heaiication Johan M. Commelin, October 15, 2013 Introduction Let C be a categor, equipped with a Grothendieck topolog J. Let 2 Ph(C) be a preshea. The purpose o these notes is to assign to a morphism to a shea! #, through which an other morphism to a shea! G actors. In doing so, a crucial r^ole is plaed b the plus construction, dened below. We rst some preliminar conventions and notation. Notation. Let U 2 C be an object. The contravariant Hom-unctor Hom( ; U ) will be denoted h U. A morphism : V! U in C will be identied with the induced morphism h V! h U. (Indeed, to avoid conusion, in the tet below we will introduce morphisms in C as morphisms between presheaves.) In the same spirit, we identi elements o (U ) with morphisms h U!. Remark. Let U 2 C be some object. A sieve on U is a subunctor o h U. Nevertheless, we will use the notation T, to denote the ``intersection'' o two sieves on U. ormall, this means z hu T. where Denition. The plus construction is given b: ( ) + : Ph(C)! Ph(C) 7! + ; + : C op! et U 7! colim 2J (U )Hom Ph(C)(; ): The plus construction is a well-deined unctor Remark. The colimit in the denition o + (U ) eists, since it is a colimit in et. Notation. Given some covering sieve on some U 2 C, and an element 2 Hom(; ), we write or the image o in + (U ) = Hom(h U ; + ). Lemma. The plus construction applied to gives a preshea +. Proo. Given a morphism : h V! h U rom C, there is an obvious candidate or + ( ). Ater all, given a covering sieve on U, we have a covering sieve { on V. There is a natural morphism {! b composition with. Consequentl we get a morphism Hom Ph(C)(; )! Hom Ph(C)( { ; ). This gives a natural transormation rom the diagram o + (U ) to the diagram o + (V ) {, and thereore an induced morphism between the colimits. In practice, this means that or :!, it maps to = j {. 1
h V h U + { We leave it to the reader to veri that + preserves identit and composition. Corollar. There is a natural morphism :! +, b mapping 2 (U ) to. Indeed, viewing as morphism h U!, we have =. + h U Corollar. or a covering sieve on U, and a morphism :!, we have = j. h U + Proo. We veri the identit b ``probing'' it with representables. Let : h V! be some morphism. We ma then identi h V with {. B the preceding lemma, we have = j { = : The preceding corollar gives, =. ince is arbitrar, we conclude that j =. Lemma. The plus construction is a unctor. Proo. Given a morphism o presheaves :! G, one obtains a natural transormation o the diagram dening + (U ) to the diagram dening G + (U ) just b composing with. This induces morphisms on the colimits, which are compatible with restriction morphisms. Thus we have a morphism +! G +, and this construction evidentl preserves identities and composition. eparatedness o + Lemma. The preshea + is separated. Proo. Let U 2 C be some object, and a covering sieve on U. We have to show that an natural transormation :! + etends to at most one natural transormation h U! +. Assume such an etension eists. Let and be two such etensions, represented b :!, and :!, where and are covering sieves on U. 2
h U + Put T =. Then we have jt = jt = jt : urther, jt represents, and jt represents. It ollows that we ma replace our setup with the ollowing: h U + T or an V 2 C, and 2 (V ), we have = = = = : Consequentl, and agree on some common renement T o h V. We use these T and the transitivit aiom o Grothendieck topologies to create a covering sieve o U, on which and agree. Recall the transitivit aiom or Grothendieck topologies: Let R be a sieve on U. I there is a covering sieve T on U, such that or all V 2 C and 2 T (V ), the sieve { R covers V, then R covers U. Let R be the sieve V; 2T (V ) ( T ). In other words, ever morphism in R(W ), when viewed as morphism h W! h U, actors via some : T! h U. Note that T { R, and thereore, b the transitivit aiom o Grothendieck topologies, R is a covering sieve on U. Observe that b construction R is a subsieve o T, and thereore we ma replace our setup with the ollowing: h U + R 3
To prove that =, it now suces to prove that =. Let W 2 C be arbitrar, and k : h W! R be an element o R(W ). B denition o R, there eists some V 2 C, and 2 T (V ), such that the composition k : h W! R! T equals h W! T! h V! T. ince jt = jt we also see that k = k. As W and k are arbitrar, we conclude that =, which implies =. I is separated, + is a shea We continue the notation o the previous section. Lemma. Assume that is separated. The + is a shea. Proo. Now we have to show that etends (and uniqueness will ollow rom the previous section). or an V 2 C, and an 2 (V ), the composition gives an element o + (V ). Represent this element b some morphism :!. These representing morphisms are compatible, in the sense that the are, up to renement, unctorial in V and. More precise: or some W 2 C, and g 2 (W ), the morphisms g and g might not be equal, but the do agree on some W -covering subsieve ;g o g. g h V + g h W g g ;g Let R be the sieve V; 2(V ) ( ). In other words, ever morphism in R(W ), when viewed as morphism h W! h U, actors via some :! h U. Note that { R, and thereore, b the transitivit aiom o Grothendieck topologies, R is a covering sieve on U. Giving a morphism ~ : R! boils down to giving its composition (an element o (W )) with ever morphism k : h W! R. B denition o R, such a morphism k actors as k = g, with 2 (V ), g 2 (W ). Now put ~ k = g. This does not depend on the actorisation k = g, since g etends k : k!, and is separated. 4
h V g ~ h W k R We are done, i we check that ~ etends. This is done b a doublelaered ``probing'' with representables. or an : h V!, we have to show that equals ~ j. We ma test this equalit on, because + is separated. But or an g 2 (W ), we have k = g 2 R(W ), and or such k we have just proven the etension. ~ h U + h V g ~ h W k R Idempotence, adjunction, and eactness The idempotence ollows immediatel rom the denition: I is a shea, then the colimit dening + (U ) has terms Hom Ph(C)(; ) = (U ), because o the shea propert, and the Yoneda lemma. It is then clear that + (U ) = (U ), which grants the idempotence. Let be a preshea, and G a shea. We have to show that We reduce this to proving that Hom(; G) = Hom( # ; G): Hom(; G) = Hom( + ; G): This is actuall not ver hard. B unctorialit o the plus construction, we have a map + : +! G associated to ever map :! G. I 2 + (U ) is represented b :!. Then + () = (), showing that + is the unique etension o along. + h U + G G id 5
Now that we have the adjunction established, it is a ormal consequence that ( ) # preserves all colimits. To show that it also preserves nite limits, again, we reduce this to showing that ( ) + preserves nite limits. Now it boils down to the observation that the colimit in the denition o + (U ) is ltered, and thereore commutes with nite limits. 6