An adjoint characterization of the category of sets

namely 90 a M0 a M! a M1 a Yset : set !Mset: We recall from [8] or [9] that a locally small category B is said to be total (abbreviating totally cocomplete) if Y : B !MB has a left adjoint, X: Considerable motivation for the terminology is given in either reference. Examples include categories of algebras, categories of spaces and categories of sheaves on a Grothendieck site. The reader is advised to keep in mind the situation when B is an ordered set and Y is replaced by its counterpart # in the 2-category, ord, of ordered sets, order-preserving functions and transformations. There #: B ! DB sends an element b to the down-closed subset of B consisting of all x such that x b: (DB is the lattice of all down-closed subsets of B ordered by inclusion.) This functor has a left adjoint, namely supremum, W; precisely when B is (co)complete. It is helpful to think of X above as a generalization of W : Continuing the analogy, we recall from [1] that W has a left adjoint precisely when B is (constructively) completely distributive. With this in mind we say that a total category is totally distributive when it has an adjoint string, W a X a Y : B !MB: The considerations in the previous paragraph show that MA is totally distributive for small A. In the ord case a left adjoint for W classi es the ; or \totally below", relation de ned by b b0 if and only if, for any D in DB, b0 WD implies b 2 D: A similar interpretation is possible for W: Its transpose, Bop B ! set; is in some respects like another hom functor. At least it makes good sense to think of its values as sets of \arrows", a priori distinct from the arrows of B. A left adjoint, V; for W expresses a universal property with respect to the new arrows and if this colimit-like functor itself has a left adjoint then ordinary limits also distribute over these colimit-like universals. The point of the heuristics of the preceding paragraph is that the adjoint strings we are considering are manifestations of \exactness". Given a suitably complete and cocomplete category B it seems possible, ab initio, that B be more distributive than set. The Theorem 3 of this paper shows that this is not the case. Exactness of a locally small category is strictly bounded by the exactness of set. Note further that while total categories B can fail to be cototal (that is, Bop can fail to be total), totally distributive categories are always cototal. This and a detailed study of the heuristics above will appear in a separate forthcoming paper. 2 The adjoint characterization Let B be a totally distributive category with adjoint string W a X a Y : B ! MB: We write ; : X a Y to indicate that is the unit and is the counit for the adjunction. Since Y is fully faithful, is an isomorphism and X is cofully faithful i. e. CAT(X;C) is fully faithful for all C. We write ; : W a X for the other adjunction. Cofully faithfulness of X implies that the unit, ; is an isomorphism and soW is fully faithful. We de ne :W ! Y to be the unique natural transformation satisfyingX = 1: Equivalently, is the unique solution of X = 1. We write I : E ! B for the inverter of : W ! Y : B !MB; i. e. E is the full subcategory of B determined by those B for which B is an isomorphism. I is the resulting inclusion. For any functor F : C ! D with D(FC;D) in set for all C;D and for any G : K ! D; we follow Street and Walters, [8], in writing D(F;G) : K !MC for the functor whose value at K in K is D(F ; GK): If D is locally small, D(F;G) is the composite K G ! D Y !MD MF !MC: Further, still assuming that D is locally small, and for any H : K ! MD; the Yoneda Lemma gives MD(Y F;H) =MF H even though MD need not be locally small. Lemma 1 A category B is equivalent to one of the form MA with A small if and only if B is totally distributive and the inverter I; as above, is dense and Kan. Proof. (only if) We have already remarked thatMA is totally distributive for small A. Here E is the Cauchy completion of A. (Since this part of the Lemma is not central to our 4 present concerns we leave the proof of this claim as an exercise for the reader. In the ord case it is discussed in [5].) It is easy to see that I is dense and Kan. (if) Given B and I as above, consider the composite B Y !MB MI !ME = B(I; 1B): Since Y and MI have left adjoints, namely X and 9I respectively, so does B(I; 1): We denote the left adjoint by I ? { ; since its value at in ME; I ? ; is the colimit of I weighted by [8]. The unit for I ? { a B(I; 1) is an isomorphism since I is dense. The following isomorphisms are justi ed by (in order): de nition of I ? { ;W a X; is inverted by I; the Yoneda lemma and fully faithfulness of 9I (which follows from fully faithfulness of I). B(I; I ? ) = B(I; (X 9I)( )) =MB(WI;9I( )) =MB(Y I;9I( )) = (MI 9I)( ) = : Thus B(I; 1) : B ! ME is an equivalence. Since both E and now ME are locally small it follows from [7] (see also [2]) that E is small as required. If C and D are total then a functor F : C ! D preserves all colimits if and only if it has a right adjoint. If, moreover, F is Kan then preservation of all colimits is equivalent to invertibility of the canonical natural transformation XD9F ! FXC as shown in the left hand diagram below. C D F MC MD 9F ? XC ?XD C MD F MC MMD 9F ? XC ?MYD = = Again, the reader is advised to think of \X" as a general counterpart of the supremum arrow for a complete ordered set. Now replace D in the immediately preceding discussion byMD; where D is an arbitrary locally small category. According to our de nition of total category 5 and again invoking [7] (or [2]) MD is total if and only if D is small. But we do haveMYD assuming only that D is locally small. If F is both Kan and a left adjoint then a canonical isomorphism as in the right hand diagram is produced by a modi cation of the calculations which establish that the canonical arrow in the left hand diagram is an isomorphism. Of course we implicitly noted in the Introduction that if D is small then MYD = XMD. The point is that for D locally small, MD has the requisite weighted colimits and they are provided by MYD. Let B be a totally distributive category with V a W . Then W : B !MB is both Kan and a left adjoint. The considerations of the previous paragraph show thatWX =MY 9W . SinceW is fully faithful, XW = 1B and we haveMY 9W W = W: (This is a formulation for totally distributive categories of the \Interpolation Lemma" for constructively completely distributive lattices as in [5].) Now a calculation shows that the natural isomorphism above, MY 9W W = ! W; admits description by both MY 9W W MY 9Y W MY 9 W = W and MY 9W W MY 9W Y MY 9W = W X Y = W; where both the rst and last un-named isomorphisms express the fully faithfulness of Y and the second un-named isomorphism is an instance of MY 9W = WX. These descriptions show that the profunctor B B determined by W : B ! MB carries an idempotent comonad structure, with counit determined by : W ! Y: It is convenient to de ne T = V Y : B ! B: Then MY 9W =MY MV =M(V Y ) =MT which shows that MT coinverts : By Lemma 4.3 of [4], T inverts . 6 Lemma 2 A category B is equivalent to one of the form MA with A a small, completeordered set if and only if B is totally distributive with V a W:Proof. (only if) A small, complete ordered set, A, is a total category. Indeed, byde nition #A : A ! DA has a left adjoint. So does the inclusion DA ! MA and itscomposite with #A is Y : A ! MA; which therefore has a left adjoint. It follows thatMA has the required adjoint string.(if) We saw above that T = V Y inverts :W ! Y:We denote the inverter I : E ! Bas above, so there exists a unique functor H : B ! E such that IH = T: We show H a Iby showing that E(H; 1) = B(1; I): NowB(1; I) = Y I = WI =MB(Y;WI) = B(V Y; I) = B(T; I) = B(IH; I) = E(H; 1)where we have the last isomorphism because I is fully faithful. From H a I we have I Kan(with 9I =MH). To see that I is dense considerI ? { B(I; 1) = X 9I MI Y = X MH MI Y = X M(IH) Y= X M(T ) Y = X B(T; 1) = X B(V Y; 1)= X MB(Y;W ) = X W =1B:By (the proof of) Lemma 1, B is equivalent to ME and the equivalence B(I; 1) identi es Iand YE: Thus H a I shows that E is total (directly, although that was already clear abovesince a full re ective subcategory of a total is total) and hence complete in the usual sense.But from Lemma 1 we also have E small so, by [3], E is an ordered set.Theorem 3 A category B is equivalent to set if and only if B is totally distributive withV a W and V preserves pullbacks. 7 Proof. (only if) This follows from the Introduction. For if we have U a V then certainlyV preserves pullbacks.(if) Now T = V Y preserves pullbacks. It follows from the construction of H in Lemma2 that H preserves pullbacks so E is \lex total", meaning that the de ning left adjointfor totality is left exact. (It necessarily preserves the terminal object.) By [6], E is aGrothendieck topos (for since E is small the size requirement in [6] is trivially satis ed).But since, by Lemma 2, E is also an ordered set it must therefore be 1: Indeed, we havetrue = false : 1 ! in E:Corollary 4 The category set is characterized by U a V a W a X a Y:References[1] B. Fawcett and R. J. Wood. Constructive complete distributivity I. Math.Proc. Cam. Phil. Soc., 107:81{89, 1990.[2] F. Foltz. Legitimite des categories de prefaisceaux. Diagrammes, 1:1{5,1979.[3] P.J. Freyd. Abelian Categories. Harper and Row, 1964.[4] R. Pare, R. Rosebrugh, and R.J. Wood. Idempotents in bicategories.Bulletin of the Australian Math. Soc., 39:421{434, 1989.[5] R. Rosebrugh and R. J. Wood. Constructive complete distributivity IV.to appear, 1992.[6] R. Street. Notions of topos. Bulletin of the Australian Math Society,23:199{208, 1981.8 [7] R. Street. Unpublished manuscript. 1979.[8] R. Street and R. F. C. Walters. Yoneda structures on 2-categories. Jour-nal of Algebra, 50:350{379, 1978.[9] R. J. Wood. Some remarks on total categories. Journal of Algebra,75:538{545, 1982.9