1. Metric Sites
 
Definition 1.1. Let C be a category. A metric pretopology  on C consists of the data 
 (a) for every object X  C, a topological space (X), and 
 (b) for every morphism f: Y --> X in C, a continuous function (f): (Y) --> (X), 
subject to the conditions 
 (1) (1X) is the identity of (X). 
 (2) (gf) = (g)(f) whenever gf is defined. 

Alternatively one can define a metric pretopology  on C to be a functor from C to the metacategory of topological spaces (cf. [Mac Lane 1971, p.9]). The set (C) of topological spaces together with all the continuous functions (f) is a category, thus t may be viewed as a functor from C to the category (C).  

Definition 1.2. A metric presite is a pair (C) consisting of a category C and a metric pretopology  on C; C is called the underlying category of (C).  

Suppose (C) is a metric presite. We often simply write C for (C). If X is an object and f: Y  --> X a morphism, we shall adopt the following notation:  
|X| (or simply X) for the underlying space (X) (a subset U of |X| is often called a subset of X); 
|f| (or simply f) for the underlying map t(f); 
|f(X)| for the subset |f|(|X|) of |Y|; 
f(U) for |f|(U) if U  |Y|, and f-1(V) for |f|-1(V) if V  |X|. 

We say f: Y  --> X is surjective, injective, homeomorphism, bicontinuous, open, closed, etc., if the underlying map |f|: |Y|  --> |X| is so. An object X is called an empty object if the underlying space |X| of X is empty.  

A morphism f: Y --> X is called active if any morphism g: Z --> X with |g(Z)|  |f(Y)| factors through f uniquely. An effective morphism is an active morphism f: Y --> X such that the continuous map |f|: |Y| -->|X| is an embedding (i.e., f is bicontinuous).  

A subset U of |X| is called effective if there is an effective morphism f: Y --> X such that |f(Y)| = U. Since Y is uniquely determined by U up to isomorphism, we often write U for Y, and call U an effective subobject of X; f is called the effective morphism of U 

More generally, a subset U of |X| is called active if there is a morphism f: Y --> X such that |f(Y)|  U, and any morphism g: Z --> X with |g(Z)|  U factors through f uniquely. The morphism f is an active morphism, uniquely determined by U (but in general |f(Y)|  U), called the active morphism of U. Any effective subset of |X| is active. 

Remark 1.3. An effective morphism plays the role of a local isomorphism in categorical geometry. It has all the properties one would expect for a local isomorphism:  
 (a) An effective morphism is uniquely determined by its image. 
 (b) An effective morphism is a monomorphism. 
 (c) An isomorphism is effective. 
 (d) A surjective effective morphism is an isomorphism. 
 (e) A composition of two effective morphisms is effective. 
 (f) Suppose f: Y ® X is an effective morphism. Then for any subset U  |Y|, U is 
effective if and only if f(U) is effective, in which case the induced morphism U --> f(U) is an 
isomorphism of effective subobjects (note that (a) - (d) hold for any active morphism). 
  
A metric presite C is effective (resp. everywhere effective) if any open subset (resp. any subset) of any object is effective. Similarly a metric presite C is called active (resp. everywhere active) if any open subset (resp. any subset) of any object is active. A metric presite is locally effective if open effective subsets of any object X form a base for |X|.  

Definition 1.4. A metric site is a locally effective metric presite satisfying the following condition: 
If f: Y --> X is a morphism in C and U an open effective subset of |X|, then f-1(U) is an open effective subset of |Y| (thus the intersection of two open effective subsets of |X| is effective). 
A metric pretopology t on a category C is a metric topology if (C, t) is a metric site. 

Example 1.4.1. (a) The empty metric presite is a metric site. 
(b) Any effective (resp. everywhere effective) metric presite is a metric site, called an effective (resp. everywhere effective) metric site. 
(c) Since the empty subset  of any object X Î C belongs to every base of |X|,  is effective if C is locally effective. Thus any non-empty locally effective metric presite has at least one empty object. This also implies that any initial object of a locally effective metric presite (if exists) must be an empty object.  
(d) Any locally effective metric presite with fibre products is a metric site (this follows from (1.7b), and the fact that, if f: Y  --> X is a morphism and U an open active subset of |X|, then f-1(U) is active). 

Suppose (CC) and (DD) are metric presites. An isometry from C to D is a functor : C --> D such that D is isomorphic to C (i.e., D((X)) is naturally homeomorphic to C(X) for any X Î C), and  
sends an open effective morphism in C to an open effective morphism in D. An isometry  is called an embedding (resp. equivalence) of metric presites if the functor j is an embedding (resp. equivalence) of the underlying categories.  

Example 1.4.2. Suppose (CC) is a metric presite, B a subcategory of C. We define the induced metric pretopology on B to be the restriction C|B of  on B. We say (BC|B) (or simply B) is a subpresite of C if the inclusion functor B ® C is an isometry of metric presites from (BC|B) to (CC).  

Example 1.4.3. Suppose (CC) is a metric site. If a subpresite (BC|B) of C is also a metric site, then we say that B is a subsite of C. 

Example 1.4.4. Suppose (C) is a metric presite. For any object X Î C denote by C/X the slice category of objects f: Y ® X over X. Let X: C/X ® C be the functor sending each f to Y. Then C/X is a metric presite with the metric pretopology C/X(f) = |Y|, and X is an isometry from C/X to C. If C is a metric site, then C/X is also a metric site. 

Definition 1.5.strict metric site is a metric site C in which the following glueing lemma for morphisms holds:  
Suppose X, Y are objects and {Ui} is an open effective cover of |X|. Suppose for each i we have a morphism fi: Ui ® Y such that the restrictions of fi and fj to Ui Ç Uj are the same. Then there exists a unique morphism f from X to Y such that the restriction of f to Ui is fi 

Example 1.5.1. Any full subsite of a strict metric site is a strict metric site.  

Example 1.5.2. The metric presite w(X) of subspaces of a topological space X is an everywhere effective, strict metric site. The metric presite W(X) of open subsets of X is an effective, strict subsite of w(X). All the morphisms in w(X) and W(X) are effective. 

Let (CC) be a metric presite. By a pointed object of C we mean a pair (X, x) consisting of an object X Î C and a point x Î |X|. A morphism of pointed objects from (Y, y) to (X, x) is a morphism f: Y ® X such that f(y) = x 

Definition 1.6. Suppose (CC) is a metric presite and D a full subcategory of C. A subset U of an object X Î C is called D-exact if the following conditions are satisfied: 
(a) For any point x Î U there is a morphism f: (Y, y) ® (X, x) of pointed objects of C such that Y Î D and |f(Y)| Í U. 
(b) For any x Î U the category D/(U, x) of pointed objects f: (Y, y) ® (X, x) over (X, x) such that Y Î D and |f(Y)| Í U is connected. 
(c) A subset V of U is open in the subspace U if for any morphism f: Y ® X such that Y Î D and |f(Y)| Í U, f-1(V) is an open subset of |Y|. 

Suppose C is a metric presite. A subset U of an object X Î C is called exact if it is a C-exact subset of |X|. A metric presite C is called exact (resp. everywhere exact) if any open subset (resp. any subset) of any object X Î C is exact.  

Proposition 1.7. Suppose C is a metric presite. 
(a) A subset U of an object X is effective if and only if U is active and exact (thus any active, exact metric presite is an effective metric site). 
(b) C is exact if the open exact subsets of any object X form a base for |X| (thus any locally effective metric presite is exact). 

Proof. (a) One direction is obvious. Suppose U is an active subset of X and f: Y ® X is the active morphism of U. Suppose U is exact. Then (1.6.a) means that |f(Y)| = U, (1.6b) implies that |f| is injective, and (1.6c) indicates that any subset V of U is open if f-1(V) is open, so |f| is an embedding. Thus f is effective and  |f(Y)| = U is effective.  
(b) Suppose open exact subsets of any object X form a base for |X|. Let U be an open subset of |X|. We verify the conditions of (1.6.) for U: 
First (1.6.a) follows from the fact that U has an open exact cover {Vi} and (1.6a) holds for each open exact subset Vi. Suppose x Î U. We have to prove that the category C/(U, x) is connected. Let V be an open exact neighborhood of x contained in U. Then C/(V, x) is a connected subcategory of C/(U, x) (1.6b). It suffices to prove that any object of C/(U, x) is connected to an object of C/(V, x). Suppose f: (Y, y) ® (X, x) is an object of C/(U, x). Then f-1(V) is an open subset of Y containing y. Applying (1.6.a) to an open exact subset of f-1(V) containing y we can find a morphism g: (Z, z) ® (Y, y) such that g(Z) Í f-1(V). Since fg: (Z, z) ® (X, x) is an object of C/(V, x), we see that the object f Î C/(U, x) is connected to an object fg Î C/(V, x) by g. Suppose V is a subset of U such that (1.6.c) holds for V. Take an open exact cover {Vi} of U. Then (1.6.c) holds for the subset V Ç Vi of Vi for each i. Thus each V Ç Vi is open in Vi. Hence V is open in U. n 

Definition 1.8. Suppose C is a category and D a full subcategory of C. Suppose D is a pretopology on D. A pretopology C on C is called an extension of D on C if D = C|D and |X| is D-exact for any X  C 

Remark 1.9. It is easy to see that C is unique up to a natural isomorphism, if it exists. We now show that such an extension C always exists: for any X Î C consider the triples (Y, y, f), where Y Î D, y Î |Y|, and f: Y ® X is a morphism. Write (Y, y, f) ~ (Z, z, g) if there is another triple (W, w, h) with morphisms p: W ® Y, q: W ® Z, such that fp = h = gq, p(w) = y and q(w) = z. Denote by C(X) the set of equivalence classes of these 
triples under the equivalence relation generated by ~. Any morphism u: X ® S in C induces a map C(u): C(X) ® C(S) sending each (Y, y, f) ΠC(Z) to (Y, y, uf) ΠC(S). If X Î D, then C(X) may be identified with the set D(X). A subset U of C(X) is open if for any Y Î D and f: Y ® X, f-1(U) is an open subset of |Y|. These open subsets turns C(X) into a topological spaces. If X Î D, then C(X) is naturally homeomorphic to the space D(X). Let C be the pretopology on C defined by X ® C(X) for any X Î C. Then we have D = 
C|D. It is easy to verify that C(X) is D-exact for any X Î C. Thus C is an extension of D on C 

Theorem 1.10. Suppose (CC) is a metric presite and D a full subcategory of C. Suppose tC is an extension of DC|D. Then (CC) is exact if and only if (DD) is exact. 

Proof. First suppose (DD) is exact. We prove that the extension (CC) is exact. Let U be an open subset of  an object X of C. We show that U is C-exact by verifying the conditions of (1.6). 
(a) Since |X| is D-exact, for any point x Î U there is a morphism f: (Y, y) ® (X, x) such that Y Î D (1.6a). Since D is exact, we can find a D-exact open neighborhood V of y contained in f-1(U). Let g: (Z, z) ® (Y, y) be a morphism with Z Î D and |g(Z)| Í V (1.6.a). Then fg: (Z, z) ® (X, x) is a morphism such that |fg(Z)| Í U. This proves (1.6.a) for U. 
(b) Suppose x Î U. Consider two pointed objects f: (Y, y) ® (X, x) and g: (Z, z) ® (X, x) over (X, x) in C/(U, x). We have to prove that f and g are connected in C/(U, x). We have seen in (a) that any pointed object of C/(U, x) is connected to a pointed object of D/(U, x). Thus we may assume Y, X  D. Then f and g are connected in D/(|X|, x) because |X| is D-exact. Using the fact that D is exact we can easily verify that f and g are connected in D/(U, x).  
(c) Suppose V is a subset of U such that for any morphism f: Y ® X with |f(Y)| Í U, |f -1(V)| is an open subset of |Y|. Suppose g: Z ® X is any morphism with Z Î D. For any h: W ® Z with W Î D and |h(W)| Í g-1(U), the subset (gh)-1(V) = h-1(g-1(V)) is open because |gh(W)| Í U. Since g-1(U) is D-exact, this implies that g-1(V) is an open subset of g-1(U), thus an open subset of |Z|. Since |X| is D-exact, this means that V is open in |X|, thus also open in U. Conversely, suppose (C, tC) is exact. Suppose U is an open subset of  an object X 
Î D. We show that U is D-exact by verifying the conditions of (1.6).  
(a') Since U is C-exact, for any point x Î U there is a morphism f: (Y, y) ® (X, x) with f(Y) Í U by (1.6.a). Since |Y| is D-exact, we can find a morphism g: (Z, z® (Y, y) with Z Î D (1.6.a). Then fg: (Z, z® (X, x) is a morphism of pointed objects of D such that |fg(Z)| Í U. This proves (1.6.a) for U. 
(b') We prove that for any x Î U, the category D/(U, x) is connected. Consider two pointed objects f: (Y, y® (X, x) and g: (Z, z® (X, x) over (X, x) in D/(U, x). Since U is C-exact, f and g are connected in C/(U, x). Using the fact that C is an extension of D we can show that f and g are also connected in D/(U, x).  
(c') Suppose V is a subset of U such that for any morphism f: Y  ® X with |f(Y)| Í U and Y Î D, f-1(V)| is an open subset of |Y|. Suppose g: Z  ® X is any morphism with g(Z) Í U. For any morphism h: Y ® Z with Y Î D, we have |gh(Y)| Í U. Thus (gh)-1(V) = h-1(g-1(V)) is open. Since |Z| is D-exact, this implies that g-1(V) is open. Since U is C-exact, this means that V is open in U. n 

Corollary 1.11. Suppose (CC) is a metric presite and D a full subcategory of C. Suppose C is an extension of DC|D. Suppose (DD) is exact and (CC) is active. Then (CC) is an effective metric site, and (D) is a subsite of (CC) if D is dense in C. 

Proof. The first assertion follows from (1.10) and (1.7.a). Now suppose D is dense in C. Suppose f: Y ® X is an open effective morphism in D; we prove that f is effective in C. Since f is bicontinuous, it suffices to prove that it is active in C. Suppose g: Z  ® X is a morphism in C such that |g(Z)| Í |f(Y)|. Then for any morphism h: W ® Z with W D, gh: W ® X has the image |gh(W)| Í |f(Y)|. Since f is effective in D, gh factors through f 
uniquely. Since D is dense in C, this implies that h factors through f uniquely, thus f is active in C. This shows that D is a subsite of C. n 

Theorem 1.12. Suppose (CC) is a metric presite and D a full dense subcategory of C. Suppose the glueing lemma (1.5) holds for the morphisms from X to Y with X Î D. Then (CC) is a strict metric site. 

Proof. We prove that C is strict by verifying the condition of (1.5) for any objects X, Y Î C and an open effective cover {Ui} of  |X|. Suppose f, g Î homC(X, Y) and f ¹ g. Since D is dense in C, We can find Z Î D and h: Z  ® X such that fh ¹ gh. Since {h-1(Ui)} forms an open cover of Z Î D, by the glueing lemma we can find an open effective subset V of |Z| contained in some h-1(Ui) such that the restrictions of fh and gh to V are different (because the collection of all such V forms an open effective cover of |Z|). It follows that 
the restrictions of f and g to Ui are different. Now suppose for any i we have a morphism fi: Ui  ® Y such that the restrictions of fi and fj to Ui Ç Uj are the same. For any object Z Î D and any morphism h: Z  ® X, let 
{Vj}be an effective open cover of |Z| such that h(Vj) Í Ui for some Ui. Denote by hi the restriction of h to Vi. Then fihi and fjhj agree on Vi Ç Vj. By the glueing lemma for morphisms from Z to Y these fihi determine a morphism h': Z ® Y. We obtain a map homC(Z, X) ® homC(Z, Y) given by h ® h' for each Z Î D. Since D is dense in C, the collection of all these maps determines a morphism f Î homC(X, Y) whose restriction to 
each Ui is fi. The uniqueness of f is obvious. This proves that (CC) is strict. n 

 
 
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