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

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

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

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

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

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

has a left adjoint namely supremum precisely when B is cocomplete It is helpful to

think of X ab ove as a generalization of Continuing the analogywe recall from that

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

In the ord case a left adjoint for classies the or totally b elow relation dened

by b b if and only if for any D in D B b D implies b D A similar interpretation is

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

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

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

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

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

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

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

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

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

comp osite with is Y A MA which therefore has a left adjoint It follows that

MA has the required adjoint string

if WesawabovethatT VY inverts W Y We denote the inverter I E B

as ab ove so there exists a unique functor H B E such that IH T Weshow H a I

by showing that EH BI Now

BI YI WI MBY W I BVYI BT I BIH I EH

where wehave the last isomorphism b ecause I is fully faithful From H a I wehave I Kan

MH To see that I is dense consider with I

X MH MI Y X MIH Y X I MI Y I BI

X BT X BVY X MT Y

X W X MBY W

By the pro of of Lemma B is equivalentto ME and the equivalence BI identies I

and Y Thus H a I shows that E is total directly although that was already clear ab ove

since a full reective sub category of a total is total and hence complete in the usual sense

But from Lemma we also have E small so by E is an ordered set

Theorem Acategory B is equivalent to set if and only if B is total ly distributive with

V a W and V preserves pul lbacks

Pro of only if This follows from the Intro duction For if wehave U a V then certainly

V preserves pullbacks

if Now T VY preserves pullbacks It follows from the construction of H in Lemma

thatH preserves pullbacks so E is lex total meaning that the dening left adjoint

for totality is left exact It necessarily preserves the terminal ob ject By E is a

Grothendieck top os for since E is small the size requirement in is trivially satised

But since by Lemma E is also an ordered set it must therefore b e Indeed wehave

true false inE

Corollary The category set is characterizedbyU a V a W a X a Y

References

B Fawcett and R J Wo o d Constructive complete distributivityI Math

Proc Cam Phil Soc

F Foltz Legitimite des categories de prefaisceaux Diagrammes

PJ Freyd Abelian Categories Harp er and Row

R Pare R Rosebrugh and RJ Wood Idemp otents in bicategories

Bul letin of the Australian Math Soc

R Rosebrugh and R J Wo o d Constructive complete distributivityIV

to app ear

R Street Notions of top os Bul letin of the Australian Math Society

R Street Unpublished manuscript

R Street and R F C Walters Yoneda structures on categories Jour

nal of Algebra

R J Wood Some remarks on total categories Journal of Algebra