{"id":4466,"date":"2021-12-20T19:04:55","date_gmt":"2021-12-20T18:04:55","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=4466"},"modified":"2022-05-17T10:36:22","modified_gmt":"2022-05-17T08:36:22","slug":"sheaves-and-streams-ii-sheafification-and-stratified-etale-maps","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=4466","title":{"rendered":"Sheaves and streams II: sheafification, and stratified \u00e9tale maps"},"content":{"rendered":"\n

Season’s greetings, first!<\/p>\n\n\n\n

Last time<\/a>, I explained how one can build the \u00e9tale space<\/em> of a presheaf F<\/em> over a topological space X<\/em>. Let me remind you how this is built. I will then show how one can retrieve a sheaf from an \u00e9tale map, leading to a nice adjunction and its associated monad, sheafification. This is all well-known (see Chapter 10 in [1], for example), but then I would like to apply all that to the presheaf of locally monotone functions of a prestream, which we had already started to examine last time.<\/p>\n\n\n\n

From presheaves to \u00e9tale spaces and maps<\/h2>\n\n\n\n

Let F<\/em> be a presheaf over a space X<\/em>. For each open set U<\/em> of X<\/em>, the presheaf gives us a set FU<\/sub><\/em>, and restriction maps rUV<\/sub><\/em> from FV<\/sub><\/em> to FU<\/sub><\/em>, for every pair of open sets U<\/em> \u2286 V<\/em>. We build the stalk Fx<\/sub><\/em> of F<\/em> at each point, as the quotient F<\/em>[x<\/em>]\/\u2261x<\/sub><\/em>, where F<\/em>[x<\/em>] is the disjoint union of the sets FU<\/sub><\/em>, where U<\/em> ranges over the open neighborhoods of x<\/em>, and (U<\/em>, f<\/em>) \u2261x<\/sub><\/em> (V<\/em>, g<\/em>), if and only if there is an open neighborhood W<\/em> of x<\/em> included in both U<\/em> and V<\/em> such that rWU<\/sub><\/em>(f<\/em>)=rWV<\/sub><\/em>(g<\/em>). I will often omit the U<\/em> and V<\/em> parts, and I will just write f<\/em> \u2261x<\/sub><\/em> g<\/em> instead of (U<\/em>, f<\/em>) \u2261x<\/sub><\/em> (V<\/em>, g<\/em>).<\/p>\n\n\n\n

The \u00e9tale space S<\/strong><\/em>(X<\/em>, F<\/em>) is the disjoint union of all the stalks Fx<\/sub><\/em>; its elements are the pairs (x<\/em>, [f<\/em>]), where x<\/em> is a point in X<\/em> and [f<\/em>] is the equivalent class modulo \u2261x<\/sub><\/em> of an element f<\/em> of a set FU<\/sub><\/em>, where U<\/em> is an open neighborhood of x<\/em>. There is a projection map p<\/em> : S<\/strong><\/em>(X<\/em>, F<\/em>) \u2192 X<\/em>, which maps (x<\/em>, [f<\/em>]) to x<\/em>, and p<\/em> is a surjective local homeomorphism.<\/p>\n\n\n\n

For this to work, we had to define the topology on S<\/strong><\/em>(X<\/em>, F<\/em>) as the finest that makes all the overlined maps f<\/em><\/span><\/em> continuous, where f<\/em><\/span><\/em> : U<\/em> \u2192 S<\/strong><\/em>(X<\/em>, F<\/em>) is defined by f<\/em><\/span><\/em>(x<\/em>) \u225d (x<\/em>, [f<\/em>]) for every x<\/em> in U<\/em>, this for every open subset U<\/em> of X<\/em> and for every f<\/em> \u2208 FU<\/sub><\/em>. Finally, we had observed that the sets f<\/span><\/em>(U<\/em>), where U<\/em> ranges over the open subsets of X<\/em>, and f<\/em> ranges over FU<\/sub><\/em>, form a base of the topology of S<\/strong><\/em>(X<\/em>, F<\/em>).<\/p>\n\n\n\n

A surjective local homeomorphism p<\/em> from a space Y<\/em> to X<\/em> is called an \u00e9tale map<\/em> on X<\/em>. As is usual in category theory, the data of a map (or a morphism) includes the specification of the domain and of the codomain. In particular, an \u00e9tale map on X<\/em> is given by a pair (Y<\/em>, p<\/em>) of a space Y<\/em> and a surjective local homeomorphism p<\/em> onto X<\/em>. We do not need to specify X<\/em> in the pair, since I said “an \u00e9tale map on X<\/em>“, but I may later say “an \u00e9tale map” to mention a triple (Y<\/em>, p<\/em>, X<\/em>).<\/p>\n\n\n\n

From \u00e9tale maps to sheaves<\/h2>\n\n\n\n

Let us try to retrieve a presheaf from an \u00e9tale map (Y<\/em>, p<\/em>) on X<\/em>. We form the sheaf LSect<\/strong>(Y<\/em>, p<\/em>) of local sections<\/em> of p<\/em>. A local section is a continuous map s<\/em> from an open subset U<\/em> of X<\/em> to Y<\/em> such that p<\/em> o s<\/em> is the identity on U<\/em>. It is easy to see that LSect<\/strong>(Y<\/em>, p<\/em>) is always a sheaf on X<\/em>\u2014not just a presheaf!\u2014, with the obvious notion of restriction: rVU<\/sub><\/em>(s<\/em>) is really the restriction s<\/em>|V<\/em><\/sub> of s<\/em> : U<\/em> \u2192 Y<\/em> to V<\/em> \u2286 U<\/em>.<\/p>\n\n\n\n

Let us say all that in more categorical terms. For a fixed topological space X<\/em>, there is a category PreSh<\/strong>(X<\/em>) of presheaves on X<\/em>. (Reminder: X<\/em> is fixed!) Since presheaves on X<\/em> are just functors from O<\/strong>X<\/em>op<\/sup> to Set<\/strong>, it should come as no surprise that the morphisms on PreSh<\/strong>(X<\/em>) are the natural transformations. Explicitly, a morphism \u03c6 of presheaves on X<\/em> from F<\/em> to G<\/em> is a collection of functions \u03c6U<\/sub><\/em> : FU<\/sub><\/em> \u2192 GU<\/sub><\/em>, one for each open subset U<\/em> of X<\/em>, which commutes with restrictions.<\/p>\n\n\n\n

There is also a full subcategory Sh<\/strong>(X<\/em>) of sheaves on X<\/em>. Its objects are the sheaves on X<\/em>, and its morphisms are the same, namely, natural transformations.<\/p>\n\n\n\n

Finally, there is a category Etale<\/strong>(X<\/em>) of \u00e9tale maps on X<\/em>. Its objects are the surjective local homeomorphisms (Y<\/em>, p<\/em>) on X<\/em>, and the morphisms from (Y<\/em>, p<\/em>) to (Z<\/em>, q<\/em>) are the continuous maps f<\/em> : Y<\/em> \u2192 Z<\/em> such that q<\/em> o f<\/em> = p<\/em>. You may have recognized a full subcategory of the slice category Top<\/strong>\/X<\/em>, whose objects are all the pairs (Y<\/em>, p<\/em>) where p<\/em> is a continuous map from Y<\/em> to X<\/em>, not just the surjective local homeomorphisms.<\/p>\n\n\n\n

There is a functor E<\/strong> : PreSh<\/strong>(X<\/em>) \u2192 Etale<\/strong>(X<\/em>). On objects, it maps every presheaf F<\/em> on X<\/em> to the projection map p<\/em> : S<\/em><\/strong>(X<\/em>, F<\/em>) \u2192 X<\/em>, the \u00e9tale map of F<\/em>. Its action on morphisms \u03c6 : F<\/em> \u2192 G<\/em> (namely, natural transformations) is defined by E<\/strong>(\u03c6) (x<\/em>, [f<\/em>]) \u225d (x<\/em>, [\u03c6(f<\/em>)]). I will let you check by yourselves that E<\/strong>(\u03c6) is indeed a continuous map from S<\/em><\/strong>(X<\/em>, F<\/em>) to S<\/em><\/strong>(X<\/em>, G<\/em>). This is just a play on symbols; just make sure that you do not get lost in notation.<\/p>\n\n\n\n

We have defined the sheaf of local sections LSect<\/strong>(Y<\/em>, p<\/em>) of an \u00e9tale map on X<\/em>, and this construction, too, extends to a functor from Etale<\/strong>(X<\/em>) to PreSh<\/strong>(X<\/em>), and in fact to the full subcategory Sh<\/strong>(X<\/em>). Given any morphism of \u00e9tale maps f<\/em> : LSect<\/strong>(Y<\/em>, p<\/em>) \u2192 LSect<\/strong>(Z<\/em>, q<\/em>), we define LSect<\/strong>(f<\/em>) \u225d \u03c6 as the natural transformation such that, for every subset U<\/em> of X<\/em>, for every local section s<\/em> : U<\/em> \u2192 Y<\/em> of p<\/em>, \u03c6U<\/sub><\/em>(s<\/em>) is the map f<\/em> o s<\/em>, which is a local section (from U<\/em> to Z<\/em>) of q<\/em>.<\/p>\n\n\n\n

Those two functors are related, as you may have guessed. For every presheaf F<\/em> on X<\/em>, there is a morphism \u03b7F<\/sub><\/em> : F<\/em> \u2192 LSect<\/strong>(E<\/strong>(F<\/em>)) of presheaves on X<\/em>, which maps every element f<\/em> of FU<\/sub><\/em> to our old friend f<\/span><\/em><\/em> : U<\/em> \u2192 S<\/strong><\/em>(X<\/em>, F<\/em>). Indeed, that is a continuous map, by definition of the topology on S<\/em><\/strong><\/strong>(X<\/em>, F<\/em>), and a section of the corresponding \u00e9tale map p<\/em>, since f<\/span><\/em><\/em>(x<\/em>)=(x<\/em>, [f<\/em>]) for every x<\/em> in U<\/em>, by definition. I will let you check that \u03b7F<\/sub><\/em> is natural in F<\/em>, too, although that is really a consequence of the following proposition.<\/p>\n\n\n\n

Proposition (E \u22a3<\/strong> LSect).<\/strong> LSect<\/strong> is right adjoint to E<\/strong>, with unit the natural transformation \u03b7 that we have just defined.<\/p>\n\n\n\n

Proof sketch.<\/em> In order to see this, it is enough to show that, given any morphism \u03c6 : F<\/em> \u2192 LSect<\/strong>(Y<\/em>, p<\/em>) of presheaves on X<\/em>, there is a unique morphism \u03c6* : E<\/strong>(F<\/em>) \u2192 (Y<\/em>, p<\/em>) of \u00e9tale maps such that LSect<\/strong>(\u03c6*) o \u03b7F<\/sub><\/em> = \u03c6.<\/p>\n\n\n\n

If \u03c6* exists, then it must map every element (x<\/em>, [f<\/em>]) of S<\/em><\/strong>(X<\/em>, F<\/em>) to an element y<\/em> of Y<\/em> such that p<\/em>(y<\/em>)=x<\/em>, first (being a morphism of \u00e9tale maps). The condition LSect<\/strong>(\u03c6*) o \u03b7F<\/sub><\/em> = \u03c6 means that, for every open subset U<\/em> of X<\/em>, for every f<\/em> \u2208 FU<\/sub><\/em>, \u03c6* o f<\/span><\/em> = \u03c6U<\/sub><\/em>(f<\/em>). In other words, for every x<\/em> \u2208 U<\/em>, we must have \u03c6*(x<\/em>, [f<\/em>]) = \u03c6U<\/sub><\/em>(f<\/em>). In particular, \u03c6* is unique, if it exists.<\/p>\n\n\n\n

Hence we define \u03c6* as mapping (x<\/em>, [f<\/em>]), for every open subset U<\/em> of X<\/em> and every f<\/em> \u2208 FU<\/sub><\/em>, to \u03c6U<\/sub><\/em>(f<\/em>)(x<\/em>). I will let you check the rest for yourselves: that this is well-defined, that this is continuous, and that p<\/em>(\u03c6*(x<\/em>, [f<\/em>])) = x<\/em>. This is not difficult, just try not to get lost in notation\u2014again. \u2610<\/p>\n\n\n\n

The counit of the adjunction E<\/strong> \u22a3 LSect<\/strong><\/h2>\n\n\n\n

The counit \u03b5 of the adjunction E<\/strong> \u22a3 LSect<\/strong> is defined at each object (Y<\/em>, p<\/em>) of Etale<\/strong>(X<\/em>) by \u03b5(Y<\/em>, p<\/em>)<\/sub> \u225d idLSect<\/strong>(Y<\/em>, p<\/em>)<\/sub>*, as is usual for every adjunction. Explicitly, \u03b5(Y<\/em>, p<\/em>)<\/sub> maps every point (x<\/em>, [s<\/em>]) in S<\/em><\/strong>(LSect<\/strong>(Y<\/em>, p<\/em>)) (where s<\/em> : U<\/em> \u2192 Y<\/em> is a local section of p<\/em>, U<\/em> is an open neighborhood of x<\/em> in X<\/em>, and the equivalence class [s<\/em>] is taken relative to \u2261x<\/sub><\/em>) to s<\/em>(x<\/em>) \u2208 Y<\/em>. The following is pretty remarkable.<\/p>\n\n\n\n

Lemma.<\/strong> The unit \u03b5(Y<\/em>, p<\/em>)<\/sub> of the adjunction E<\/strong> \u22a3 LSect<\/strong> is a (natural) isomorphism, in Etale<\/strong>(X<\/em>), from the \u00e9tale map q<\/em> : S<\/em><\/strong>(LSect<\/strong>(Y<\/em>, p<\/em>)) \u2192 X<\/em> that maps (x<\/em>, [s<\/em>]) to x<\/em>, to the \u00e9tale map p<\/em> : Y<\/em> \u2192 X<\/em>.<\/p>\n\n\n\n

This may seem extraordinary, since there does not seem to be any way of retrieving (x<\/em>, [s<\/em>]) from just one single value s<\/em>(x<\/em>) in the range of s<\/em>, not even considering that s<\/em> is itself unknown. However, the lemma does not say that \u03b5(Y<\/em>, p<\/em>)<\/sub> is an isomorphism of S<\/em><\/strong>(LSect<\/strong>(Y<\/em>, p<\/em>)) into Y<\/em> in Top<\/strong>, but an isomorphism in the category Etale<\/strong>(X<\/em>) of \u00e9tale maps on X<\/em>.<\/p>\n\n\n\n

Proof.<\/em> We will build the inverse to \u03b5(Y<\/em>, p<\/em>)<\/sub> explicitly. That must be a continuous map \u03b5\u20131<\/sup> : Y<\/em> \u2192 S<\/em><\/strong>(LSect<\/strong>(Y<\/em>, p<\/em>)) such that, for every y<\/em> in Y<\/em>, if p<\/em>(y<\/em>) is a point x<\/em> of X<\/em>, then q<\/em>(y<\/em>)=x<\/em>. In other words, \u03b5\u20131<\/sup>(y<\/em>) must be equal to (x<\/em>, [s<\/em>]) where x<\/em>\u225dp<\/em>(y<\/em>), for some local section s<\/em> : U<\/em> \u2192 Y<\/em> of p<\/em>, for some open neighborhood U<\/em> of x<\/em>. This is where we need the fact that p<\/em> is a local homeomorphism. By definition, there is an open neighborhood V<\/em> of y<\/em> such that p<\/em> restricts to a homeomorphism of V<\/em> onto some open neighborhood U<\/em> of x<\/em>. On U<\/em>, p<\/em> has exactly one section, and that is s<\/em>\u225dp<\/em>\u20131<\/sup>|U<\/em><\/sub>.<\/p>\n\n\n\n

We therefore define \u03b5\u20131<\/sup> as mapping every point y<\/em> of Y<\/em> to (x<\/em>, [p<\/em>\u20131<\/sup>|U<\/em><\/sub>]), where x<\/em>\u225dp<\/em>(y<\/em>), V<\/em> is a small enough neighborhood of y<\/em>, U<\/em> is the image of V<\/em> by p<\/em>\u2014by small enough, we mean that V<\/em> is small enough around x<\/em> so that p<\/em> restricts to a homeomorphism of V<\/em> onto its image U<\/em>.<\/p>\n\n\n\n

For every point (x<\/em>, [s<\/em>]) in S<\/em><\/strong>(LSect<\/strong>(Y<\/em>, p<\/em>)), we have \u03b5\u20131<\/sup>(\u03b5(Y<\/em>, p<\/em>)<\/sub>(x<\/em>, [s<\/em>])) = \u03b5\u20131<\/sup>(x<\/em>) = (x<\/em>, [p<\/em>\u20131<\/sup>|U<\/em><\/sub>]). We claim that this is equal to (x<\/em>, [s<\/em>]). It suffices to show that s<\/em> and p<\/em>\u20131<\/sup>|U<\/em><\/sub> agree on a sufficiently small open neighborhood of x<\/em>. Since s<\/em> is a local section of p<\/em>, s<\/em> is a continuous map from some open neighborhood U’<\/em> of x<\/em> such that p<\/em> o s<\/em> = id. The sufficiently small open neighborhood of x<\/em> is U<\/em> \u2229 U’<\/em>: for every x’<\/em> in U<\/em> \u2229 U’<\/em>, p<\/em>(s<\/em>(x’<\/em>))=x’<\/em>, so s<\/em>(x’<\/em>)=p<\/em>\u20131<\/sup>|U<\/em><\/sub>(x’<\/em>). All this shows that \u03b5\u20131<\/sup> o \u03b5(Y<\/em>, p<\/em>)<\/sub> = id.<\/p>\n\n\n\n

For every point y<\/em> in Y<\/em>, we have \u03b5(Y<\/em>, p<\/em>)<\/sub>(\u03b5\u20131<\/sup>(y<\/em>)) = \u03b5(Y<\/em>, p<\/em>)<\/sub>(x<\/em>, [p<\/em>\u20131<\/sup>|U<\/em><\/sub>]), where x<\/em>\u225dp<\/em>(y<\/em>), V<\/em> is a small enough neighborhood of y<\/em>, and U<\/em> is the image of V<\/em> by p<\/em>. Hence \u03b5(Y<\/em>, p<\/em>)<\/sub>(\u03b5\u20131<\/sup>(y<\/em>)) = p<\/em>\u20131<\/sup>|U<\/em><\/sub>(x<\/em>) = y<\/em>. This shows that \u03b5(Y<\/em>, p<\/em>)<\/sub> o \u03b5\u20131<\/sup> = id.<\/p>\n\n\n\n

Finally, we claim that \u03b5\u20131<\/sup> is continuous. For this, we do not need to know the definition of \u03b5\u20131<\/sup>; it suffices to remember that q<\/em> o \u03b5\u20131<\/sup> = p<\/em>, which follows from the fact that q<\/em> = p<\/em> o \u03b5. We take an arbitrary open neighborhood W<\/em> of \u03b5\u20131<\/sup>(y<\/em>) in S<\/em><\/strong>(LSect<\/strong>(Y<\/em>, p<\/em>)), and we claim that there is an open neighborhood V<\/em> of y<\/em> whose image by \u03b5\u20131<\/sup> is included in W<\/em>. Because q<\/em> is a local homeomorphism, we may replace W<\/em> by a smaller open neighborhood of \u03b5\u20131<\/sup>(y<\/em>) such that q<\/em> restricts to a homeomorphism of W<\/em> onto its image by q<\/em> (in X<\/em>). We let V<\/em> be p<\/em>\u20131<\/sup>(q<\/em>(W<\/em>)). For every y’<\/em> in V’<\/em>, we show that \u03b5\u20131<\/sup>(y<\/em>‘) is in W<\/em>. Since q<\/em> is a homeomorphism of W<\/em> onto q<\/em>(W<\/em>), it suffices to show that q<\/em>(\u03b5\u20131<\/sup>(y<\/em>‘)) is in q<\/em>(W<\/em>). This is now obvious: since q<\/em> o \u03b5\u20131<\/sup> = p<\/em>, this boils down to p<\/em>(y’<\/em>) \u2208 q<\/em>(W<\/em>), which follows from the fact that y’<\/em> is in V<\/em> = p<\/em>\u20131<\/sup>(q<\/em>(W<\/em>)). \u2610<\/p>\n\n\n\n

The adjunctions whose counit is a natural isomorphism are a well known object, see Section 3 of the nCatLab entry on reflective subcategories<\/a>. In particular, we learn that the monad LSect<\/strong> o E<\/strong>, from PreSh<\/strong>(X<\/em>) to PreSh<\/strong>(X<\/em>), with unit \u03b7, is idempotent; that LSect<\/strong> is fully faithful; and that Etale<\/strong>(X<\/em>) is equivalent to its so-called essential image<\/a> by LSect<\/strong>, which then turns out to be a reflective subcategory of PreSh<\/strong>(X<\/em>). As usual with category theory, this involves quite a lot of notions, which will benefit from some concrete explanations.<\/p>\n\n\n\n

The sheafification functor LSect<\/strong> o E<\/strong><\/h2>\n\n\n\n

Let me recall that LSect<\/strong>(Y<\/em>, p<\/em>) is always a sheaf, not just a presheaf. It so happens that the essential image I just mentioned is the full subcategory Sh<\/strong>(X<\/em>) of sheaves on X<\/em>. This can be checked by hand by verifying the following series of lemmata.<\/p>\n\n\n\n

Lemma A.<\/strong> Let F<\/em> be a sheaf on X<\/em>, with restriction map rUV<\/sub><\/em> : FV<\/sub><\/em> \u2192 FU<\/sub><\/em> for all pairs of open sets U<\/em>\u2286V<\/em>. Let U<\/em>, V<\/em> be two open subsets of X<\/em>, f<\/em> \u2208 FU<\/sub><\/em>, g<\/em> \u2208 FV<\/sub><\/em>, and let us assume that for every y<\/em> in U<\/em> \u2229 V<\/em>, f<\/em> \u2261y<\/sub><\/em> g<\/em> (namely, there is an open neighborhood W<\/em> of y<\/em> included in both U<\/em> and V<\/em> such that rWU<\/sub><\/em>(f<\/em>)=rWV<\/sub><\/em>(g<\/em>)). Then r<\/em>(<\/sub>U<\/em><\/sub><\/em> \u2229 <\/sub>V<\/em><\/sub><\/em>)<\/sub>U<\/sub><\/em>(f<\/em>) = r<\/em>(<\/sub>U<\/em><\/sub><\/em> \u2229 <\/sub>V<\/em><\/sub><\/em>)<\/sub>V<\/sub><\/em>(g<\/em>).<\/p>\n\n\n\n

Proof.<\/em> For every y<\/em> in U<\/em> \u2229 V<\/em>, there is an open neighborhood W<\/em> of y<\/em> included in both U<\/em> and V<\/em> such that rWU<\/sub><\/em>(f<\/em>)=rWV<\/sub><\/em>(g<\/em>), and we call it W<\/em>(y<\/em>). The family of open sets W<\/em>(y<\/em>) forms an open cover of U<\/em> \u2229 V<\/em>, when y<\/em> varies over U<\/em> \u2229 V<\/em>. The family C<\/strong> of elements rW<\/sub><\/em>(<\/sub>y<\/sub><\/em>)<\/sub>U<\/sub><\/em>(f<\/em>), when y<\/em> varies over U<\/em> \u2229 V<\/em>, is trivially consistent, since it is the family of restrictions of a single element f<\/em>. By the sheaf property, it must be the collection of restrictions to each W<\/em>(y<\/em>) of a unique element h<\/em> of F<\/em>U<\/em> \u2229 V<\/em><\/sub>. Since rW<\/sub><\/em>(<\/sub>y<\/sub><\/em>)<\/sub>U<\/sub><\/em>(f<\/em>) is equal to rW<\/sub><\/em>(<\/sub>y<\/sub><\/em>)(<\/sub>U<\/em><\/sub><\/em> \u2229 V)<\/sub>(r<\/em>(<\/sub>U<\/em><\/sub><\/em> \u2229 <\/sub>V<\/em><\/sub><\/em>)<\/sub>U<\/sub><\/em>(f<\/em>)) for every y<\/em>, that unique element h<\/em> must be r<\/em>(<\/sub>U<\/em> \u2229 V<\/em><\/sub><\/em>)<\/sub>U<\/sub><\/em>(f<\/em>). C<\/strong> is also the family of elements rW<\/sub><\/em>(<\/sub>y<\/sub><\/em>)<\/sub>V<\/sub><\/em>(g<\/em>), when y<\/em> varies over U<\/em> \u2229 V<\/em>, since rW<\/sub><\/em>(<\/sub>y<\/sub><\/em>)<\/sub>U<\/sub><\/em>(f<\/em>)=rW<\/sub><\/em>(<\/sub>y<\/sub><\/em>)<\/sub>V<\/sub><\/em>(g<\/em>). By a symmetric argument, the unique element h<\/em> must be equal to r<\/em>(<\/sub>U<\/em><\/sub><\/em> \u2229 <\/sub>V<\/em><\/sub><\/em>)<\/sub>V<\/sub><\/em>(g<\/em>), and the claim follows. \u2610<\/p>\n\n\n\n

Lemma B.<\/strong> Let F<\/em> be a sheaf of X<\/em>, and let s<\/em> : U<\/em> \u2192 S<\/strong><\/em>(X<\/em>, F<\/em>) be any local section of p<\/em> : S<\/strong><\/em>(X<\/em>, F<\/em>) \u2192 X<\/em>, where U<\/em> is any open subset of X<\/em>. There is a unique f<\/em> \u2208 FU<\/sub><\/em> such that s<\/em>=f<\/span><\/em>.<\/p>\n\n\n\n

Let me recall that f<\/span><\/em><\/em>(x<\/em>) \u225d (x<\/em>, [f<\/em>]) for every x<\/em> in U<\/em>. Let me also recall from last time that the sets f<\/span><\/em>(U<\/em>), where U<\/em> ranges over the open subsets of X<\/em>, and f<\/em> ranges over FU<\/sub><\/em>, form a base of the topology of S<\/strong><\/em>(X<\/em>, F<\/em>).<\/p>\n\n\n\n

Proof.<\/em> By definition, s<\/em> maps every point x<\/em> of U<\/em> to (x<\/em>, [fx<\/sub><\/em>]), for some element fx<\/sub><\/em> of FU<\/sub><\/em>(<\/sub>x<\/sub><\/em>)<\/sub>, for some sufficiently small open neighborhood U<\/em>(x<\/em>) around x<\/em>. The equivalence class [fx<\/sub><\/em>] of fx<\/sub><\/em> is with respect to \u2261x<\/sub><\/em>, and I should probably write it as [fx<\/sub><\/em>]x<\/sub><\/em> to reflect this, but the notation would start to feel somewhat redundant. Still, we will have to be careful about this point below.<\/p>\n\n\n\n

We will pick U<\/em>(x<\/em>) so small that it is included in U<\/em>. Since p<\/em> is a local homeomorphism, we can also pick U<\/em>(x<\/em>) so small that p<\/em> restricts to a homeomorphism from some open neighborhood V<\/em>(x<\/em>) of s<\/em>(x<\/em>) to U<\/em>(x<\/em>): there is an open neighborhood V<\/em> of s<\/em>(x<\/em>) such that p<\/em> restricts to a homeomorphism from V<\/em> onto p<\/em>(V<\/em>), and if needed, we replace U<\/em>(x<\/em>) by its intersection with the open set p<\/em>(V<\/em>), and we define V<\/em>(x<\/em>) as V<\/em> \u2229 p<\/em>\u20131<\/sup>(U<\/em>(x<\/em>)). Since p<\/em> o s<\/em> = id, the inverse of p<\/em>|V<\/em>(x<\/em>)<\/sub> is s<\/em>|U<\/em>(x<\/em>)<\/sub>.<\/p>\n\n\n\n

Finally, we can replace V<\/em>(x<\/em>) by any basic open subset of it that still contains s<\/em>(x<\/em>). This means that we can assume that V<\/em>(x<\/em>) is actually of the form f<\/span><\/em>(U’<\/em>) for some open subset U’<\/em> of X<\/em> and some element f<\/em> of FU’<\/sub><\/em>. If that is so, then the image of f<\/span><\/em>(U’<\/em>) by p<\/em> is U<\/em>, and since the image of V<\/em>(x<\/em>) by p<\/em> is U<\/em>(x<\/em>), we must have U’<\/em>=U<\/em>(x<\/em>). Also, since s<\/em>(x<\/em>) is in V<\/em>(x<\/em>)=f<\/span><\/em>(U<\/em>(x<\/em>)), we must have [fx<\/sub><\/em>]=[f<\/em>], so V<\/em>(x<\/em>) is, in fact, equal to f<\/span>x<\/sub><\/em>(U<\/em>(x<\/em>)).<\/p>\n\n\n\n

This implies that, for every point x<\/em> of U<\/em>, for every y<\/em> \u2208 U<\/em>(x<\/em>), s<\/em>(y<\/em>)=(y<\/em>, [fx<\/sub><\/em>]y<\/sub><\/em>). (Note that the definition of s<\/em>(y<\/em>) is (y<\/em>, [fy<\/sub><\/em>]y<\/sub><\/em>), with a “y<\/em>” as subscript to “f<\/em>“, not “x<\/em>“; I am claiming that you can put an “x<\/em>” there, and this will make no difference.) Explicitly, s<\/em>(y<\/em>)=(y<\/em>, [fy<\/sub><\/em>]y<\/sub><\/em>); since y<\/em> is in U<\/em>(x<\/em>), s<\/em>(y<\/em>)=s<\/em>|U<\/em>(x<\/em>)<\/sub>(y<\/em>) is in V<\/em>(x<\/em>)=f<\/span>x<\/sub><\/em>(U<\/em>(x<\/em>)), and is therefore of the form (y’<\/em>, [fx<\/sub><\/em>]y’<\/sub><\/em>), for some point y’<\/em>, which must of course be equal to y<\/em>.<\/p>\n\n\n\n

Hence, given any two points x<\/em> and x’<\/em> in U<\/em>, we obtain that for every y<\/em> in U<\/em>(x<\/em>) \u2229 U<\/em>(x’<\/em>), s<\/em>(y<\/em>) is both equal to (y<\/em>, [fx<\/sub><\/em>]y<\/sub><\/em>) and to (y<\/em>, [fx’<\/sub><\/em>]y<\/sub><\/em>), so that fx<\/sub><\/em> \u2261y<\/sub><\/em> fx’<\/sub><\/em>. Lemma A tells us that r<\/em>(<\/sub>U<\/em><\/sub><\/em>(x<\/em>) \u2229 <\/sub>U<\/em><\/sub><\/em>(<\/sub>x’<\/em><\/sub><\/em>))<\/sub>U<\/sub><\/em>(<\/sub>x<\/sub><\/em>)<\/sub>(fx<\/sub><\/em><\/em>) = r<\/em>(<\/sub>U<\/em><\/sub><\/em>(x<\/em>) \u2229 <\/sub>U<\/em><\/sub><\/em>(<\/sub>x’<\/em><\/sub><\/em>))<\/sub>U<\/sub><\/em>(<\/sub>x’<\/sub><\/em>)<\/sub>(fx’<\/sub><\/em><\/em>). Therefore, as x<\/em> varies in U<\/em>, the family of elements fx<\/sub><\/em> \u2208 FU<\/sub><\/em>(<\/sub>x<\/sub><\/em>)<\/sub> forms a consistent family. Since the open sets U<\/em>(x<\/em>) form an open cover of U<\/em>, the sheaf property implies that there is a unique element f<\/em> of FU<\/sub><\/em> such that fx<\/sub><\/em>=rU<\/sub><\/em>(<\/sub>x<\/sub><\/em>)<\/sub>U<\/sub><\/em>(f<\/em><\/em>) for every x<\/em> in U<\/em>.<\/p>\n\n\n\n

In particular, for every x<\/em> in U<\/em>, fx<\/sub><\/em> and f<\/em> have the same restriction to U<\/em>(x<\/em>), showing that fx<\/sub><\/em> \u2261x<\/sub><\/em> f<\/em>. Therefore s<\/em>(x<\/em>) = (x<\/em>, [f<\/em>]), with the same f<\/em> for all the points x<\/em> in U<\/em>. In other words, s<\/em>=f<\/span><\/em>. The fact that this f<\/em> is unique is obvious. \u2610<\/p>\n\n\n\n

Theorem.<\/strong> For every presheaf F<\/em> on X<\/em>, the unit \u03b7F<\/sub><\/em> : F<\/em> \u2192 LSect<\/strong>(E<\/strong>(F<\/em>)) at F<\/em> of the adjunction E<\/strong> \u22a3 LSect<\/strong> is an isomorphism of presheaves if and only if F<\/em> is a sheaf.<\/p>\n\n\n\n

Proof.<\/em> Let me recall that a morphism in the category of presheaves on X<\/em> is a natural transformation. Hence an isomorphism of presheaves is a natural isomorphism. If F<\/em> is a sheaf, it is easy to see that any isomorphic presheaf will also be a sheaf. The interesting direction is to show that if F<\/em> is a sheaf, then \u03b7F<\/sub><\/em> is an isomorphism of (pre)sheaves.<\/p>\n\n\n\n

For every open subset U<\/em> of X<\/em>, the component \u03b7F<\/sub><\/em>,<\/sub>U<\/sub><\/em> of the natural transformation \u03b7F<\/sub><\/em> at U<\/em> maps every f<\/em> \u2208 FU<\/sub><\/em> to f<\/span><\/em><\/em> : U<\/em> \u2192 S<\/strong><\/em>(X<\/em>, F<\/em>); let me recall that f<\/span><\/em><\/em>(x<\/em>) \u225d (x<\/em>, [f<\/em>]) for every x<\/em> in U<\/em>. We need to show that \u03b7F<\/sub><\/em>,<\/sub>U<\/sub><\/em> is a bijection. To this end, we consider an arbitrary local section s<\/em> : U<\/em> \u2192 S<\/strong><\/em>(X<\/em>, F<\/em>) of p<\/em>, and we must show that s<\/em>=f<\/span><\/em> for some unique f<\/em> \u2208 FU<\/sub><\/em>. This is exactly what we proved in Lemma B. \u2610<\/p>\n\n\n\n

In particular, if the presheaf F<\/em> we start with is already a sheaf, then F<\/em> is already isomorphic to its sheafification LSect<\/strong>(E<\/strong>(F<\/em>)), through \u03b7F<\/sub><\/em>. The adjunction E<\/strong> \u22a3 LSect<\/strong> then restricts to an adjunction equivalence of categories between the category Sh<\/strong>(X<\/em>) of sheaves on X<\/em> and the category Etale<\/strong>(X<\/em>) of \u00e9tale maps on X<\/em>.<\/p>\n\n\n\n

Because it arises from an adjunction, LSect<\/strong> o E<\/strong> is the functor part of a monad, the sheafification monad<\/em>. It replaces a presheaf by the best sheaf that approximates it, and shows that the category Sh<\/strong>(X<\/em>) of sheaves on X<\/em> is a reflective subcategory of the category of presheaves on X<\/em>.<\/p>\n\n\n\n

What about streams, then?<\/h2>\n\n\n\n

Everything I have said until now is well known. I would like to apply this to streams and prestreams.<\/p>\n\n\n\n

Last time<\/a>, we had seen that we could start from a prestream (X<\/em>, (\u2291U<\/sub><\/em>)U<\/em> \u2208 O<\/strong>X<\/em><\/sub>), build its presheaf F<\/em> of locally monotone maps to any prescribed poset P<\/em>, and that F<\/em> is a sheaf if and only (X<\/em>, (\u2291U<\/sub><\/em>)U<\/em> \u2208 O<\/strong>X<\/em><\/sub>) is a stream. I had also drawn pictures of the \u00e9tale space S<\/strong>(X<\/em>, F<\/em>) of that sheaf in the case where the given stream is the directed circle, and that P<\/em> is the Sierpi\u0144ski poset {0 < 1}:<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

The \u00e9tale space itself is the collection of four stacked circles on top, and the \u00e9tale map p<\/em> is projection down to the circle at the bottom.<\/p>\n\n\n\n

Last time, we had observed that the stalks Fx<\/sub><\/em> = p<\/em>\u20131<\/sup>({x<\/em>}), at any point x<\/em>, are ordered, by [f<\/em>] \u2264x<\/sub><\/em> [g<\/em>] if and only if, for some sufficiently small open neighborhood W<\/em> included in the domains of f<\/em> and g<\/em>, f<\/em>|W<\/em><\/sub> \u2264 g<\/em>|W<\/em><\/sub>. May this is the start of some form of additional structure on the \u00e9tale map (S<\/strong>(X<\/em>, F<\/em>), p<\/em>) that would allow us to retrieve the whole stream, right? In other words, if I give you an arbitrary \u00e9tale map (Y<\/em>, p<\/em>), plus some additional structure, surely you can reconstruct a sheaf from that, namely, LSect<\/strong>(Y<\/em>, p<\/em>), but maybe one could recognize that sheaf of local sections precisely as the sheaf of locally monotone maps for some circulation on X<\/em>.<\/p>\n\n\n\n

My impression is that the additional structure is not that ordering. Instead, it should simply be a map from Y<\/em> to P<\/em>, with some extra properties. Let us discover which. I am not completely happy with what follows, but that is certainly a solution.<\/p>\n\n\n\n

Etale streams<\/h2>\n\n\n\n

When Y<\/em>=S<\/strong>(X<\/em>, F<\/em>) and F<\/em> is the sheaf of locally monotone maps from a prestream (X<\/em>, (\u2291U<\/sub><\/em>)U<\/em> \u2208 O<\/strong>X<\/em><\/sub>) to P<\/em>, there is an obvious map from Y<\/em> to P<\/em>, which I will call eval<\/em>: for every point (x<\/em>, [f<\/em>]) of S<\/strong>(X<\/em>, F<\/em>), let me define eval<\/em>(x<\/em>, [f<\/em>]) as f<\/em>(x<\/em>). In the example of the directed circle, where we had in fact taken P<\/em> to be {0 < 1}, eval<\/em> maps the top two circles (“far-right” and “right-strict”) to 0, and the next two circles (“right-slack” and “anywhere”) to 1.<\/p>\n\n\n\n

The definition of eval<\/em>(x<\/em>, [f<\/em>]) as f<\/em>(x<\/em>) is independent of the choice of representative f<\/em> in the equivalence class [f<\/em>], because any pair of such representatives agrees on some open neighborhood of x<\/em>, in particular on x<\/em> itself. Diagrammatically, the situation is as follows.<\/p>\n\n\n\n

\"\"<\/a><\/figure><\/div>\n\n\n\n

For every open subset U<\/em> of X<\/em>, the local sections s<\/em> : U<\/em> \u2192 Y<\/em> of p<\/em> are exactly the maps f<\/span><\/em> : x<\/em> \u21a6 (x<\/em>, [f<\/em>]), where f<\/em> ranges over the monotone maps from (U<\/em>,\u2291U<\/sub><\/em>) to P<\/em>. We retrieve f<\/em> as eval<\/em> o s<\/em>. In particular, this allows us to compare local sections by defining s<\/em> \u2264U<\/sub><\/em> s’<\/em> if and only if eval<\/em> o s<\/em> \u2264 eval<\/em> o s’<\/em>, for all local sections s<\/em>, s’<\/em> : U<\/em> \u2192 Y<\/em> of p<\/em>; this mimics the usual pointwise ordering of monotone maps from (U<\/em>,\u2291U<\/sub><\/em>) to P<\/em>. Although I will not mention that ordering explicitly below, it really subtends everything.<\/p>\n\n\n\n

From now on, we will assume that P<\/em> is a complete lattice. (This is certainly the case of {0 < 1}!) Then the collection of all the maps eval<\/em> o s<\/em>, where s<\/em> : U<\/em> \u2192 Y<\/em> ranges over the local sections of p<\/em> with domain U<\/em>, is a complete lattice that is isomorphic to the complete lattice of all monotone maps from (U<\/em>,\u2291U<\/sub><\/em>) to P<\/em>. Moreover, for every x<\/em> \u2208 U<\/em> and for every a<\/em> \u2208 P<\/em>, there is a local section s<\/em> : U<\/em> \u2192 Y<\/em> such that (eval<\/em> o s<\/em>)(x<\/em>)=a<\/em>. There is an obvious one, which is the one such that eval<\/em> o s<\/em> is the constant map with value a<\/em>. Perhaps more importantly, there is a smallest one; explicitly, for s<\/em> that smallest local section, eval<\/em> o s<\/em> is characterized as mapping every x<\/em><\/em>‘ \u2208 U<\/em><\/em> to a<\/em> if x<\/em> \u2291U<\/sub>x’<\/em>, and to the least element of P<\/em> otherwise. We arrive at the following definition.<\/p>\n\n\n\n

Definition.<\/strong> Let P<\/em> be a fixed complete lattice, and X<\/em> be a fixed topological space. A stratified \u00e9tale map<\/em> on X<\/em> is a triple (Y<\/em>, p<\/em>, e<\/em>), where Y<\/em> is a topological space, and:<\/p>\n\n\n\n