filtered category if $f$ 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 $(\mathcal{C}, J)$ be a small site and let $(\mathcal{D}, K)$ be a small-generated site. Then a geometric morphism
is induced by a morphism of sites $(\mathcal{D}, K) \leftarrow (\mathcal{C}, J) : F$ precisely if the inverse image functor $f^*$ respects the Yoneda embeddings, i.e. there is a functor $F$ making the following diagram commute:
In the special case when $\mathcal{C}$ and $\mathcal{D}$ have finite limits and $\mathcal{D}$ 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 $f$, the factorization $F$ is, if it exists, necessarily a morphism of sites: because since $f^*$ is left adjoint and thus preserves all colimits and every object in $Sh(C)$ is a colimit of representables, $f^*$ 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. 1.
So we show that $F$ is necessarily a morphisms of sites:
since the Yoneda embedding and sheafification as well as inverse images preserve finite limits, so does $f^* j_{\mathcal{C}}$ and hence $F$ preserves finite limits, hence is a flat functor;
$f^* h_{\mathcal{C}}$ preserves coverings (maps them to epimorphisms in $Sh(D, K)$) and since $K$ is assumed to be subcanonical it follows from prop. \ref{CharacterizationOfSubcanonicalSites} that $j_{\mathcal{D}}$ also reflects covers. Therefore $F$ preserves covers.
Let $(\mathcal{C},J)$ be a small site and let $\mathcal{E}$ be any sheaf topos. Then we have an equivalence of categories
between the geometric morphisms from $\mathcal{E}$ to $Sh(\mathcal{C}, J)$ and the morphisms of sites from $(\mathcal{C}, J)$ to the big site $(\mathcal{E}, C)$ for $C$ the canonical coverage on $\mathcal{E}$.
This appears as (Johnstone, cor. C2.3.9).
Since for the canonical coverage the Yoneda embedding is the identity, this follows directly from prop. 2.
Corollary 1 leads to the notion of classifying toposes. See there for more details.
If $C$ and $D$ are κ-ary sites, then a functor $f:C\to D$ is a morphism of sites if and only if it preserves finite local $\kappa$-prelimits.
See (Shulman, Prop. 4.8)
Kashiwara-Schapira, Categories and Sheaves, section 17.2 (in terms of local isomorphisms).
Saunders MacLane Ieke 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