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An Adjoint Characterization of the Category of Sets

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An Adjoint Characterization of the Category of Sets An adjointcharacterization of the category of sets Rob ert Rosebrugh Department of Mathematics and Computer Science Mount Allison University Sackville N B EA C Canada R J Wood Department of Mathematics Statistics and Computing Science Dalhousie University Halifax N S BH J Canada Abstract op If a category B with Yoneda emb edding Y B CATB set has an adjoint string U a V a W a X a Y then B is equivalenttoset The authors gratefully acknowledge nancial supp ort from NSERC Canada Diagrams typ eset using M Barrs diagram macros Intro duction The statement of the Abstract was implicitly conjectured in Here we establish the conjecture We will see that it suces to assume that B has an adjoint string V a W a X a Y with V pullback preserving Aword on foundations and our notation is necessaryWe write set for the category of small sets and assume that there is a Grothendieck top os SETofsetswhich contains the set of arrows of set as an ob ject The category of category ob jects in SET whichwe write CAT is cartesian closed and set is an ob ject of CAT Thus for C a category in CAT op CAT C set is also an ob ject of CAT and we abbreviate it by MC it was written coop coop P C in Substitution gives a functor M CAT CAT where CAT is the dual which reverses b oth arrows of CAT functors and cells natural transformations A op category B in CAT is said to b e local ly smal l if it has a hom functor B B set or equivalently a Yoneda emb edding Y Y B MB Wesay that a category A is smal l B if the set of arrows of A is an ob ject of set All categories under consideration other than SET and CAT are ob jects of CAT A functor F A B is said to b e Kan if MF MB MA has a left adjoint denoted F If A is small and B is lo cally small then F is Kan but neither condition ML Smallness of A is necessary if saywehave L a F then ML aMF and F and lo cal smallness of B also ensures that MF has a right adjoint whichwe denote by F In particular for small A the Yoneda emb edding Y A MA yields Y a A A MY aY MA MMA and it is shown in that Y is isomorphic to A A A Y We can apply these considerations to A the empty category which is the initial MA ob ject of CAT The unique functor M is necessarily Y and gives rise to Y aMY a Y M But M is isomorphic to set and is terminal in CAT so the adjoint string is more conveniently lab elled a a set A further application of the result quoted from gives an adjoint string of the kind mentioned in the Abstract namely aM aM aM a Y set Mset set We recall from or that a lo cally small category B is said to b e total abbreviating total ly cocompleteifY 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 category ord of ordered sets orderpreserving functions and transformations There B DB sends an element b to the downclosed subset of B consisting of all x such that x b D B is the lattice of all downclosed subsets of B ordered by inclusion This functor W has a left adjoint namely supremum precisely when B is cocomplete It is helpful to W think of X ab ove as a generalization of Continuing the analogywe recall from that W has a left adjoint precisely when B is constructively completely distributive With this in mind wesay that a total category is total ly 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 W In the ord case a left adjoint for classies the or totally b elow relation dened W by b b if and only if for any D in D B b D implies b D A similar interpretation is op p ossible for W Its transp ose B B set is in some resp ects like another hom functor At least it makes go o d 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 prop erty with resp ect to the new arrows and if this colimitlike functor itself has a left adjoint then ordinary limits also distribute over these colimitlike universals The p oint of the heuristics of the preceding paragraph is that the adjoint strings we are considering are manifestations of exactness Given a suitably complete and co complete category B it seems p ossible ab initio that B b e more distributive than set The Theorem of this pap er shows that this is not the case Exactness of a lo cally small category is strictly b ounded by the exactness of set Note further that while total categories B can fail to b e op cototal that is B can fail to b e total totally distributive categories are always cototal This and a detailed study of the heuristics ab ove will app ear in a separate forthcoming pap er The adjointcharacterization Let B b e 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 CWe write W a X for the other adjunction Cofully faithfulness of X implies that the unit is an isomorphism and so W is fully faithful We dene W Y to b e the unique natural transformation satisfying X Equivalently is the unique solution of X We write I E B for the inverter of W Y B MB i e E is the full sub category of B determined by those B for which is an isomorphism B I is the resulting inclusion For any functor F C D with DFCDinset for all C D and for any G K D we follow Street and Walters in writing DF G K MC for the functor whose value at K in K is DF GK If D is lo cally small DF G is the comp osite G Y MF K D MD MC Further still assuming that D is lo cally small and for any H K MD the Yoneda MF H even though MD need not b e lo cally small Lemma gives MDYFH Lemma Acategory B is equivalent to one of the form MA with A smal l if and only if B is total ly distributive and the inverter I as above is dense and Kan Pro of only if Wehave already remarked that MA 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 present concerns we leave the pro of of this claim as an exercise for the reader In the ord case it is discussed in It is easy to see that I is dense and Kan if Given B and I as ab ove consider the comp osite Y MI B MB ME BI B Since Y and MI have left adjoints namely X and I resp ectivelysodoes BI We denote the left adjointby I since its value at in MEI is the colimit of I weighted by The unit for I a BI is an isomorphism since I is dense The following isomorphisms are justied by in order denition of I W a X is inverted by I the Yoneda lemma and fully faithfulness of I which follows from fully faithfulness of I MI I MBYII MBWI I BI X I BI I Thus BI B ME is an equivalence Since b oth E and now ME are lo cally small it follows from see also 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 equivalentto invertibility of the canonical natural transformation X F FX as shown in the left D C hand diagram b elow F F MC MD MC MMD X X X MY C D C D C D C MD F F 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 by MD where D is an arbitrary lo cally small category According to our denition of total category and again invoking or MD is total if and only if D is small But wedohave MY D assuming only that D is lo cally small If F is b oth Kan and a left adjoint then a canonical isomorphism as in the right hand diagram is pro duced by a mo dication of the calculations which establish that the canonical arrow in the left hand diagram is an isomorphism Of course we implicitly noted in the Intro duction that if D is small then MY X The D MD p oint is that for D lo cally small MD has the requisite weighted colimits and they are provided by MY D Let B b e a totally distributive category with V a W ThenW B MB is b oth Kan and a left adjoint The considerations of the previous paragraph show that WX MY W Since W is fully faithful XW and wehave MY W W W This is a formulation for B totally distributive categories of the Interp olation Lemma for constructively completely distributive lattices as in Now a calculation shows that the natural isomorphism ab ove MY W W W admits description by b oth MY W MY W W MY Y W W and MY W MY W W MY W Y W W X Y where b oth the rst and last unnamed isomorphisms express the fully faithfulness of Y and WX These descriptions the second unnamed isomorphism is an instance of MY W show that the profunctor B B determined by W B MB carries an idemp otent comonad structure with counit determined by W Y It is convenient to dene T VY B B Then MY W MY MV MVY MT which shows that MT coinverts By Lemma of T inverts Lemma Acategory B is equivalent to one of the form MA with A a smal l complete ordered set if and only if B is total ly distributive with V a W Pro of only if A small complete ordered set A is a total category Indeed by denition A DA has a left adjoint So do es the inclusion D A MA and its A
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