aa r X i v : . [ m a t h . L O ] M a y BROUWER FIXED POINT THEOREM AS ACOROLLARY OF LAWVERE
RUPERT M c CALLUM
Abstract.
It is investigated in what sense the Brouwer fixedpoint theorem may be viewed as a corollary of the Lawvere fixedpoint theorem. A suitable generalisation of the Lawvere fixed pointtheorem is found and a means is identified by which the Brouwerfixed point theorem can be shown to be a corollary, once an ap-propriate continuous surjective mapping A ′ → X A ′′ has been con-structed for each space X in a certain class of “nice” spaces foreach one of which the exponential topology on X A ′′ exists, andhere A ′ and A ′′ have the same carrier set and the topology on A ′ is finer than on A ′′ . It is shown that there is a certain natural wayof attempting to derive Brouwer as a corollary of Lawvere whichis not possible, namely that is there is no space A for which theexponential topology on [0 , A exists and there is a continuoussurjection A → [0 , A . We then examine the range of contextsin which phenomena like those described in the first result occur,from a broadly model-theoretic perspective, with a view towardsapplications for the original motivation for the problem as a prob-lem in decision theory for AI systems, suggested by the MachineIntelligence Research Institute. The Brouwer fixed point theorem, whose first published proof was in[1], states that if D is a closed ball in a finite-dimensional Euclideanspace then every continuous mapping D → D has a fixed point. TheLawvere fixed point theorem first appeared in [2]. Let us recall thestatement of the Lawvere fixed point theorem and its various applica-tions. The theorem states that, in a Cartesian closed category in whichthere is a point-surjective morphism A → X A , every endomorphism of X has a fixed point. This is in some sense “the essence of” all diag-onal arguments, having as corollaries Cantor’s theorem, the diagonallemma which is used in the proof of G¨odel’s theorem, unsolvabilityof the halting problem, and existence of a program in any computerlanguage which outputs its own source code.It is of interest to know whether the Brouwer fixed point theorem intopology can be recovered as a special case of the Lawvere fixed point c CALLUM theorem; the purpose of this note is to examine various senses in whichthis is and is not the case.One might first naturally ask whether there exists some space A , andsome class of spaces including every closed ball in a finite-dimensionalEuclidean space, with the property that, if X is a space in the class,then the exponential topology on X A exists and there is a continuoussurjection A → X A . We shall later show that it is provable in ZFCthat this is not the case even if we just restrict to X = [0 , A ′ and A ′′ could be identified, with the samecarrier set and the topology on A ′ finer than on A ′′ , that is to saythere is a point-bijective morphism h : A ′ → A ′′ . Suppose that someclass of spaces could be found, including every closed ball in a finite-dimensional Euclidean space, with the following properties. If X is aspace in the class, then the exponential topology on X A ′′ exists andthere is a continuous surjection g : A ′ → X A ′′ , such that, if U ⊆ X is open, and eval: X A ′′ × A ′′ → X is the evaluation map, and π isthe projection from X A ′′ × A ′′ onto the first factor, then if we let V = g − ( π (eval − ( U ))), then both V is open in A ′ and h ( V ) is open in A ′′ . If such a surjection can be found for every space in our class, thenthe proof of the Lawvere fixed point theorem applies to show that ev-ery continuous endomorphism of every space X in the class has a fixedpoint and so the Brouwer fixed point theorem is recovered as a specialcase. This shall be our strategy in the next two sections, where we willindeed recover Brouwer as a corollary of an appropriate generalisationof Lawvere in this way. We must begin by defining the appropriateclass of spaces.1. Defining the appropriate class of topological spaces
Let us begin by stating the appropriate generalisation of Lawvere; wemust translate the previously given discussion into category-theoreticterms.
Theorem 1.1.
Suppose that C is a category with a terminal object andclosed under products, and S is a class of objects of C . Suppose that A ′ and A ′′ are objects of C with a point-bijective morphism h : A ′ → A ′′ . Suppose that, for every X ∈ S , the exponential object X A ′′ exists(exponential object relative to some full Cartesian-closed subcategoryof C , held fixed throughout). Suppose that for each such X there is apoint-surjective morphism g : A ′ → X A ′′ , with the property that everymorphism A ′ → X which factors through g × h : A ′ → X A ′′ × A ′′ also ROUWER FIXED POINT THEOREM AS A COROLLARY OF LAWVERE 3 factors through h : A ′ → A ′′ . Then every object X is such that everymorphism from X to itself has a fixed point.Proof. Same proof as proof of standard Lawvere fixed point theorem. (cid:3)
In order to recover the Brouwer fixed point theorem as a special caseof this generalisation of Lawvere, we must define the class of objects S , and the objects A ′ and A ′′ for which we intend to apply it.Let ω denote the first uncountable ordinal, and consider the completeinfinite binary tree of height ω , viewed as a generalised Cantor space.So this means that the carrier set is the set of all strings of length ω ofsymbols from the set { , } , with the topology for which the basic opensets are sets consisting of all the strings starting with a fixed string ofcountable length as an initial fragment. This space will be denoted by A ′ . The space with the same carrier set, with the product topologyobtained when the space is viewed as a product of ℵ many discretetwo-element spaces in the natural way, will be denoted by A ′′ . Nowconsider the class S of all spaces X satisfying the following conditions. Definition 1.2.
A topological space X is said to be nice and a memberof S if the following hold.First, the space X is compact and contractible and is the image ofthe generalised Cantor space described above under a continuous sur-jection. Secondly, the space X is the disjoint union of an open densesubset U and the complement V , and both U and V admit a “homoge-nous” metric, where what we mean by this is as follows. Firstly, withregard to U , there exists some ǫ >
0, with the property that, for all δ such that 0 < δ < ǫ , every pair of distinct open balls of radius δ centredat a point in U such that the closure of the balls does not intersect V ,has the property that the balls in the pair are isometric. Then, withregard to V , we require that sufficiently small open balls in X , centredat points of V , of the same radius, are isometric.Clearly the class S of nice spaces so defined includes all closed ballsin finite-dimensional Euclidean spaces. Since A ′′ and all spaces in S are all k -spaces the existence of all the needed exponential topologiesis clear (as the category of k -spaces is in fact a full Cartesian-closedsubcategory of Top).In an earlier version of this argument, we thought that the condition ofcontractibility would end up playing an essential role in the proof. Itnow appears that the condition involving the existence of a certain type M c CALLUM of metric is really the key consideration, and most likely the conditionof contractibility follows from this anyway, and is therefore redundant.In any case the inclusion of the condition of contractibility can bedispensed with in what follows; no essential use of that condition willbe made.In addition, we must show that there is a continuous surjection A ′ → X A ′′ for every nice space X , satisfying the requirements given in thestatement of the generalised Lawvere fixed point theorem given above.It will then follow by the generalised Lawvere fixed point theorem thatevery nice space X is such that every continuous function X → X has a fixed point, including all closed balls in finite-dimensional Eu-clidean spaces as a special case, thereby completing the first part ofour argument. We can now state our main theorem with regard to theapplication of the generalisation of Lawvere given above.2. Brouwer as a direct corollary of a generalisation ofLawvere
Theorem 2.1.
Let A ′ and A ′′ be as in the previous section, supposethat X is a nice topological space. Then there is a continuous surjection A ′ → X A ′′ , where X A ′′ has the exponential topology, satisfying therequirements given in the statement of the generalised Lawvere fixedpoint theorem stated in the previous section. As a corollary of this,the Brouwer fixed point theorem can be recovered as a corollary of thegeneralised Lawvere fixed point theorem.Proof. We must construct the continuous surjection g : A ′ → X A ′′ on the stated hypotheses. Suppose that x ∈ A ′ ; note that x can bethought of as a bit-string of length ω , and we must describe how tochoose g ( x ). Let us begin by making some observations about how onemight code for an element of X A ′′ .It is evident that given that X is a nice space, the space X has thecardinality of the continuum. It is also possible to construct an ω -sequence T := { C n : n ∈ ω } of coverings of X by finitely many openballs, each covering C n ∈ T being such that it can be partitioned intotwo collections of open balls with each collection having the propertythat all of the balls in it are pairwise isometric, and also such that themesh of the covering C n tends towards zero as n goes to infinity. Nowsuppose that we have a mapping which sends each point of a countablesubset C ⊆ A ′′ to a centre of some open ball appearing in some C n ∈ T ,with every centre of every such open ball appearing in the range of the ROUWER FIXED POINT THEOREM AS A COROLLARY OF LAWVERE 5 mapping. If an extension of this mapping to a continuous element of X A ′′ exists, there will be some countable ordinal α , with the propertythat, given any bit-string of length α and considering the set of allelements of A ′′ which have this bit-string as an initial fragment, theextension in question will be constant on this set.So we see from this that a coding scheme can be constructed wherebyevery element of X A ′′ can be coded for by a countable bit-string (notnecessarily unique given the initial choice of element of X A ′′ ). Firstwe describe the general form that the data to be used in this codingwill take, where again, data of this form can always be constructedfor any given element of X A ′′ , but not uniquely. Consider a map θ whose domain B is a countable collection B of countable bit-strings,closed under taking initial fragments, and such that every element of2 ω has a fragment which is the union of a branch of B , and whoseco-domain is X . We can further require that in the case of bit-stringsof zero or successor length, the value of the map at these bit-stringsis always a centre of an open ball from some C n ∈ T . Next, use theaxiom of choice to construct a function ρ defined on the set of allcountable limits of limit ordinals, whose value at each such ordinal α is an ω -sequence of limit ordinals cofinal in α . We can now alsomake the further requirement that given any branch in B , the trace ofthe mapping θ along this branch, a well-ordered countable sequence ofelements of X is “generalised Cauchy of degree n ” for some positiveinteger n which is the same for all branches. What this means is that,for each fragment of the branch of limit length, we obtain an ω -sequenceof ordinals cofinal in the length of the fragment, either in the obviousway, if it is a limit ordinal which is not a limit of limit ordinals, or via thepreviously constructed function ρ otherwise, and then we require thatthe trace of the mapping θ along the nodes of the fragment of the branchof B indexed by this ω -sequence of ordinals, is Cauchy relative to themetric on X which we have been holding fixed throughout, and withthe speed of convergence having a uniform lower bound determined by n , say for example that if m, m ′ ≥ k and x m , x ′ m ∈ X are the elementsof X corresponding to the m -th and n -th ordinals in the sequence then d ( x m , x ′ m ) < n + k where d is the metric on X . (We are requiring n to be positive for the moment, but shall later need to generalise tosituations where we have the same criterion with n allowed to be zeroor negative.) Let E be the set of all tuples ( B, θ, n ) where
B, θ and n are as described above. We now wish to describe a coding schemewhereby every such tuple can be coded for by a countable well-orderedbit-string. M c CALLUM
The integer n can clearly be coded for by a finite bit-string. Eachpoint of X which is a centre of one of the open balls appearing in oneof the coverings C n can be coded for by a finite bit-string according tosome fixed coding scheme, and in the case where we are dealing withan arbitrary point of X , our requirements entail that there is no needto include a code for this point of X , since it will be possible to infer itfrom data occurring earlier in the bit-string that is coding for our tuple( B, θ, n ). The original sequence T = { C n : n ∈ ω } of coverings can bechosen in such a way that it is indeed always possibile to be able tosatisfy the “generalised Cauchy criterion” at each limit stage of eachbranch of B ; this is a consequence of the homogeneity assumptions wemade on the metric.Thus, our coding scheme will be such that under the coding scheme acountable well-ordered bit-string codes for a positive integer n togetherwith a map θ from a countable collection B of countable well-orderedbit-strings into points of X with the constraints described before, wherethe only points of X that actually need coding can be coded for by finitebit-strings.The exact details of the coding scheme are not all that important butwe shall introduce a couple of extra requirements on it later on whichwill be easily seen to be possible to fulfil. Firstly, we will want torequire that the set D of countable well-ordered bit-strings which canserve as codes in the coding scheme is such that no two countablebit-strings in D are such that one is a fragment of the other, everybit-string of length ω has a bit-string in D as a fragment, and everybit-string in D is infinite. It is clear that this requirement can befulfilled. Then our energies will be occupied with showing that everyelement of E does code for an element of X A ′′ , and at that point it willbe clear that a continuous surjection A ′ → X A ′′ can be constructedfrom the coding scheme in a natural way (namely, given an element of A ′ which is a bit-string of length ω , find the unique element b ∈ D which is a fragment of it, and map the element of A ′ to the elementof X A ′′ corresponding to the element of E which is coded for by b ).We shall want the coding scheme to be constructed not only in sucha way as to satisfy the previously given requirements, but also in sucha way that the surjection constructed from the coding scheme in theobvious way meets the requirements stated in the statement of thegeneralised Lawvere fixed point theorem, relative to the space X . Thatis to say, if one takes an open subset of X , then takes the pre-image in X A ′′ × A ′′ under the evaluation map, the projection of that onto the first ROUWER FIXED POINT THEOREM AS A COROLLARY OF LAWVERE 7 factor, and then the pre-image of that under the surjection A ′ → X A ′′ ,which one views as both a subset of A ′ and A ′′ , this set must be openrelative to both topologies. In the case of A ′ , the basic open sets for thetopology are countable intersections of sets whose defining condition isgiven by specifying the value of just one bit at the α -th point in thesequence for some countable ordinal α , whereas in the case of A ′′ , thebasic open sets are finite intersections of such sets. We must requirethat the set we obtain be open in the latter topology (the former onebeing finer). Once we have established that every element of E doesindeed code for an element of X A ′′ , the possibility of constructing thecoding scheme in such a way that this requirement is fulfilled easilyfollows from the details already given.Suppose again that ( B, θ, n ) ∈ E . The values of the mapping θ at theends of the branches of B determine a mapping A ′′ → X in an obviousway, and clearly for every element of X A ′′ there is indeed at least oneelement of E that represents it under this coding scheme. We shallsay more presently of how we might be able to ensure that the inducedmapping A ′′ → X so defined is continuous if we are starting with anarbitrary element of E .Given any element of E , there exists a countable ordinal α such thatthe mapping θ coded for by the said element of E extends uniquely toa mapping 2 ≤ α → X , satisfying the aforementioned generalisation ofthe Cauchy criterion along each fragment of a branch of limit length.Uniquely, that is, if we introduce the extra constraint that for elementsof 2 ≤ α not in B the mapping has the same value as for the union ofthe branch of B consisting of fragments which are in B . What we nowneed to see is that, given the hypothesis of compactness of X , and the“generalised Cauchy criterion” on the mapping θ , every element of E does indeed determine an element of X A ′′ , that is the function from A ′′ into X which is thereby defined is indeed continuous.Using our chosen coding scheme, every element of E clearly gives riseto a mapping 2 ≤ α → X , for an appropriately chosen countable ordinal α , as previously noted, and proving continuity of the restriction of thisto 2 α (with the obvious product topology on 2 α ) is sufficient. Supposethat β ≤ α and consider the induced mapping 2 β → X . We mustshow by transfinite induction that this is always continuous, relativeto the generalised Cantor space topology on 2 β , given the hypothesisof compactness and contractibility on X and the generalised Cauchycriterion on θ . So, suppose that β ≤ α and the desired conclusion has M c CALLUM been established for all ordinals less than β . Clearly we only need toconsider the case where β is a limit ordinal, and we have a mapping2 ≤ β → X available, which we shall use freely in what follows, satisfyingthe generalised Cauchy criterion which we have introduced.The induction hypothesis states that the map 2 γ → X is continuousfor all γ < β , and since X is compact, this holds with a modulus ofuniform continuity depending only on γ . Here the modulus of uniformcontinuity can be a constant of Lipschitz continuity relative to anymetrics on 2 γ and X that we wish. Naturally the metric on X that wewish to employ is the one that we have been holding fixed all along,and the choice of metric on 2 γ does not matter as long as it is com-patible with the obvious product topology on 2 γ , and chosen so thatthere is a choice of Lipschitz constant which works for all γ < β . Fromthis consequence of the induction hypothesis together with our “gen-eralised Cauchy criterion”, it can be concluded that the map 2 β → X is continuous and so now our transfinite induction goes through, yield-ing the final conclusion that the mapping 2 α → X is continuous, andtherefore that the induced mapping A ′′ → X is. So this means thatour coding scheme is indeed such that every element of E does indeedgive rise to an element of X A ′′ , and conversely every element of X A ′′ arises from (at least one) element of E . Now we need to construct acoding scheme whereby elements of E can be coded for by countablewell-ordered bit-strings from an appropriate collection D , satisfying allthe previously mentioned constraints. With it now established thatevery element of E does indeed code for an element of X A ′′ there is nodifficulty in seeing that all the constraints on the coding scheme canindeed be satisfied.The existence of a surjection A ′ → X A ′′ with all of the desired prop-erties is now clear. It follows by the generalised Lawvere fixed pointtheorem that every nice space X is such that every continuous endo-morphism has a fixed point. Thus the Brouwer fixed point theoremcan be recovered as a corollary of this generalisation of Lawvere. (cid:3) Brouwer as a direct corollary of ordinary Lawvere isnot possible
It is natural to ask whether there exists a space A with the propertythat for all spaces X in an appropriate class, the exponential topologyon X A exists and there is a continuous surjection A → X A . Establish-ing this for a class that included every closed ball in a finite-dimensional ROUWER FIXED POINT THEOREM AS A COROLLARY OF LAWVERE 9
Euclidean space would give us a way of recovering the Brouwer fixedpoint theorem as a direct corollary of the ordinary Lawvere fixed pointtheorem. But in fact we shall now see how to prove in ZFC that evenjust in the case X = [0 ,
1] there is no such space.
Theorem 3.1.
It is provable in ZFC that there does not exist any topo-logical space A for which the exponential topology on [0 , A is definedand such that there is a continuous surjection g : A → [0 , A .Proof. Suppose that the space A and the surjection g exist in V anddefine B := ( A, g ). Consider the inner model M := L ( R )[ B ]. A Skolemhull argument shows that we can find a B ′ = ( A ′ , g ′ ) with the sameproperties relative to L ( R ), such that B ′ ∈ L α ( R ) for some countablelimit α . Let ρ be the function constructed earlier, restricted to α .There exists a real number r such that each continuous endomorphism f of [0 ,
1] in L [ B ′ , ρ ] is in L α [ r ]. We follow arguments given by StephenSimpson in [3] for equivalence of Brouwer’s fixed point theorem with W KL over RCA . Choose a structure S for the second-order lan-guage of arithmetic which is an ω -model and which includes a codefor every element of L α [ r ] in the domain of the number variables anditems coding for the structure L α [ r ] relative to this coding scheme ap-pear in the domain of the set variables, and such that the structureis a model for RCA but not W KL . This is possible by the model-theoretic results about RCA and W KL presented in Chapter VIII of[3]; namely, one selects the least “Turing ideal” in P ( ω ) which satisfiesthe constraints just given, and this will model RCA but not W KL .Then, following the proof of Theorem IV.7.7 in [3], construct a contin-uous endomorphism of [0 ,
1] in S which has no fixed point in L α [ r ]. Butthis is a contradiction because of the constructive nature of the proofof the Lawvere fixed point theorem. If B ′ occurs in L α [ r ] and thereforehas its transitive closure (as an ∈ -structure) fully coded for in S , and B ′ has the stated properties relative to L ( R ), then there should be nodifficulty in the proof of an existence of a fixed point in S , but in factexistence of the fixed point fails in S . (cid:3) This proof basically says that, since the Lawvere fixed point theoremis constructive, the non-constructive nature of any procedure for con-structing the fixed point in any proof of the Brouwer fixed point the-orem would have to be in some sense “entirely coded for” by the sur-jection g . This can work where g : A ′ → X A ′′ for two spaces A ′ and A ′′ with the same carrier set and different topologies, but not when A ′ and A ′′ are the same space. c CALLUM The original motivation for the problem
The Machine Intelligence Research Institute originally became inter-ested in this problem motivated by concerns in decision theory. Sup-pose that we have an agent with two actions A and B available tothem, and they must choose a probability distribution for which actionto perform which depends in a continuous way on which observationthey make from the space X of all possible observations. Then theset of policies available is [0 , X (let us assume that the exponentialtopology does indeed exist). It may be that the agent is observinganother agent in the environment so the space X of all possible ob-servations is equal to the space of all possible agents. If each agenthas a well-defined policy, then there is a mapping X → [0 , X whichwe might reasonably require to be continuous. It is of interest from adecision-theoretic point of view to know whether for some spaces X itis possible for such a continuous mapping to be surjective.We have obtained the answer “No, such spaces do not exist, but ifyou are happy with there being a finer topology on the X that occurson the left than on the X that occurs in the exponent on the right,then yes the existence of such a continuous surjection is possible”. Wewere dealing with spaces of cardinality 2 ℵ which are perhaps too bigto be plausible candidates for practical applications, so it may be ofinterest to explore whether other examples can be given of more modestcardinality. Let us attempt to survey the range of examples that canoccur, looking at it from the perspective of model theory.Suppose that S ⊆ P ( ω ) is a Turing ideal. Countable well-orderedbit-strings can be coded for by elements of P ( ω ), and a certain set A of countable well-ordered bit-strings can be coded for by elements of S . Note that the supremum of the lengths of bit-strings in A may bestrictly less than the ω of “the real world”. Consider the ω -modelfor the second-order language of arithmetic determined by S . Theset A can clearly be coded as a “definable sub-class” of the range ofthe second-order variables on this structure. The “generalised Cantorspace topology” and “compact product topology” on A are likewiseboth clearly definable by means of “iterated predicative comprehen-sion” on this structure. However, if RCA is our base theory thenwe will not go beyond being conservative over RCA by introducingiterated predicative comprehension for higher types. Clearly, some ad-ditional axiom is needed along the lines of “an uncountable subtree ofthe complete binary tree of height ω , every level of which is countable, ROUWER FIXED POINT THEOREM AS A COROLLARY OF LAWVERE 11 has an uncountable branch”. (Of course, an investigation from thepoint of view of Reverse Mathematics naturally suggests itself here.)If we introduce an axiom like that in the third-order part of the lan-guage, then the desired surjection g : A ′ → X A ′′ will be found in thedomain of the third-order variables. But of course there is no guar-antee that every real number recursively constructible from g occursin the original Turing ideal S . But if S is chosen to have the ap-propriate closure properties, then this will be the case and then anappropriate “space of agents” can be constructed from S which hasthe desired properties relative to a particular model (and also relativeto any strictly larger Turing ideal with the same closure properties,including all of P ( ω )). Relative to such a model you will be able tofind a “space of agents” (possibly countable “in the real world”) anda continuous surjection (relative to the model, and also many strictlylarger models as just indicated) from the space of agents onto the spaceof those policies with a certain upper bound on their complexity. The“space of agents” is also uniformly definable across the whole class ofmodels for which the construction works, and the bound on complexityof the bit-strings occurring in the space of agents can be strictly largerthan the bound on complexity of the ordinals which index the entriesin the bit-string. So you can indeed enlarge the space of agents so thatit has cardinality 2 ℵ and arbitrary elements of R are definable fromthe bit-strings that occur in it, but the gap between the upper boundon complexity of elements of X A ′′ and the complexity of g will stillhold.The hierarchy of complexity can be refined further, going to consider-ations of computational complexity rather than just descriptive com-plexity. We can go to a weakened version of RCA which only allowsfor elementary recursive comprehension and then substitute “Turingreducibility” with “Turing reducibility via elementary recursive func-tions”. Or for “elementary recursive” one could substitute “polynomial-time computable”. (In these cases, clearly the height of the binary tree A would become considerably shorter. I previously thought that thiskind of considerations might lead to situations where the height of thebinary tree is only a recursive ordinal but Alex Mennen has sketched aproof that this is not the case and one only obtains shortening of theheight to ω CK , thereby possibly reducing the interest of these consid-erations.) c CALLUM
So, for example, one could have a function from the space of agentsto the space of polynomial-time computable policies, but without anypossibility of organising things so that every agent is coded for by areal number that is itself polynomial-time computable, or so that thefunction from the space of agents to the space of polynomial-time com-putable policies is polynomial-time computable. This starts to sound abit more like the kind of application that would be of interest from thepoint of view of the study of artificial intelligence. The Machine Intel-ligence Research Institute may wish to explore further ramifications ofthe results presented here along the lines just suggested.