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C AHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES RONALD B ROWN P ETER N ICKOLAS Exponential laws for topological categories, groupoids and groups, and mapping spaces of colimits Cahiers de topologie et géométrie différentielle catégoriques, tome 20, no 2 (1979), p. 179-198 <http://www.numdam.org/item?id=CTGDC_1979__20_2_179_0> © Andrée C. Ehresmann et les auteurs, 1979, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Vol. XX -2 (1979) CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE EXPONENTIAL LAWS FOR TOPOLOGICAL CATEGORIES, GROUPOIDS AND GROUPS, AND MAPPING SPACES OF COLIMITS by Ronald BROWN and Peter NICKOLAS INTRODUCTION. One aim of this paper is give a topological version of the well known exponential law for categories, which states that for categories A and B there is a functor category ( A, B) such that for any C, D and E , there is a natural isomorphism of categories to topological version we use as underlying category of spaces the category K of topological spaces X and k-continuous maps, by which we For our mean tinuous maps the f: functions -> F such that f a: C - Y is continuous for all con- C - X of compact Hausdorff spaces C into We give Y) of k-maps X - Y a topology with a subbase of sets a : set K (X, W(a, U ) X o f such that f a (C) C U, Y . 7hen K has an exponential of those functions above and U open in with a : C - X as law generalizing to non-Hausdorff spaces the exponential law of [4] , (The corresponding category of k-spaces and continuous maps is well known - see for example [10] - but gives less precise results than ours.) Our objects must in K, prove version of topological and this (D, E) prove that the a brings k-category structure it is convenient to (1) is for us to our maps of k-categories, second if D and E (D, E) are have these maps defined that expository point are. This k-continuous, as is, category that we means we must and to do this induced maps. We there- Ehresmann [2] involving the double k-category 0 E of commuting squares of E , with its horizontal and vertical compositions 00 and B . M. and MI"’e Ehresmann have pointed out to us that our Theorem 2 is very close to a special case of Proposition 4.10 fore use methods of A. and C. 179 L2j, of in the (or groupoid ) objects cartesian closed category is itself cartesian closed. However a that which proves that the category of category of this exposition simplicity and utility our special case and of its applications we hope will show of these results and methods. exponential isomorphism ( 1 ) is also valid for k-groupoids C, D, E and hence also for k-groups (though (D, E) is of course a groupoid and not a group). Our main application of the exponential law is to the space M (D, E) of morphisms of k-groups with the compact-open topology. We prove that if D is a colimit I£m Dx of k-groups, then the natural bijection The is if D is k-homeomorphism. Also, k-group and E is any k-group, is a a k-homeomorphism. ducts and free k-groups. In Section 3 the map 4J As of (3) is we a a consequence we are prove that if D is homeomorphism. previous sections, U D given in [5]. For further references and k-groupoid, U D groupoids categories bibliography of [6]. we is its universal then the natural map These results than that of of a the on free k-pro- in Section 2. a Hausdorff kw -groupoid, then The and makes to obtain results theory approach here is more direct use of the explicit construction and refer the reader applications to of topological the 80 papers listed in the 1. AN EXPONENTIAL LAW FOR k-CATEGORlES. object in this section is to set up an exponential law for topological categories. To do this we need to start with a cartesian closed category of topological spaces. A number of such categories are available, Our but for purposes it is convenient to use a modification of the k-continuous maps of [4] to allow for non-Hausdorff spaces. our 180 say Let X and Y be topological spaces and f : X- Y a function. "We is k-continuous if for all compact Hausdorff spaces C and conti- f nuous maps composite f a : C - Y C - X the a : spaces and functions form K(X, Y ) is the maps X - of k-continuous set we denote by K, so that Y . The usual product and give the product and sum in K . Although the analogous k-spaces and continuous maps has been considered by a number of spaces sum theory of of writers, ces category which a is continuous. These to the reader is warned that the literature contains many referen- and «k-continuous functions* defined in k-spaces » senses dif- ferent from those used here. There is k ( X) k: K - K where, for X a topological space, topology with respect to all continuous maps a : C - X Hausdorff. Then k (X) is an identification space of a functor a has the final for C compact and locally compact space ing for each non-open aA : f: CA , X set such that the A of k(X) al (A) is now define sub-base of open a sets continuous map from a sets CA obtained compact Hausdorff a open in not compact-open the of spaces sum X - Y is k-continuous if and only if We a namely, - CA CA by choosand a map ( cf. [10] ). A function f: k ( X ) - Y is continuous. topology on K (X , Y) by taking W(a, U) for U open in Y and compact Hausdorff space C to X ; here a: as C - X W(a, U) maps f: X - Y such that f a( C ) C U . (This [10].) The exponential law stated in [4] for consists of the k-continuous topology is considered in Hausdorff spaces and extended tion 5.6, can now be stated for all spaces (1.1 ) Exponential is well-defined, is a The function are k-continuous, non-Hausdorff spaces in to some law in K . bijection The and is a space K(X, Y) as follows: exponential correspondence homeomorphism. is functorial in the sense that if f *:K( X, Y ) -K (W, Y) is g* : K ( X , Y ) -K ( X , Z ) is k-con- then the induced function continuous and the induced function [3], Sec- 484 tinuous. is continuous continuous in the are g* Further, if g also is g* . (The so circumstances given are proofs easy; that g* is al- ways k-continuous follows from the exponential law (1.1) as in is also easy to prove that if g: Y - Z is a homeomorphism into, is g*: f* , that [4]. ) It so also K(X, Y > - K(X, Z). proved in [4] is that if f : W- X is a k-identificaf : k( W ) - X is an identification map), then Another result tion map (that is a It then the and U c on that, sets W(a, U) for a form is injective and its inverse is k- if a: 11 is a sub-base for the open of Y, C - X continuous, C compact Hausdorff sub-base for the open a sets sets of K (X, Y) . From this we standard way : natural map homeomorphism. Suppose further that Y X Z is and Y X Z has its ( 1.3 ) The a is, f * its domain). be shown ( 1 ,2 ) The is into (that can ’U , deduce in a if k-homeomorphism continuous is is, topology as a a pull-back subspace as in the of Y X Z . Then diagram we have: natural map homeomorphism where the latter space is the pull-back of g* and h* . objective now might be to extend the exponential law (1.1) to the cases of topological categories and groupoids. Since the morphisms are only to be k-continuous rather than continuous, however it seems reasonable to deal instead with k-categories and k-groupoids, in which the Our 182 structure To fix the of a set C of k-continuous. are maps notation, arrows and the of composition pairs ( p , q) C k-category and a’, a, u spaces if in addition it is A a respectively), C - the usual axioms for is such a set C of groupoid category. It is usual a arrows. category C in which C and and m are k-continuous. morphism f: with functions objects, together map) C , where C X C is the subset of C X C, a p = a’ q , and where m ( p , q ) is written p q ; X confuse the category with the A (small) category consists and satisfy must of a final and initial maps, unit such that these functions to m : O b (C) a set a’, a : C- Ob(C) (the u:Ob (C)->C (the first recall that we Further, C p-1 and the inverse map p- C - D of k-categories is Ob (C) a are k-groupoid is k-continuous. consists of a pair of k-con- tinuous functions which commute is written with the category M ( C, D). This set can structure. The set be identified with a of these subset morphisms of K ( C, D),, the space of k-continuous functions between the spaces of arrows, and give M(C, D) We object now the compact-open wish of k-category ( C, D) having M ( C, D) space and with k-continuous natural transformations commuting of topology. to construct a this purpose it is convenient arrows squares in D to to [2] defining subspace of D4 follow be the of D such that p q, rs are defined and arrow space of two 183 as arrows. as For first the space 0 D of quadruples equal. This k-categories with object space D . the horizontal category In D , has initial and final maps the we space is One of these, respectively, and unit map composition The vertical category D has initial and final maps B unit map and vertical composition let X be the domain of the composition Eel , with its topology subspace of DDxDD. Then X is the arrow space of a k-category Now as a with object space D X D and with composition composition E3: X - D D is then a morphism of k-categories. effect, we are expressing the statement that o D is a double k-category. The vertical In PROPOSITION 1. For with o ture maps induced bject k-categories C, space M (C, by D) and arrow D there is space M ( ture 13 D The on initial, m k-category (C,D) D), and with struc- the vertical category structure on 0 D . I f D is a C, D) with the induced inverse map. Finally, if D groupoid, so also is ( is a topological category or topological groupoid, P ROOF. C, a so also is ( C, D). final and unit maps of the vertical category struc- D D are morphisms between the horizontal category 03D and 184 D itself. They therefore induce k-continuous maps between M ( C, D) , and these we take as ( C , D) . The composition in ( C , D) is and the M ( C, m D) initial, final and unit maps of the k-continuous map homeomorphism given by (1.3). The axioms for a k-category are easily verified. Similarly, if D is a k-groupoid the inverse map on B D is a morphism for (11 D , and induces a k-continuous map where is the a making ( C , D) If the ( C, D) . a k-groupoid. functions of D structure This proves are continuous so also are those of Proposition 1. T H EO R EM 2 ( T he exponential law for k-categories). I f C , D and E, k-categories, there is a natural isomorphism of k-categories which is continuous and has k-continuous inverse. logical category, then O -1 are Further, if E is a topo- is continuous. complete proof of the theorem is quite lengthy, and we therefore omit a number of straightforward details. For k-categories A, B an application of (1.2) shows that we may regard the space (A , B) as a subspace of K ( A , B)4. In particular, we can regard ( C, (D, E)) as a subspace of K(C, K(D, E)4)4 and hence P ROOF. A of K(C, K(D, E))16 . To define C X D - mE, so 9, , note that that for each f of ( C X D, E ) is (c,d):(x,y)-(w,z) morphism : an arrow a arrow of C X D we may write Note that q, of f with morphisms C X D - E since they are the composites d’D: m E , E respectively. We require that 0 f be a mor- r are dD 185 phism C - D( D, E) , so for any given commuting square in (D , E ) where fined, for d:y-z an arrow of D , by a Certainly each ki(c)(d) c : x - w the of C define ki ( c ) is in DE since q, in r are M ( D , D E) morphisms de- and are f maps into 0 E . To show that (say) k2 ( c)E M (D, D E) we note that the four components of k2 ( c) are k-continuous ( since q is k-continuous) and so k2 (c) is k-continuous. Also, if d d, d2 in D , then we have = and so As for the verifications that a s required. to lVl (D, ill E ), for that for k3 (c) k1 (c), k4 ( c), though Again, the proofs that a is similar little k1 (c), k3 ( c), k4(c) belong to that for k2(c) , while those different, are Of E(C, (D, E)) ; equally straightforward. and that (C X D, E) , are routine though rather lengthy, and we To check continuity (rather than just k-continuity) of 0 , in omit details. note that by (1.2 ) the map of (1.4) embeds M(CXD, coE) as a 186 subspace of K(CXD, E)4 ; then E) , ma pinto K ( c, K ( D, E )) 16 , can be written as a where the four components of each on C are the ( u d’)* , ( u d)* D . Since each or of the exponen- composites of (1.1) and the maps on K(C, K(D, E)) tial map 0 ua Bi O1 x 02 x 03 x 04 as is by u a’ or continuity induced continuous, the of 0 follows. To construct 4J = S -1 , first define TT, o , y : D E - E to be the maps which take respectively. Thus and 7T is k-continuous. Now c : x - w in ( D, E), It is C, of write given g (c) and define for straightforward Q to are the initial and final maps of m E and y an arrow g as a commuting is a ( C, (D, E)) and an arrow in C check that (D is the inverse of 0 . The topological 7r, Q, an arrow square ( c, d ) : (x, y) - ( w, z) of 4Y follows from the continuity of and if E of O-1 X D : k-continuity k-continuity of y ; continuous, and hence so and the category, then y is is 4). COROLL ARY 1. tion For any k-categories C, D, a natural bijec- M(C, (D, E))- M(D, (C, E)). From Theorem 2 and the facts about we E there is groupoids in Proposition 1, deduce: COROLLARY 2 k-groupoids, (The exponential law for k-groupoids). 1 f C, D , E there is a natural are isomorphism of k-groupoids which is continuous and has k-continuous inverse. Further, 187 if E is a to- pological groupoid, then 9 is a homeomorphism. particular, we obtain an exponential law for the case when C , D, E are topological groups, although of course (D, E) remains a topological groupoid and not a group. In isomorphism 0 of Theorem 2 and its kcontinuity could be proved a little more easily by first constructing a natural bijection Jl (C X D, E)- M ( C, ( D, E)) and applying a standard argument using associativity of the product. Such an argument does not easily give continuity of 0 or (for the case when E is a topological groupREMARK. oid) of The existence of the 0-1 . 2. APPLICATIONS TO COLIMITS. application is to determine up to k-homeornorphism the space M (D, E) , where E is a k-group and D is a colimit of k-groups. The use of k-groupoids rather than just k-groups, however, is not orrly required by our method of proof, but has the advantage of easily giving results on free k-groups. By generalising still further to k-categories i+e obtain results on k-monoids as well as k-groups. Our main kind of The existence of arbicrary colimits of k-categories, k-proupoids that for the topological proved k-groups case of [6]. However, the results of this section do not require knowledge of the co-completeness of these categories. may be or in a manner similar to l’e first need: functor Ob from the category of k-categories, or category of k-groupoids, to the category K preserves limits and PROPOSITION 3. from the The colimits. PROOF. There are functors P, T , respectively left and right adjoints to Ob - namely: point-like functor P : X - X (where the topological groupoid X object space X and only identity arrows ), and the tree functor T : X - X X X ( where the topological groupoid the has 188 X X X has object space X , arrow d’, space X X X and are the projec- tions). Hence Ob preserves limits and colimits. THEOREM 4. Let D, E be k-categories and lim DB of a diagram of k-categories DX. suppose that D is a colimit Then the natural map x isomorphism of categories, is continuous and has k-continuous verse. The analogous ’result also holds for colimits of k-groupoids. is an P ROOF. The k-continuous continuous morphisms To prove that O is morphisms DB - D given by (D, E) - (DB, E) and hence of categories an isomorphism Corollary 1 to Theorem 2, k-category C natural bijections verse we use for any a in- the colimit induce continuous morphism with k-continuous in- and standard arguments, to obtain The result follows from the Yoneda Lemma. Theorem 4 and Proposition 3 COROLL ARY 1. Let the k-category now yield: D be a colimit lim DB of diagram a A o f k-categories DÀ. is a Then for any k-category bijection with k-continuous for colimits of k-groupoids. continuous also holds For our next topological category Let D be a E the natural map inverse. The corollary, we rephrase the definition [6] for the case of k-categories. k-category, and or: 189 analogous result of the universal Ob(D)- Y a k-continuous func- tion. The universal category of k-category Va ( D) is k-categories as in the diagram cation map, then a pushout Y is a in the k-identifi- injection whose inverse is k-continuous on its domain, the set of morphisms f: D - E for which Ob (f) factors k-con- tinously through a. PROOF. Since o is a be the continuous which is is to 1 f D is a k-category, and a: Ob(D)for any k-category E the induced map COROLLARY 2. ; defined continuous a k-identification map, with k-continuous inverse. Let V be the injection pull- back of o * and projection V- M (D, E) is also injection with kcontinuous inverse. Since Q is the composite of this projection with the natural map b : M(Uo(D), E)- V , the result follows from Corollary 1. Then the Bt1hen the above space Y is called the universal k-monoid COROLLARY 3. induced by If D is morphism continuous singleton, and is denoted k-category a the universal of D , a a and E D - a U(D) U a (D) is a k-monoid, by U ( D ). k-monoid, then the is a continuous map bijection with k-continuous inverse. This follows immediately COROLLARY 4. Let X be a from Corollary 2. space and i: X - 190 F+(X) the canonical map from X the to free k-monoid on X . Then for any k-monoid E the induced map is a continuous with k-continuous inverse. bijection point-like category on X (cf. the proof of Proposition 3), and 2 denotes the topological category with two objects 0, 1 , one non-identity arrow 0 - 1 , and the discrete topologies on objects and arrows, then the composite If X denotes the PROOF. injection x - (x, 0 ) k-monoid i : X - F+ (X). So of the and the universal the result follows morphism defines from Corollary 3. the free k-groupoid version of Corollary 1 to deduce some on free k-groups and free k-products of k-groups. These results sharpened for kw-groups in Section 3. We results will be by G is If COROLLARY 5. induced the use now a the universal k-groupoid morphism and K is G - U ( G) a is k-group, a then the map continuous bijection with k-continuous invers e. This follows from of k-groupoids If is a as is B GÀ of the COROLLARY 6. If a family of in the category the universal group union II since the colimit k-categories k-groupoid (the analogous topological result is in [6]). {GB }BE A coproduct G = * GÀ Corollary 3 U(G) of the k-groups, their free k-product is the of k-groups. This k-group can be taken k-groupoid ë, a continuous which is the disjoint k-groups Gx . {GB IXCA is family of k-groups, a K the natural map is as bijection with k-continuous inverse. 191 then for any k-group PROOF. This follows from is easily proved to Suppose K It can such that i since the natural map homeomorphism. a that X is pointed space. The Graev free k-group a k-group FGK(X) and a pointed map i : X - FGK(X) is universal for pointed maps in K from X to k-groups. now on X consists of in be Corollary 5 be shown that if e a is the base point of X, i may be taken as the composite of the injection j: xI (x, e) of X into the tree topological groupoid X X X with the universal morphism X X X - U(X X X) (cf. [6]). COROLLARY 7. For any space X and k-group K the map of pointed maps X,K, with i * induced by i : continuous bijection with k-continuous inverse. into the space is a P ROOF. This follows from is a continuous Y where is a from Y ( f): bijection once we have with k-continuous inverse. proved F G K ( X), that Now j * has inverse morphism (x, y)- f (x) f (y)-1 . So j* clearly continuous. The k-continuity of Vf comes composite XXX- K is the bijection, and is regarding it as the where m * Corollary 5 X - is induced by the map abelian m ’: (a, b)I a b-1 of K X K into K . pointed space X consists of an abelian k-group A GK ( X) and pointed m ap j : X - AGK(X) in K such that j is universal for pointed maps in K from X to abelian k-groups. The map p: FGK(X)- AGK(X) is a k-identification map. The Graev free k-group on a X , Y be paracompact, Hausdorff pointed spaces such that AGK(X ), AGK(Y) are isomorphic k-groups. Then, for any abelian group 17 and n > 0 , the Cech reduced cohomology groups COROLLARY 8. Let 192 are isomorphic. K be PROOF. Let abelian an k-group. The induced map k-homeomorphism (since p is a k-identification map). So by Corollary 7 a k-isomorphism AGK( X) - AGK(Y) induces a k-isomorphism is a bijection of pointed homotopy classes [X, K] -[Y, K] . Let K be the Filenberg-Mac Lane k-group K(TT, n). By [8] (cf. also [1] chapter 6) the abelian group [ X, K] is isomorphic to Hn(X; 7T ) , and hence a and the result follows. E XAMPL E. and C is is Let X be the «Cech circle >> - that is, X an arc isomorphic to in R2B( A UB) joining (0, 0 ) to AuBuC, (1, 0 ) . Z , and it follows that AGK (X) is AGK([0 ,1]) . Similarly AGK(S1) = not Then where H1 (X; k-isomorphic Z) to k-isomorphic to AGK ([0,1]). However this method does not distinguish between AGK(X) and AGK(S1) and we do not know if these are k-isomorphic. is not 3. THE UNiVERSAL TOPOLOGICAL GROUP OF A HAUSDORFF k(ù -G ROU P. In this section we consider only topological groupoids and topolo- gical groups. Recall also that a kw -space X is one which has the weak topology with respect to some increasing sequence {Xn} I of compact subspaces with union X . A Hausdorff k-continuous map X - k,,-space X has the property that any Y is continuous. topological groupoid which as a topological space is a k,)-space is called a kw -groupoid. It is proved in [5] that if G is a Hausdorf f kwgroupoid, then its universal topological group U(G) is a Hausdorff kw group. We shall use the explicit description of the topology of U (G) given in [5] to prove : A 193 G be THEOREM 5. Let Hausdorff k(ù-groupoid a and K topological a group. Then the map induced by the universal i * is PROOF. It is clear that i : G - U ( G) is morphism a homeomorphism. bijection and, as an induced tinuous, and we need only prove the continuity of ( i *)-1 . Let f: U(G)- K be a morphism and let W(A, U) pact in U ( G) be Let a a sub-basic open neighbourhood of f, so map, is that A is con- com- and U is open in K . Since G is a kw -groupoid, { Gn} it has the weak of compact topology subspaces. For with respect to each pair of integers m , n > 0 there is a continuous map p : (Gn)m-U(G) sending each m-tuple of elements of Gn to its reduced form in U ( G) ([5], page 432). The methods of [5] also show that the compact set A is contained an increasing in some sequence p ( Gn)m, and the definition of multiplication in U (G) is such that Setting V f -1 (U ), = put , so that N is Hausdorff, an each open x set. is compact Since in C has a compact neighbourhood contained in N . Since C is closed and hence compact in finite is a we have set Ux m g ( Cxj) there and F in C such that Since U is open and each ..., (Gn)m is compact, there in K such that 194 are open sets Ux1,... Let W j = 1, ... , m , and take so we This F . Then x E and we prove that have completes the proof From Theorem 5 of gE W , for hc W, and set h *= ( i *)-1 ( h) : we must prove h *( A)C U . If y E A , then y = p ( c) for some c E C , and then CE C’ for an x E F, Let we sets W ( Cxj, Uxj), be the intersection of the sub-basic open kw -groups, and on we can of ( i*) -1 . deduce results topological on countable free groups applications, namely products k,)-spaces [7], on Theorem 4. Rather than detail to different line of a continuity Graev free in Corollaries 6 and 7 pursue of the these, abelian to we as shall topological groups. {GB}JEA COROLL ARY 1. If dorff kw-groups, and G = E is a countable collection o f abelian, Hauscategory of topological GÀ is their sum in the a Hausdorff kw-group, and for any abelian groups, then G is logical group K the natural map is a topological isomorphism. PROOF. Clearly G group and k,)-spaces is a abelian topo- so is a is the Hausdorff is compact quotient But any kw-space. covering, homeomorphism, by G’= * GB of and the footnote so to of a kw-groups is naturally isomorphic to the sub- the induced map countable 195 commutator map of Hausdorff quotient Proposition 3.5 The dual group COROLLARY 2. its by sum of [4]. of abelian Hausdorff pro duc t of their duals. This is the case K for countable also of sums = S’ of Corollary 1. locally compact groups are Similar results due to to this Kaplan [9] (see [11] ). Our are two remaining results such free groups group is are on ([14] and free abelian [7]). continuous map i : X - topological groups. There The Markov free abelian topo- AM (X) into topological abelian group such that i is universal for maps of X into topological abelian groups. The Graev free abelian topological group is a pointed continuous m ap j : X - AGr(X) universal for pointed maps into topological abelian groups. If X is a Hasudorff kw -space, then so also are AM(X) and AG(X) [12]. logical If a X, Y are spaces, then C (X , Y) a will denote the space of con- tinuous functions X - Y with the compact open topology. If X, Y are pointed spaces, then C-(X, Y) is the subspace of C ( X, Y) of pointed maps. COROLLARY 3. If X is pointed Hausdorff koi-space, and K is an abel- group, then the maps ian topological- are homeomorphisms. P ROOF. The a proof is similar to that of quotients by closed subgroups, gical group U (X X 2’ ) , where 2’ is are Corollary 2, and the FM(X) tree since the maps is the universal groupoid on suggests a topolo- the discrete space {0,1}. for i: EXAMPLE. Corollary 3 examples Theorem 11 of to X - AM(X) [15], which the spaces X, Y and the states class of counter- that under mild conditions topological abelian group K , an algebraic isomorphism between C(X, K) and C( Y, K) which preserves constant functions induces a homeomorphism between X and Y . Let EX : AM(X ) - Z ( w ith Z the discrete group of integers) be the homomorphism determined on 196 by the constant map X - Z with value 1 . Let 4J : AM( X) - AM( Y) topological isomorphism such that 6y$ = EX. Then the composite is be a algebraic isomorphism preserving constant functions, and is indeed homeomorphism, by Corollary 3, if X and Y are k. -spaces. However, it is easy to give examples of non-homeomorphic X , Y for which such a (D exists (cf. [7 , Section 5) and X, Y and K satisfy the conditions of L 151 ( see the erratum to [13] ). an a School of Mathematics and Computer Science, University College of North Wales, BANGOR LL57 2UW, Gwynedd, U.K. and Department of Mathematics, University of Queensland, ST. LUCIA, Queensland 4067, AUSTRALIA 197 REFERENCES. 1. BARRATT, M.G., Track groups II, Proc. London Math. Soc. (3) 5 ( 285-329. 1955 ), 2. BASTIANI-EHRESMANN, A. and EHRESMANN, C., Catégories de foncteurs structurés, Cahiers Topo. et Géo. Diff. XI ( 1969), 329-384. 3. BROWN, R., Elements of modem Topology, McGraw Hill (Maidenhead), 1968. 4. BROWN, R., Function spaces and product topologies, Quart. J. Math. Oxford ( 2) 15(1964), 238- 250. 5. BROWN, R. and HARDY, J.P.L., Subgroups of free topological groups and free products of topological groups, J. London Math. Soc. ( 2) 10 (1975), 431440. 6. BROWN, R. and HARDY, J.P.L., Topological groupoids I: Universal morphisms, Math. 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