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UNIVERSAL ARROWS TO FORGETFUL FUNCTORS

arXiv:funct-an/9208001v1 20 Aug 1992UNIVERSAL ARROWS TO FORGETFUL FUNCTORSFROM CATEGORIES OF TOPOLOGICAL ALGEBRAVladimir G. PestovDepartment of MathematicsVictoria University of WellingtonP.O. Box 600Wellington, New Zealandvladimir.pestov@vuw.ac.nzAbstract.

We survey the present trends in theory of universal arrows to forgetfulfunctors from various categories of topological algebra and functional analysis tocategories of topology and topological algebra.Among them are free topologicalgroups, free locally convex spaces, free Banach-Lie algebras, and more. An accent isput on relationship of those constructions with other areas of mathematics and theirpossible applications.

A number of open problems is discussed; some of them belongto universal arrow theory, and other may become amenable to the methods of thistheory.Introduction (2)1. Major classical examples (3)2.

Structure of free topological groups (7)3. M-equivalence and dimension (11)4.

Applications to general topological groups (15)5. Free products of topological groups (17)6.

Free Banach-Lie algebras and their Lie groups (19)7. Lie-Cartan theorem (20)8.

Locally convex Lie algebras and groups (21)9. Supermathematics (23)10.

C⋆-algebras and noncommutative mathematics (25)Acknowledgments (26)Bibliography (28)1991 Mathematics Subject Classification. 18A40, 22A05, 15A75, 16B50, 16W50, 17B01, 17B35,22A30, 22E65, 43A40, 46A03, 46B04, 46H20, 46L05, 46M15, 54H11, 54A50, 54B30.Tt bAMS T X

2VLADIMIR G. PESTOVIntroductionThe concept of a universal arrow was invented by P. Samuel in 1948 [Sa] andput in connection with his investigations on free topological groups. The followingdefinition is taken from the book [MaL].Definition.

If S : D →S is a functor and c an object of C, a universal arrowfrom c to S is a pair < r, u > consisting of an object r of D and an arrow u : c →Srof C, such that to every pair < d, f > with d an object of D and f : c →Sd anarrow of C, there is a unique arrow f ′ : r →d of D with Sf ′ ◦u = f.In other words, every arrow f to S factors uniquely through the universal arrowu, as in the commutative diagramcu−−−−→SrySf ′cf−−−−→Sd□This notion bears enormous generality and strength, and at present it is an es-sential ingredient of category theory [MaL] and theory of toposes [Joh]. In fact,many mathematical constructions can be interpreted as universal arrows of one ofanother kind.

Examples are: quotient structures and substructures, products andcoproducts, including algebraic and topological tensor products, universal envelop-ing algebras, transition from a Lie algebra to a simply connected Lie group and viceversa, compactifications of all kinds (Stone-ˇCech, Bohr, and others), completions,prime spectra of rings, and much more.We are interested in the particular case where S is a forgetful functor from somecategory of topological algebra or functional analysis, D, to another category oftopological algebra or functional analysis or a category of topology, C. Historicallythe first, and studied in most detail, is the construction of the free topological group,F(X), over a topological space X, where C is the category of Tychonofftopologi-cal spaces and continuous mappings and D is the category of Hausdorfftopologicalgroups and continuous homomorphisms. A number of similar constructions have re-ceived a comprehensive treatment, among them are free Abelian topological groups,free compact groups, free locally convex spaces.

At the same time, in recent yearssimilar constructions have arisen — either explicitely or implicitely — in other ar-eas of mathematics. In some cases no attempt has been made to establish a bridgebetween those and former types of universal arrows — although seemingly such aconnection would facilitate a study of new constructions.

Among the disciplineswhere new types of universal arrows to forgetful functors are likely to play a no-ticeable role, are infinite-dimensional Lie theory, supermanifold theory, differentialgeometry, C⋆-algebras and “quantized” functional analysis.We do not aim at a comprehensive presentation of the subject outlined in the titleof this paper, nor we give detailed proofs of the results: such an elaborate approachwould lead to a voluminous treatise. Instead, we discuss a few carefully selectedlines of development which, as we see it, dominated the research over more than50 years.

We are focussing on the most interesting unsolved problems. Also, we doour best in forecasting the future directions of the theory paying special attention

UNIVERSAL ARROWS TO FORGETFUL FUNCTORS3to recent germs of it in areas of mathematics bordering topological algebra (Lietheory, functional analysis and mathematical physics).This small survey inevitably tends to the results and ideas coming from theRussian (or, in a more politically correst language, ex-Socialist, to cover Ukrainian,Moldavian, Georgian, Bulgarian and other contributors) school of universal arrowtheorists, where the author himself comes from. Most probably and to the author’sregret, the contributions from the other two major centers — the Australian andthe American schools — were underrepresented in this article.

As a matter of fact,the author’s personal tastes and research work of his own were prevalent in selectingtopics for discussion.Our bibliography, although (intentionally) not complete, is hopefully “every-where dense” in the subject (a comparison due to Kac [Kac1]).1. Major classical examplesThe following are major examples of universal arrows to forgetful functors fromcategories of topological algebra, which are subject of a traditional study in thisarea.

We are marking with a lozenge (♦) those notions which will be later consideredin this survey to some extent.By abuse of terminology and notation, we willsometimes identify a universal arrow with its target object (no confusion shouldhowever result from that).1.♦C = Tych (the category of Tychonofftopological spaces and continuous map-pings) and D = TopGrp (the category of Hausdorfftopological groups andcontinuous homomorphisms). The universal arrow from an object X ∈C (a Ty-chonoffspace) to the forgetful functor S : D →C is the (Markov) free topologicalgroup over X, F(X).This notion was introduced in 1941 by Markov [Mar1] who presented his resultsin most detail somewhat later [Mar2].

Among those mathematicians who respondedfirst to the new concept, were Nakayama [Nak], Kakutani [Kak], Samuel [Sa] andGraev [Gr1]; the latter work has had an enormous impact on later investigations inthe area, and the paper by Samuel, as we have already mentioned, has produced adeep methodological insight.2.C = Tych∗(the category of pointed Tychonofftopological spaces and contin-uous mappings preserving base points) and D = TopGrp (the base point ofa topological group being e, the identity). The universal arrow from an objectX ∈C (a pointed Tychonoffspace) to the forgetful functor S : D →C is theGraev free topological group over X, FG(X).In fact, the Markov and Graev free topological groups are very closely relatedto each other by means of the following short exact sequence:e →Z →FM(X) →FG(X) →eThe choice of a basepoint ∗∈X does not affect the topological group FG(X).The Markov free group of X is isomorphic to the Graev free group of the disjointsum X ⊕{∗}.

[Gr1,2]. This is why we consider the Markov free topological groupsonly.

Anyway, the Graev approach seems more convenient in some other cases suchas free Banach spaces and free Banach-Lie algebras over metric spaces.3.C = Met∗(the category of pointed metric spaces) and D = MetGrp (thecategory of groups endowed with bi invariant metric) The universal arrow from

4VLADIMIR G. PESTOVan object (X, ρ, ∗) ∈C (a pointed metric space) to the forgetful functor S : D →C is the free group over X\{∗} endowed with the Graev metric ¯ρ.This concept is due to Graev [Gr1,2]. The metrized group (F(X), ¯ρ) is of noparticular interest by itself; it deserves attention as an auxiliary device for study-ing the free topological group F(X).An amazing example of such kind is theArhangel’ski˘ı’s theorem from [Arh4].

If one wants to consider Graev matrics on aMarkov free group then one should start with a fixed metric ρ on the set X ⊕{e}.4.C = Tych and D = V is a variety of Hausdorfftopological groups, consideredas a subcategory of TopGrp. The universal arrow from an object X ∈C (aTychonoffspace) to the forgetful functor S : D →C is the free topological groupover X in the variety V, FV(X).Varieties of topological groups can be understood in different sense (cf.

[Mo1,2,10]and [Pr2,3,PrS]). It would not be clear what is the “right” notion until a non-disputable version of the Birkhofftheorem for topological groups is obtained (see,however, [Ta]).

Anyway, all of the most important classes of topological groups fitboth definitions. Examples of varieties are: the variety of SIN groups (topologicalgroups with equivalent left and right uniformities) [MoTh1]; that of topologicalgroups with quasi-invariant basis [Kats1] (= ℵ0-balanced groups in [Arh5]); of to-tally bounded, or precompact, groups; of ℵ0-bounded groups [Gu, Arh5] etc.

Thereis a survey on free topological groups in varieties [Mo10]. A free topological group,FV(X), in a variety V is actually the composition of the universal arrow F(X) andthe universal arrow from F(X) to the natural embedding functor V →TopGrp.The notion of a free topological group relative to classes of topological groups,considered by Comfort and van Mill [ComvM], can be redefined in terms of freetopological groups in relevant varieties, and the questions of existence of such freetopological groups completely reduces to certain questions about free topologicalgroups in varieties.The following is the most important particular case.5.C = Tych and D = AbTopGrp (the category of Abelian topological groupsand continuous homomorphisms).

The universal arrow from an object X ∈C (aTychonoffspace) to the forgetful functor S : D →C is the (Markov) free Abeliantopological group over X, A(X).6.C = Tych∗and D = AbTopGrp. The universal arrow from an object X ∈C(a Tychonoffspace) to the forgetful functor S : D →C is the Graev free Abeliantopological group over X, AG(X).Of course, A(X) (resp.AG(X)) is just the abelianization of F(X) (resp.,FG(X)).7.C = Tych and D = CompGrp (the category of compact topological groupsand continuous homomorphisms).

The universal arrow from an object X ∈C (aTychonoffspace) to the forgetful functor S : D →C is the free compact groupover X, FC(X).Remark that the free compact group FC(X) is nothing but the Bohr compacti-fication, bF(X), of the free topological group, F(X). (The Bohr compactification,bG, of a topological group G [Mo9] is the universal arrow from G to the embed-ding functor CompGrp →TopGrp.) We do not touch free compact groups inour survey, and refer the reader to the series of papers by Hofmann and Morris[Hf HfMo1 5]Also free compact groups may be viewed as completions of free

UNIVERSAL ARROWS TO FORGETFUL FUNCTORS5precompact groups (or, just the same, free totally bounded groups), that is, freetopological groups in the correspondent variety. Free precompact groups have beenstudied recently in connection with some questions of dimension theory [Sh].Of course, the notion of the free compact Abelian group over X also makes sense,and the structure of such groups has been described in detail (loc.

cit. ).8.♦C = Unif is the category of uniform spaces and D = TopGrp.There areat least four “natural” forgetful functors from D to C; our choice as S is thefunctor S assigning to a topological group G the two-sided uniform structure onit; we will denote the resulting uniform space by Gt.

The universal arrow froman object X ∈Unif (a uniform space) to the functor S is the free topologicalgroup over X, or the uniform free topological group, F(X).This was an invention of Nakayama [Nak]. Free topological groups over uni-form spaces later proved to be a most natural framework for analysing some as-pects of free topological groups, see [Nu2].

Free topological groups over uniformspaces provide a straightforward generalization of free topological groups over Ty-chonoffspaces, because for a Tychonoffspace X the free topological grop over X iscanonically isomorphic to the free topological group over the finest uniform spaceassociated to X.9.By replacing the category Tych by Unif in the items 2,4,6 one comes to theobviously defined concepts of a (Graev) free (Abelian) topological group over auniform space.10.C = Tych (resp., Unif) and D = LCS (the category of locally convex spacesand continuous linear operators). The universal arrow from an object X ∈C(a Tychonoffspace; resp., a uniform space) to the forgetful functor S : D →C(which in the second case is also defined unambiguously, unlike in item 8) is thefree locally convex space over a topological (uniform) space X, and is denoted byL(X).This concept is also an invention of Markov [Mar1].

However, for some reasonit received no immedeate attention from mathematical community until the paperby Ra˘ıkov [Rai2]. The most important of later developments is due to Uspenski˘ı[U2].

A particular case of this construction — the notion of a vector space endowedwith finest locally convex topology — is well known in functional analysis [Sch]; itis actually the free locally convex space over a discrete topological space X.11.As in item 4, one can consider universal arrows from an object of Tych to theforgetful functor V →Tych where V is a variety of locally convex spaces in oneor another sense. We denote the resulting free locally convex space over X in thevariety V by LV(X).We refer the reader to a very solid paper [DmOS] by Diestel, Morris and Saxon,and a survey [Mo10] by Morris.

Other references include [Ber].12.♦If V is the variety of locally convex spaces with weak topology then the resultingfree locally convex space with weak topology over a Tychonoffspace X is denotedby Lp(X).This concept seemingly was well known in functional analysis for decades, be-cause the space Lp(X) is the weak dual of the space Cp(X) of continuous functionson X in the topology of pointwise (simple) convergence. See, e.g., [Wh] and refer-ences therein

6VLADIMIR G. PESTOV13.C = Met∗and D = Ban is the category of complete normed linear spaces andlinear operators of norm ≤1. The universal arrow from an object X ∈Met∗tothe forgetful functor S : D →C (the origin is a base point) is the free Banachspace over a pointed metric space, B(X).This object first appeared in the paper by Arens and Eells [ArE]; see also [Rai2;Pe9].

However, it was considered by functional analysts independently and at adifferent angle of view: the normed space B(X) is known as the predual of thespace Lip (X) of Lipshitz functions on a pointed metric space X.14.C = Tych and D is the category of universal topological algebras of a givensignature Ω. In this case the universal arrow from a space X to the forgetfulfunctor D →C is the free universal topological algebra over X.Such algebras were first considered by Mal’cev [Mal] and others [Ta, Pr2, PrS].We will not touch them in our survey.15.C = Tych and D is the category of topological associative rings or associativealgebras.

The resulting free topological rings and free topological algebras havebeen also considered by Arnautov, Mikhalev, Ursul and others [AMV].Later in our survey we will consider also a number of less traditional examples ofuniversal arrows to forgetful functors. All of them are universal arrows to forgetfulfunctors of one or another kind.The following notion, that of free product oftopological groups, at first seems not to fit into this scheme.16.♦Let C = TopGrp × TopGrp, D = TopGrp, and let S be the diagonal functorTopGrp →TopGrp × TopGrp.

(That is, S(G) = (G, G).) The universalarrow from a pair (G, H) of topological groups to the functor S is called the freeproduct of G and H and denoted by G ∗H.

In other terms, G ∗H is just thecoproduct of G and H in the category TopGrp.Anyway, it is well known that this notion (belonging to Graev [Gr3]) is of thesame nature as that of a free topological group, those constructions share a numberof common properties and indeed, it can be (if necessary) reshaped as a universalarrow to an appropriate forgetful functor. Let C = TopGrp×TopGrp be as above,and let D denote the category of all topological groups with two fixed subgroups.Then G∗H can be viewed as the universal arrow from a pair (G, H) to the forgetdulfunctor from D to C which forgets the first group and sends a triple (F, G, H) to(G, H).17.♦In an obvious way, the concept of the free product can be defined for arbi-trary families of topological groups, {Gα : α ∈A}.

This product is denoted by∗α∈AGα.In all the aforementioned cases, similar methods, which are actually of a cate-gorial nature, are used to prove the existence, uniqueness and a number of otherproperties of universal arrows. We will summon those results as follows.1.1.

Theorem. (1) In all cases 1–17 the universal arrow exists and is unique.

(2) In all cases apart from 4, the universal arrow is an isomorphic embedding. (3) In case 4, the universal arrow is a homeomoprhic embedding if the variety Vcontains at least one non totally path-disconnected topological group.

(4) In all cases apart from 4 and 7, the image of the iniversal arrow is topologicallyclosed□

UNIVERSAL ARROWS TO FORGETFUL FUNCTORS72. Structure of free topological groupsAmong the first, and most vital, questions to be asked about any universalarrow to forgetful functor from a category of topological algebra, is the question ofdescription of the algebro-topological structure of the target object of this arrow.In some cases such a description poses no serious problems, but for most (especiallynoncommutative) examples it is rather challenging.

Since this question seems tobe best investigated for free topological groups, we find it necessary — and veryinstructive — to survey the state of affairs in this area.1. Description of topology.

The topology of a free topological group F(X) israther complicated, and among the achievements of Graev [Gr1,2] was a descrip-tion of the topology of F(X) in the case where X is a compact space. Later hisdescription was transferred to the so-called kω-spaces by Mack, Morris and Ordman[MaMoO], which result has substantially widened the sphere of applicability of theoriginal description.

We will give it in the strongest form.Denote by ˜X = X ⊕−X ⊕{e} the disjoint sum of a Tychonoffspace X, ofits topological copy −X = {−x : x ∈X}, and a one-point space {e}. For eachn = 0, 1, 2, .

. .

there is an obvisouly defined canonical continuous mapping in :˜Xn →F(X). Denote by Fn(X) the subspace of F(C) image of in; it is closed.A topological space X is called a kω-space if it can be represented as a union ofcountably many compact subsets Xn in such a way that the topology of X is aweak topology with respect to the cover {Xn : n ∈N}, that is, a subset A ⊂X isclosed iffso are al intersections A ∩Xn, n ∈N.

Not only every compact space isa kω space; so is every countable CW-complex, every locally compact space withcountable base etc.2.1. Theorem (Graev-Mack-Morris-Nickolas).

Let X be a kω-space. Then everymapping in is quotient, and a subset A of F(X) is closed if and only if so are allintersections A ∩Fn(X).

In particular, F(X) is a kω-space.□The above theorem does not admit any noticeable further generalization, apartfrom some openly pathological cases, such as the spaces X where every Gδ set isopen (the author, unpublished, 1981). In fact, it was shown in [FOST] that themapping i3 is not quotient even for X = Q. Answering two question raised in thispaper, the author had proved the following result [Pe7].2.2.

Theorem. Let X be a Tychonoffspace.

The mapping i2 is quotient if andonly if X is a strongly collectionwise normal space (that is, every neighbourhood ofthe diagonal in X × X is an element of the universal uniform structure of X).□However, the following property of the mappings in proved to be extremelyuseful.2.3.Theorem (Arhangel’ski˘ı [Arh2,3]). Let Y be any subset of ˜Xn such thati−1n in(Y ) = Y .

Then in|Y is a homeomorphism.□A very substantial body of results concerning the structure of free topologicalgroups over kω spaces have been deduced (mostly by Australian and Americanmathematicians) from Theorem 2.1 [Br, BrH, F, FRST2, HMo, Katz, KatzMo1,2,KatzMoN1-4, Mo10,11, Nic1,3, OrST].The following charming theorem of Zarichny˘ı [Zar1,2] puts free topological groupsin connection with infinite dimensional topology The original result was stated for

8VLADIMIR G. PESTOVfree Graev topological groups, but it extends to free Markov groups immedeatelybecause topologically the group F(X) is a disjoint sum of countably many copiesof FG(X).2.4.Theorem (Zarichny˘ı [Zar1]). Let X be a compact absolute neighbourhoodretract and 0 < dim X < ∞.

Then the free topological group F(X) and the freeAbelian topological group A(X) are homeomorphic to an open subset of the locallyconvex space with finest topology Rω = lim−→Rn.□Returning back to general Tychonoffspaces X, one can still describe the topologyof F(X) with the help of mappings in, but in a rather non-constructuve way.The following construction have been performed by Mal’cev [Mal]. Denote by T0the quotient topology on F(X) with respect to the direct sum of the mappingsin, n ∈N from the space ⊕n∈N ˜Xn.

It is Hausdorffbut not necessarily a grouptopology. Now construct recursively a transfinite chain of topologies Tλ on F(X)by defining Tλ+1 as the quotient of the topology on F(X) × F(X) with respect tothe mapping (x, y) 7→x−1y, and Tτ for a limit cardinal τ as the infimum of thechain of topologies Tλ, λ < τ.

It is clear that for some λ large enough, the topologyTλ coincides with the topology of F(X). Denote the least λ with this property byλ(X).

The following question is open for more than 30 years.Problem (Mal’cev [Mal]). Which values can λ(X) assume?Seemingly, all one knows is that λ(X) = 1 for kω-spaces, and λ(X) > 1 for mostspaces beyond this class (for instance, for X = Q).Another long-standing problem asked by Mal’cev in the same paper [Mal] — thatof finding a constructive description of a neighbourhood system of identity of a freetopological group — had been solved by Tkachenko [Tk4].

Later simpler versionsof the Tkachenko’s theorem have been obtained by the author [Pe7] and Sipacheva[Si1]. We will give one of the possible forms of the result.

It is more reasonable toput it for free topological groups F(X) over uniform spaces (bearing in mind thatfor a Tychonoffspace X the free topological grop over X is canonically isomorphicto the free topological group over the universal uniform space associated to X). LetX = (X, UX) be a uniform space.

Denote by j2 a mapping from X2 to F2(X) of theform (x, y) 7→x−1y, and by j∗2 — a similar mapping of the form (x, y) 7→xy−1. IfΨ ∈(UX)F (X) is a family of entourages of diagonal indexed by elements of the freegroup over X, then we putVΨ =def ∪{x · [j2(Ψ(x)) ∪j∗2(Ψ(x))] · x−1 : x ∈F(X)}If Bn is a sequence of subsets of some group then, following [RoeD], we denote[(Bn)] =def ∪n∈N ∪π∈Sn Bπ(1) · Bπ(2) · .

. .

· Bπ(n),where Sn is a symmetric group.2.5. Theorem (Pestov [Pe7]).

Let (X, UX) be a uniform space. A base of neigh-bourhoods of identity in the free topological group F(X) is formed by all sets ofthe form [(VΨn)], where {Ψn} runs over the family of all countable sequences ofelements of (UX)F (X).□

UNIVERSAL ARROWS TO FORGETFUL FUNCTORS92. Free subgroups.

If X is a subset of a set Y , then the free group over the setof generators X is a subgroup of the free group over Y . Now let X is a topolog-ical subspace of a Tychonoffspace Y ; there is still a canonical continuous groupmonomorphism F(X) ֒→F(Y ), but it need not be a topological embedding.

Forthe first time it was noticed by Hunt and Morris [HuM], and the example wasX = (0, 1), Y = [0, 1]. Earlier Graev has shown [Gr1,2] that if Y is compact and Xis closed in Y then F(X) ֒→F(Y ) is in isomorphic embedding of topological groups.This result was transferred to kω-spaces.

In [Pe1,2,4] and [Nu2] it was noticed in-dependently that a necessary condition for the monomorphism F(X) ֒→F(Y ) tobe topological is the property that the restriction UY |X of the universal uniformityUY from Y to X coincides with the universal uniformity UX of X. (It is just an im-medeate consequence of the fact that both left and two-sided uniformilies on F(X)induce on X its universal uniform structure — the fact which in its turn followsfrom existence of Graev’s pseudometrics on F(X) and was essentially known toGraev.) In the same works [Pe1,2,4] and [Nu2] it was shown, answering a questionby Hardy, Morris and Thompson [HMoTh] that the above condition UY |X = UX issufficient in the case where X is dense in Y .

A final positive answer was obtainedby Uspenski˘ı [U5] after a series of results of intermediate strength [U3].2.6. Theorem (Uspenski˘ı [U5]).

Let X be a topological subspace of a Tychonoffspace Y . Then the monomorphism F(X) ֒→F(Y ) is a topological embedding if andonly if UY |X = UX.□A different problem has been treated by Australian and American universal arrowtheorists for a long time.

Let X and Y be some particular topological spaces; inwhich cases the free (Abelian) topological group over X can be embedded (notnecessarily in a “canonical” way) as a topological subgroup into the free (Abelian)topological group over Y ? The main device under this approach was the aboveTheorem 2.1.

We will mention just one astonishing result in this direction.2.7. Theorem (Katz and Morris [KatzMo2]).

If X is a countable CW-complex ofdimension n, then the free Abelian topological group on X is a closed subgroup ofthe free Abelian topological group on the closed ball Bn.□3.Completeness. Our next topic can be also traced back to Graev’s papers[Gr1,2].

Graev has deduced from his description of topology of the free group overa compact space that any such free topological group is Weil complete (that is,complete with respect to the left uniform structure). The result remains true forfree topological groups over kω-spaces.Examples of topological groups which are complete in their two-sided uniformitybut not Weil complete (and therefore admit no Weil completion at all) are knownfor decades, but seemingly it remains unclear whether free topological groups ad-mit Weil completion.This question was asked by Hunt and Morris [HuM].

Anobvious necessary condition for a free topological group to be Weil-complete is theDieudonn´e completeness of X, that is, completeness of X w.r.t. the finest uni-formity UX.

The state of affairs with Weil completeness is still unclear and onehas only a series of partial results stating the Weil completeness of free topologicalgroups over particular spaces [U3].However, it seems in a sense more natural to examine free topological groups foranother form of completeness — the completeness w.r.t. the two-sided uniformity(sometimes also called Ra˘ıkov completeness) [Rai1]There exists a fascinating

10VLADIMIR G. PESTOVcomprehensive result for the completeness of this kind, and the question about thevalidity of such a result was first asked independently by Nummela [Nu2] and theauthor (in oral form, talk at the Arhangel’ski˘ı’s seminar on topological algebra atMoscow University, February 1981).2.8.Theorem (Sipacheva, [Si2]). The free topological group F(X) over a Ty-chonoffspace X is complete if and only if X is Dieudonn´e complete.□The idea of the proof is based on the notion of a special universal arrow, Fρ(X),introduced by Tkachenko [Tk3].

Say that a subspace Y of a topological group G isTkachenko thin if for every neighbourhood of identity, U, the set ∩{yUy−1 : y ∈Y }is a neighbourhood of identity. Consider the category of pairs (G, Y ) where G is aHausdorfftopological group and Y is a Tkachenko thin subset of G, and obviousmorphisms between them, and let S be the functor from this category to Tychof the form (G, Y ) 7→Y .

Now by Fρ(X) one denotes the universal arrow froma Tychonoffspace X to the functor S. There is a canonical continuous algebraicisomorphism F(X) →Fρ(X), and it can be shown without serious difficulties thatthe topological group Fρ(X) is complete if and only if X is Dieudonn´e complete[Tk3]. Sipacheva has proved that the free topological group F(X) has a base ofneighbourhoods of identity that are closed in the topology of the topological groupFρ(X).Let X be a set, and let V and W be any two uniformilities on X generatingthe same Tychonofftopology.

(Such a triple (X, V, W) is termed sometimes a bi-uniform space.)Question. Does there exist a topological group F(X, V, W) algebraically generatedby X (free over X) such that V is the restriction to X of the left uniform structureof G, and W is the restriction to X of the right uniform structure?This question can be obviously reformulated in terms of universal arrows toforgetful functors.This concept may help to understand how the completenessworks.Among other results on the algebro-topological structure of the free topologicalgroups, let us mention a nice theorem of Tkachenko [Tk1,2] stating that the freetopological group over a compact space has the c.c.c.

property (together with itssubsequent generalization due to Uspenski˘ı [U1]), and a characterization of suchTychonoffspaces X that the free topological group F(X) embeds into a directproduct of a family of separable metrizable groups [Gur].5. Abelian case.

All of the above results have, of course, their analogs for freeAbelian topological groups. Moreover, one can also give a very convenient and sim-ple description of topology of A(X) which has no analog (yet?) in non-commutativecase.

One can define Graev metrics on A(X) in the same way as for F(X), and itturns out that they describe the topology of A(X). It follows from this observationthat the canonical morphism from A(X) to the free locally convex space L(X) is anembedding of A(X) as a closed topological subgroup [Tk3].

Both completeness ofA(X) over Dieudonn´e complete spaces X and the Abelian analog of the subgrouptheorem were established much earlier than their non-Abelian counterparts [Tk3].The embedding A(X) ֒→L(X) enables one to describe the topology of A(X) asthe topology of uniform convergence on all equicontinuous families of characters ofA(X) and this way Pontryagin van Kampen duality comes into being For the first

UNIVERSAL ARROWS TO FORGETFUL FUNCTORS11time the Pontryagin-van Kampen duality for free Abelian topological groups wasstudied by Nickolas [Nic2] who has shown, answering a question by Noble [No], thatthe topological group A[0, 1] is non-reflexive (that is, does not verify the statementof Pontryagin duality theorem). Later the author had obtained the following result.2.9.Theorem (Pestov [Pe8]).

Let X be a Dieudonn´e complete k-space withdim X = 0. Then the free Abelian topological group A(X) is reflexive.□The free topological group F(X) is so “regularly shaped” that one is wonderingwhether it satisfies any known version of noncommutative duality (Tannaka-Kreinduality is known to be insufficient) or, at the very least, whether its topology can bedescribed with the help of equicontinuous families of homomorphisms from F(X)into some fixed topological group — say, GL(∞) = lim−→GL(n, R).3.

M-equivalence and dimensionIn 1945 Markov in one of his important papers [Mar2] asked whether any twoTychonofftopological spaces, X and Y , with isomorphic free topological groupsF(X) and F(Y ) are necessarily homeomorphic. Soon Graev in his no less importantpapers [Gr1,2] answered in the negative by constructing a whole series of pairs X, Yof spaces with F(X) ∼= F(Y ), therefore the resulting relation of equivalence betweenTychonoffspaces turned out to be substantial.

Graev called such spaces X and YF-equivalent; we follow the terminology due to Arhangel’ski˘ı [Arh3,5,6,8] and callsuch spaces Markov equivalent or M-equivalent. Graev paid special attention tothe pairs of spaces X, Y with Graev free topological groups isomorphic, FG(X) ∼=FG(Y ); however, the distinction between the two relations of equivalence is —from the viewpoint of their topological properties — inessential.

With the help ofArhangel’ski˘ı’s terminology, one of the central results of the Graev’s paper [] canbe formulated like this.3.1. Theorem (1948, Graev).

If X and Y are M-equivalent compact metrizablespaces then dim X = dim Y .□(Here dim X stands for the Lebesgue covering dimension of a space X. )This result — as well as technique of the proof — has received a lot of attentionlater.

The generalizations of the result came in two directions: firstly, the equiva-lence relation was being replaced by more and more loose ones, and secondly, thetopological restrictions on the spaces X, Y were weakened.In 1976 Joiner [Joi] noticed that the conclusion dim X = dim Y remains trueif X and Y are both locally compact metrizable spaces such that the free Abeliantopological groups, A(X) and A(Y ), are isomorphic. (Following Arhangel’ski˘ı, wecall such spaces X, Y A-equivalent.

)Of course, A-equivalence of two topologi-cal spaces follows from their M-equivalence, because the universal arrow A(X)is a composition of the universal arrow F(X) and the functor of abelianizationTopGrp →AbTopGrp.Consider the universal arrow from the free Abelian topological group A(X) to theforgetful functor from the category of locally convex spaces with weak topology toAbTopGrp. The composition of two universal arrows is obviously the free locallyconvex space in weak topology, Lp(X).

Therefore, we come to a still looser relationof equivalence between two spaces: X and Y are l-equivalent if Lp(X) ∼= Lp(Y ). In1980 Pavlovski˘ı [Pa] had shown that dim X = dim Y if X and Y are l-equivalentspaces which are either locally compact metrizable or separable complete metrizable

12VLADIMIR G. PESTOVSo far all proofs relied on a suitable refinement of the original Graev’s techniques.A basically new method — that of inverse spectra — was invoked and applied tothis problem by Arhangel’ski˘ı [Arh3,6] who deduced from the Pavlovski˘ı’s theoremthe following landmark result.3.2. Theorem (Arhangel’ski˘ı 1980).

Let X and Y be l-equivalent compact spaces.Then dim X = dim Y .□Independently a weaker version was obtained by Zambakhidze [Zam1]: the cov-ering dimension of any two M-equivalent compact spaces is the same. Later thisresult was generalized by him to the class of ˘Cech complete, scaly, normal, totallyparacompact spaces [Zam2] (it remained not quite clear how wide this class ac-tually was).

About the same time the result had been independently somewhatgeneralized by Valov and Pasynkov [VP].Further efforts have been boosted by a question asked by Arhangel’ski˘ı [Arh5]: isit true that for TychonoffM-equivalent spaces X and Y one has dim X = dim Y ?The answer “yes” came from the author, who proved in late 1981 [Pe3] thefollowing result by combining and adjusting both Graev’s lemma and the spectraltechnique of Arhangel’ski˘ı:3.3. Theorem (Pestov, 1981).

If X and Y are l-equivalent Tychonoffspaces thendim X = dim Y .□As a matter of fact, the aforementioned Graev’s lemma, which forms the coreof the proofs, is not a single result but rather a scheme of results, improved andadjusted from one situation to another. We present it as it appears in [Pe5], not inthe most general form possible, but in a quite elegant one.3.4.

Graev’s Lemma. If X and Y are M-equivalent Tychonoffspaces then X isa union of countably many subspaces each of which is homeomorphic to a subspaceof Y .□Then one is using addition theorems for covering dimension valid for spaces withcountable base; to proceed from such spaces to a general situation, the Tychonoffspaces X and Y are decomposed in inverse spectra of spaces with countable baseand the same dimension as dim X and dim Y ; the property of l-equivalence ofthe two limit spaces is partially delegated to the spectrum spaces, in a form strongenough to ensure a version of the Graev’s lemma.It was shown by Burov [Bu] that the result and the scheme of the proof remaintrue also for cohomological dimension dimG where the group of coefficients G is afinitely generated Abelian group (in particular, dimZX ≡dim X).The weak dual space to Lp(X) is the space of continuous functions on X withthe topology of simple (pointwise) convergence, Cp(X) (it follows actually from aversion of the Yoneda lemma).

The theory of linear topological structure of theLCS Cp(X) has grown out of Banach space theory, after the following observationproved useful [Cor]: any Banach space E in weak topology is a subspace of Cp(X)where X is the closed unit ball of the dual to E with weak⋆topology. This theoryis developing now on its own, and a good survey is [Arh9].A bridge betweentheory of spaces Cp(X) and universal arrow theory is erected by means of thefollowing observation: since the two LCS’s in weak topology, Lp(X) and Cp(X),are in duality, then two topological spaces X and Y are l-equivalent if and only ifC (X) and C (Y ) are isomorphic

UNIVERSAL ARROWS TO FORGETFUL FUNCTORS13Arhangel’ski˘ı was first to suggest an even weaker realtion of equivalence betweentwo Tychonofftopological spaces, X and Y : two such spaces are called u-equivalentif the locally convex spaces Cp(X) and Cp(Y ) are isomorphic as uniform spaces(with the natural additive uniformity). Surprisingly, it was possible to make onemore step in extending the original Graev’s result.3.5.

Theorem (Gul’ko, [Gu]). If X and Y are l-equivalent Tychonoffspaces thendim X = dim Y .□The proof of Gul’ko’s result [Gu] develops along the same lines as the author’searlier theorem, but technically it is considerably more complicated.One can consider even weaker realtion of equivalence: two topological spaces,X and Y , are said to be t-equivalent [GuKh] if the locally convex spaces Cp(X)and Cp(Y ) are homeomorphic as topological spaces.

It is not known whether thedimension is preserved under the relation of t-equivalence. It is worth mentioningthat all the aforementioned equivalence relations (those of M-, l-, u-, t-equivalence)have been distinguished from each other.What remains still unclear, is the existence of a reasonable straightforward char-acterization of dimension of a Tychonoffspace X in terms of the additive uniformityof the LCS Cp(X), or the linear topological structure of the space Lp(X), or — atthe very least — the algebro-topological strucuture of F(X).

The existing proofsare in a sense obscure and do not reveal the real machinery keeping dimensionpreserved by the equivalence relations.It is an opinion of the author that emerged from discussions with Gul’ko in 1991that a complete understanding of the phenomenon of preservation of dimension isto be saught on the following way.Conjecture. The Lebesgue dimension of X can be expressed in terms of a certain(co)homology theory associated with the LCS in weak topology Lp(X).It is not clear if one can use any of the already existing (co)homology theoriesfor locally convex spaces, because a desired theory should make a sharp distinctionbetween week and normable topologies.

For instance, the space C(X) endowedwith the topology of uniform convergence on compacta instead of the pointwisetopology carries essentially no information about the dimension of X, according tocelebrated Milyutin isomorphic classification theorem [Muly].The following remarkable theorem by Pavlovski˘ı may be also sugggestive; toour knowledge, no attempt has been made yet to generalize it to arbitrary CW-complexes.3.6. Theorem (Pavlovski˘ı [Pa]).

Two polyhedra (simplicial complexes) X and Yare l-equivalent if and only if dim X = dim Y .□It is well known that dim X ≤n if and only if every continuous mapping from Xto a sphere Sn+1 is homotopically trivial [vM]. The structure of the free topologicalgroups on spheres is well understood [KatzMoN2], so the following sounds sensible.Conjecture.

Let X be a Tychonofftopological space. Then dim X ≤n if and onlyif every continuous homomorphism from the free topological group F(X) to the freetopological group F(Sn+1) is homotopically trivial.Of course, similar considerations no longer work for l-equivalence because anyLCS is contractible, but the above conjecture may help to reach a deeper under-standing for the relation of M equivalence

14VLADIMIR G. PESTOVIn addition to Gel’fand-Na˘ımark duality, general interest to the problem ofpreservation of properties of topological spaces by different functors from the cate-gory Tych to the categories of topological algebra has been heated for a long timeby the following result of Nagata [Nag].3.7.Theorem (Nagata). Two Tychonoffspaces X and Y are homeomorphicif and only if the topological rings Cp(X) and Cp(Y ) are isomorphic.In otherterms, the functor Cp(·) from Tych to TopRings is a (contravariant) inclusionfunctor.□By considering for every Tychonoffspace X the universal arrow from X to aforgetful functor from the category TopGrp to Tych sending a topological groupto a topological subspace consisting of all elements of order 2, one comes to thefollowing result [Pe12].3.8.

Theorem. There exists a (covariant) inclusion functor Tych →TopGrp.□There is no full inclusion functor of such kind [Pe12].The following question seems very natural in connection with our problematics,and it was asked independently by many (for example, by Zarichny˘ı in Baku-1987):Question.

Is it true that K-groups of M-equivalent Tychonofftopological spacesare isomorphic?An obvious idea, to obtain the affirmative answer with the help of universalclassifying groups, fails, because if G is a non-Abelian topological group and Xand Y are M-equivalent, then it follows (from the Yoneda’s lemma, actually) thatK(X) and K(Y ) are isomorphic as sets, not groups: the set Homc(F(X), G) doesnot carry a natural groups structure because of non-commutativity of G — andthe universal classifying groups in K-theory are noncommutitive. (This is why acorresponding statement in [VP] is wrong.

)The general classification of topological spaces up to an M-equivalence (as wellas l-equivelence and other relations mentioned in this section) seems a totally hope-less problem. For numerous results on preservation and non-preservation of partic-ular properties of set-theoretic topology by M-equivalence, l-equivalence etc.

see[Arh3,5,7,8, Gr1,2, Ok, Tch1,2]. From our point of view, there are at least two caseswhere such a classification may be achieved.

The first is the case of l-equivalenceof CW-complexes (in view of the Pavlovski˘ı’s theorem), and the second is the caseof M-equivalence of the so-called scattered spaces [ArhPo] (in view of the completeclassification of all countable metric spaces up to M-equivalence obtained by Graev[Gr1,2]).4. Applications to general topological groupsIn this section we consider some applications of free topological groups to generaltheory of topological groups.

Remark that perhaps one owes the very existence ofthe concept of free topological group to a stimulating applied problem of such kind:in his historical note [Mar1] Markov was openly guided by the idea of constructingthe first ever example of a Hausdorfftopological group whose underlying space wasnot normal. (The free topological group over any Tychonoffnon-normal space Xis such.

)Free topological groups provide flexible “building blocks” for erecting more so-phisticated constructions Also the following theorem is of crucial importance

UNIVERSAL ARROWS TO FORGETFUL FUNCTORS154.1. Theorem (Arhangel’ski˘ı [Arh1]).

Let f be a quotient mapping from a topo-logical space X onto a topological group G. Then the continuous homomorphismˆf : F(X) →G extending f is open and therefore G is a topological quotient groupof F(X).□Seemingly, analogs of this theorem exist for other types of universal arrows aswell, and one is wondering whether this result can be given a universal categorialshaping. This result (and its analogs) are invaluable for examining questions ofexistence of couniversal objects of one or another kind.1.

NSS property. Our first example is the NSS property.

A topological groupG has no small subgroups if there is a neighbourhood U of the identity element esuch that the only subgroup in U is {e}. This is abbreviated to NSS.

The crucialrole of the NSS property in Lie theory (especially in connection with Hilbert’s FifthProblem) is well known.In 1971 Kaplansky wrote ([Kap], p.89): “The following appeaps to be open: if Gis NSS and H is a closed normal subgroup of G, is G/H NSS? This is true if inaddition G is locally compact, but we shall only be able to prove it late in the game.

(Of course it is an old result for Lie groups. )”Very soon Morris [Mo3] answered in negative by constructing a counter-example,and later he and Thompson [MoTh2] have presented the following4.2.

Theorem. Let X be a submetrizable Tychonofftopological space (that is, aTychonoffspace admitting a continuous metric).

Then the Markov free topologicalgroup F(X) over X is an NSS group.□It was asked in [MoTh2] whether the following result is true.4.3. Theorem.

Each topological group is a quotient group of an NSS group.□The author [Pe1,4] has deduced Theorem 4.3 from Theorems 4.2 and 4.1 (andlater it turned out that such a deduction follows at once from the above Theorems4.2 and 4.1 in conjunction with [Ju], see [Arh4,5]).It was shown by Sipacheva and Uspenski˘ı [SiU] that both the original proofof Theorem 4.2 by Morris and Thompson [MoTh2] and the later proof proposedby Thompson [Th] are not free of certain deficiencies.In the same work [SiU]an elaborate proof of Theorem 4.2 (definitely “hard” — it relied on combinatorialtechnique of words in free groups) was given. Thus, both results remain valid.

Theconcept of free Banach-Lie algebra enables us to provide a purely Lie-theoretic (andcertainly “soft”) proof of Theorem 4.2 (see Section 7 below).2. Zero-dimensionality.

Our next story is about quotient groups of zero-dimensionaltopological groups, and it is strikingly similar to the preceding development. In1938 Weil (see the note [Arh4] for this and the next references) claimed that opencontinuous homomorphisms of topological groups do not increase dimension.

Thisstatement was later refuted by Kaplan by means of a counterexample. Arhangel’ski˘ı[Arh2] has shown that every topological group with countable base is a quotient groupof a zero-dimensional group.

(Zero-dimensionality here and in the sequel is under-stood in the sense of Lebesgue covering dimension dim.) Possible ways to representany topological group as a quotient group of a zero-dimensional one were discussedby Arhangel’ski˘ı in [Arh1], but it was until late 1980 that the above conjectureremained open

16VLADIMIR G. PESTOV4.4.Theorem (Arhangel’ski˘ı [Arh4,5]). Any topological group is a topologicalquotinet group of a group G with dim G = 0.□Subtle topological considerations involving Graev metrics on free groups playeda crucial role in the proof of the main auxiliary result: if a submetrizable topologicalspace X is a disjoint union of a family of spaces each of which has a unique non-isolated point then dim F(X) = 0.

Then the fact that every Tychonoffspace is aquotient of a space with the above property is used, together with Argangel’ski˘ı’sTheorem 4.1.This result brought to life a variety of satellite theorems and examples refiningthe statement. Of them the most important one is, from the author’s viewpoint,the following.4.5.

Theorem (Sipacheva [Si2]). If X is a Tychonoffspace and dim X = 0 thendim F(X) = 0.□3.

Topologizing a group. As the last example, we discuss a problem by Markov[Mar2] remaining open for 40 years.

A subset X of a group G is called uncondition-ally closed in G if X is closed with respect to every Hausdorffgroup topology onG. Markov asked [Mar2] whether a group G admits a connected group topology ifand only if every unconditionally closed subgroup of G has index ≥c.

(Obviously,this condition is necessary. )The first counterexample was constructed by the author in [Pe13].

Denote byL♭(X) the universal arrow from a uniform space X to the forgetful functor from thecategory of pairs (E, Y ), E a LCS and Y a bounded subset of E (with obviouslydefined morphisms), to Unif, of the form (E, Y ) →Y where Y inherits the additiveuniformily from E. If G is a topological group and H a closed subgroup, then the leftaction of G on the quotient space G/H with a natural quotient uniform structure[RoeD] lifts to a continuous action of G on L♭(G/H). The double semidirect productG↑= (G ⋉L♭(G)) ⋉L♭(X),where X is the disjoint sum of a family of copies of a quotient space of G ⋉L♭(G),serves as a counterexample to the Markov question in case where G is an uncount-able totally disconnected topological group.Later it was observed by Remus [Re] that the infinite symmetric group S(X) withpointwise topology provides another — much more transparent — counterexampleto the Markov’s conjecture.The author’s techniques was also used by him to construct an example of agroup admitting a nontrivial Hausdorffgroup topology but admitting no non-trivialHausdorffmetrizable topology [Pe11].Another problem of Markov still remains open.

A subset X of a topologicalgroup G is called absolutely closed if it is closed in the coarsest topology on Gmaking all mappings of the formx 7→w(x)continuous as soon as w(x) is a word in the alphabet formed by all elements of Gand a single variable x. This topology is an analog of the Zariski topology in affinespaces; we think it is natural to call it the Markov topology on a group

UNIVERSAL ARROWS TO FORGETFUL FUNCTORS17Problem (Markov [Mar2]). Prove of refute the conjecture: every unconditionallyclosed subset of a group is absolutely closed.Denote by TM(G) the Markov topology on a group G, and by T∧(G) — thetopology intersection of all Hausdorffgroup topologies on G.It is clear thatTM(G) ⊂T∧(G).

The Markov’s problem can be be now put in other terms: isit true that for an arbitrary group G one has TM(G) = T∧(G)?5. Free products of topological groupsGraev [Gr3] presented a constructive description of the topology of the freeproduct G∗H of two compact groups; also he proved a version of Kurosh subgrouptheorem in the same paper.

Later both results have been generalized to kω-groups(or, more precisely, topological groups whose underlying spaces are kω) [MoOTh].It is known that those results are no longer true beyond the class of such topologicalgroups.One can ask about the free products of topological groups almost the samenatural questions as for free topological groups: to give a reasonable descriptionof topology in general case, to prove (or refute) that the free product of two (anarbitrary family of) complete topological groups is a complete group; to prove (orrefute) that if Hα is a topological subgroup of Gα for every α ∈A then ∗α∈AHαis a topological subgroup of ∗α∈AGα. However, here is a question deserving, fromour viewpoint, a special attention — and not only because of its respectable age.As it is well known, the construction of free product of groups is a generalizationof the construction of a free group: indeed, the free group F(X) over the set Xof free generators is just the free product ∗x∈XZx of |X| copies of the infinitecyclic group Z.

This is obviously not the case with free topological groups and freeproducts of topological groups — unless X is discrete. In 1950 Graev mentionedthis and remarked that “the question of existence of a natural construction whichwould embrace both free topological groups and free products of topological groupsstill remains open.” It does — for some 42 years already.Let {Gx : x ∈X} be a family of topological groups indexed with elements ofa topological space X.

One would like to define the free product ∗x∈XGx as anappropriate universal arrow in such a way that 1) in case where Gx ∼= Z for allx ∈X, the group ∗x∈XGx was (naturally) topologically isomorphic to the Markovfree topological group F(X); 2) in case where X is a discrete topological space,∗x∈XGx was a usual free product of topological groups.Our suggestion is that a clue to the above problem might be the space L(G) of allclosed subgroups of a topological group G, endowed with an appropriate topology.This space (and, moreover, a topological lattice) has been thoroughly studied [Pr1]in connection with extending the Mal’cev Local Theorems to the case of locallycompact groups. It is known that there exist numerous “natural” topologies on theset L(G), including the Vietoris, Chabeauty, and other topologies (loc.

cit.).Conjecture. The free product of a family of topological groups {Gx : x ∈X}indexed with elements of a Tychonofftopological space X can be defined as theuniversal arrow from {Gx : x ∈X} to the functor from the category TopGrp tothe category of all families of topological groups indexed with elements of Tychonoffspaces (with relevant morphisms between them), which assigns to a topological groupG the family {H : H ∈L(G)}, the space L(G) being endowed with an appropriatetopology

18VLADIMIR G. PESTOVThe Graev problem can be put in connection with deformation theory and quan-tum groups. In quantum physics, one considers deformations of algebro-topologicalobjects (such as Lie groups) as families of objects, Aℏ, depending on a continuousparameter ℏ, which is assumed to be a “very small” real number approaching zero.Physically, ℏis the Planck’s constant, and the case ℏ= 0 corresponds to the (quasi)classical limit of a theory; what is deformed, is the object A0.

The absence of non-trivial deformations for classical simple Lie groups and algebras was a reason forintroducing new kind of objects — the quantum groups [Drin, Man2, RTF, Ros,Wo].While there exists a rich mathematically sound deformation theory for Lie alge-bras, deformations of Lie group are often treated at a heuristic level. The conjecturalGraev construction would enable one to consider the family Gℏ, ℏ≥0 of Lie groupsas a veritable continuous path in the topological space L(∗x∈XGx).Quantum groups were introduced in mathematical physics to describe the so-called broken symmetries of physical systems.

The concept of a quantum groupis not something accomplished, and its development is still in progress. It is onlynatural, in search of more interrelations between newly explored categories of math-ematical physics, to look for universal arrows between them.

Does the notion of afree quantum group over a “quantum space” make sense?7. Free Banach-Lie algebras and their Lie groupsThe free Banach-Lie algebra, lie(E), over a normed space E is the universalarrow from E to the forgetful functor S from the category BLA of complete Liealgebras endowed with submultiplicative norm to the category Norm of normedlinear spaces.7.1.Theorem (Pestov [Pe18]).

The free Banach-Lie algebra exists for everynormed space E, and E ֒→lie(E) is an isometric embedding.The Lie algebralie(E) is centerless and infinite-dimensional if dim E > 0.□One can also define the free Banach-Lie algebra over an arbitrary pointed metricspace X (we will denote it lieX) as the universal arrow from X to the forgetfulfunctor from BLA to Met∗(zero goes to the marked point). Obviously, it is justthe composition of the free Banach space and free Banach-Lie algebra arrows.A Banach-Lie algebra g is called enlargable if it comes from a Banach-Lie group.Every free Banach-Lie algebra is enlargable, and we will denote the correposndingsimply connected Banach-Lie group by LG(E) (resp.

LGX). Since every Banach-Lie algebra g is a quotient Banach-Lie algebra of the free Banach-Lie algebra overthe underlying Banach space of g, then we come to an independent proof of a resultdue to van Est and ´Swierczkowski [´S3]: every Banach-Lie algebra is a quotient ofan enlargable Banach-Lie algebra.This result can be strengthened.The couniversality of the Banach space l1among all separable Banach spaces is well-known [LiT] (actually, it is due to thefact that l1 is the free Banach space over a discrete metric space).Therefore,lie(l1) is a couniversal separable Banach-Lie algebra, and the universality propertyis transfered to the Lie group LG(l1).7.2.

Theorem. There exists a couniversal connected separable Banach-Lie group.□Of course, the same is true for groups containing a dense subset of cardinality≤τ

UNIVERSAL ARROWS TO FORGETFUL FUNCTORS19One can show using results of Mycielski [My] and an idea of Gel’baum [Gel]that for any metric space X, the exponential image of X\{0} in the Lie groupGLX generates an algebraically free subgroup.Now let Y be a submetrizablepointed space admitting a one-to-one continuous mapping to X. The compositionof this mapping and the exponential mapping expGLX determines a continuousmonomorphism FG(Y ) →GLX, and since any Banach-Lie group has NSS propertythen it is shared by FG(Y ).

This is the promised “soft” proof of Morris-Thompson-Sipacheva-Uspenski˘ı theorem.In view of the existence of a couniversal separable Banach-Lie group, the follow-ing question seems most natural.Question. Does there exist a universal separable Banach-Lie group?One should compare it with the following fascinating result of Uspenski˘ı [U4].7.3.

Theorem (Uspenski˘ı). The group of isometries of the Banach space C(Iℵ0)endowed with the strong operator topology is a universal topological group withcountable base.□However, the general linear group GL(E) of any Banach space E, endowed withthe uniform operator topology, cannot serve as a universal Banach-Lie group be-cause there exist enlargable separable Banach-Lie algebras g which do not admit afaithful linear representation in a Banach space [vE´S].The universal arrow from a Lie algebra, g, to the forgetful functor from thecategory of associative algebras to the category of Lie algebras is well-known; thisis the universal enveloping algebra, U(g), of g [Dr].It seems that little is known about a topologized version of this, that is, theuniversal arrow from a locally convex Lie algebra, g, to the forgetful functor fromthe category of locally convex associative algebras to the category of locally convexLie algebras.

Let us denote this arrow by ig : g →UT (g). Is ig an embedding oftopological algebras?

(That is, does a topological version of the Poincar´e-Birkhoff-Witt theorem hold?) Is UT (g) algebraically isomorphic to U(g)?

What about theconvergence of the exponential mapping for UT (g)?The only result I am aware of in this connection is the following.7.4. Theorem [Bou].

The universal enveloping algebra U(g) of a finite-dimensionalLie algebra g can be made into a normed algebra if and only if g is nilpotent.□This means that, firstly, a metric version of the universal arrow makes no senseand, secondly, in general the algebra UT (g) is non-normable even if g is finite-dimensional.A detailed analysis of the structure of the locally convex associative algebraUT (g) would be helpful in connection with enlargability problems for g.8. Lie-Cartan theoremThe Lie-Cartan theorem says that finite-dimensional Lie algebras are enlargable,and it seems that the question on existence of a “direct” proof of the Lie-Cartantheorem, which would be independent of both known proofs (the cohomological oneby Cartan [C] and the representation-theoretic one by Ado [Ad]), is still open.

Fora detailed discussion, see the book [Po], where it is claimed that the above questionfor a long time received an attention from both French and Moscow schools of Lietheorists (including Serre)

20VLADIMIR G. PESTOVIn this Section we discuss the idea of a conjectural proof based entirely on univer-sal arrows type constructions (free topological groups and free Banach-Lie algebras).It is well known how by means of the Hausdorffseries H(x, y) one can associate inthe most natural and straightforward way a local Lie group (or, rather, a Lie groupgerm in the sense of [Ro]) to any Banach-Lie algebra g [Bou]. This is why, accordingto a result by ´Swierczkowski [´S1], the problem of enlarging a given Banach-Liealgebra g is completely reduced to the problem of embedding a local Banach-Liegroup U into a topological group G as a local topological subgroup.Let g be a Banach-Lie algebra.

Fix a neighbourhood of zero, U, such that theHausdorffseries H(x, y) converges for every x, y ∈U. (For example, set U equalto a closed ball of radius less than (1/3)log (3/2) [Bou].) Denote by Ng a closednormal subgroup generated by all elements of the form x−1[x.

(−y)]y, x, y ∈U.Clearly, the subgroup Ng is normal in F(g) and does not depend on the particularchoice of U. Denote by Gg the topological group quotient of F(g) by Ng, and byφg : g →Gg the restriction of the quotient homomorphism πg : F(g) →Gg to g.One can prove that πg is a universal arrow of a certain type.It is well known (in different terms, though — [´S2]) that the enlargability of g isequivalent to any of the following conditions: a) the intersection Ng ∩g is discretein g; b) the restriction of φg to a neighbourhood of zero in g is one-to-one; c) thetopological group Gg can be given a structure of an analytical Banach-Lie group insuch a way that φg is a local analytical diffeomorphism; in this case Lie (Gg) ∼= g,φg = expGg, and Gg is simply connected.Although one can show that the closedness of Ng in general is not sufficent forany of these conditions to be fulfilled, it is so in the following particular case.8.1. Theorem [Pe19].

A Banach-Lie algebra g with finite-dimensional center isenlargable if and only if the subgroup Ng is closed in F(g). In this case the quotienttopological group Gg carries a natural structure of a Banach-Lie group associatedto g.□The proof of this result goes as follows: firstly, it is reduced to separable Banach-Lie algebras with the help of a local theorem [Pe14], and then certain perfectly directand functorial constructions are used, including the free Banach-Lie algebra overthe underlying Banach space of g, the Banach-Lie group associated to it, and theirquotients.Now only one obstacle remains between us and a direct proof of the Lie-Cartantheorem.Conjecture.

The closedness of the subgroup Ng in the free topological group F(g)over the underlying topological space of a finite-dimensional Lie algebra g can beproved relying solely on the description of topology of free topological groups overfinite-dimensional Euclidean spaces.The subgroup Ng is compactly generated; since the compact set generating itis in F3(g) rather than F1(g) then one should single out some additional algebro-topological property of the group Ng which would ensure the closedness (or com-pleteness).We already know that Ng is always closed in F(g) for g finite-dimensional (it fol-lows from the Lie-Cartan theorem), and the problem looks so natural in this setting.It is so tempting to think that the genuine reason why the statement of Lie-Cartantheorem is always true for finite dimensional Lie algebras is not (co)homological but

UNIVERSAL ARROWS TO FORGETFUL FUNCTORS21entirely in the realm of general topology, namely: finite dimensional Lie algebrasare kω spaces, while infinite dimensional ones are not.9. Locally convex Lie algebras and groupsInfinite-dimensional groups play a major role in the contemporary pure andapplied mathematics [Kac1,2].Many of them cannot be given a structure of aBanach-Lie group (for example, groups of diffeomorphisms of manifolds, some oftheir subgroups preserving a certain differential-geometric structure, Kac-Moodygroups).

At the same time, in all particular examples to an infinite-dimensionalgroup there is associated in some natural way an infinite-dimensional Lie algebra,and therefore it is appealing, to try to develop a version of Lie theory with all itsattributes general enough to embrace all particular examples of infinite-dimensionalgroups.Such attempts have lead to the theory of Lie groups modeled over locally convexspaces (bornological and sequentially complete [Mil]), especially over Lie groupsmodeled over Fr´echet spaces [KoYMO]. We will call by a Fr´echet-Lie group a groupobject in the category of smooth Fr´echet manifolds, that is — in this case — justa smooth manifold modeled over a Fr´echet space which carries a group structuresuch that the group operations are Fr´echet C∞.There is a striking difference between the Banach and Fr´echet versions of Lietheory.

For example, although there is a well-defined notion of the Lie algebra,Lie (G), of a Fr´echet-Lie group G (which is a Fr´echet-Lie algebra), the exponentialmapping expG : Lie (G) →G need not be C∞nor a local diffeomorphism; thereforethere is in general no canonical atlas on a Fr´echet-Lie group. Moreover, the followingquestion seems to be still open:Question [Mil, KoYMO].

Does the exponential map expG : Lie (G) →G alwaysexist for a Fr´echet-Lie group G?Because of such misbehaviour of Fr´echet-Lie theory, some mathematicians arequestioning its ability to serve as a basis for infinite-dimensional group theory.Among them is Kirillov who once (Novosibirsk, January 1988) even expressed theopinion that obtaining an answer to the above question either in positive or innegative sense would be disadvantageous all the same!Nevertheless, we believe that this question should be answered in order to un-derstand the proper place of Fr´echet-Lie theory, and now we want to present a new,universal arrow type, construction of locally convex Lie algebras, which may give aclue.It is convenient to present the results in the spirit of ∆-normed spaces andalgebras belonging to Antonovski˘ı, Boltyanski˘ı and Sarymsakov [ABS].Let ∆be a directed partially ordered set. A vector space E is said to be ∆-normed if there is fixed a family of seminorms p = {pδ : δ ∈∆} with the propertypδ ≤pγ ⇔δ ≤γ.

(The family p is called a ∆-norm because it can be treated asa single map E × E →R∆where R∆is the so-called topological semifield, and itsatisfies close analogs of all three axioms of a usual norm. )Let A be an algebra.

We will say that a ∆-norm p = {pδ : δ ∈∆} on A issubmultiplicative if(i) for every δ, γ ∈∆such that δ < γ and for every x, y ∈A one has pδ(x ∗y) ≤p (x) p (y) where ∗denotes the binary algebra operation;

22VLADIMIR G. PESTOV(ii) for every δ ∈∆there is a γ such that for every x, y ∈A one has pδ(x ∗y) ≤pγ(x) · pγ(y).One can show that the topology of every locally convex topological algebra isgiven by an appropriate submultiplicative ∆-norm. For example, the locally multi-plicatively convex topological algebras introduced by Arens and Michael [Ar, Mic]are characterized by existence of a ∆-norm with the property pδ(x∗y) ≤pδ(x)·pδ(y)for all x, y ∈A and every δ ∈∆.For a fixed directed set ∆the class of all complete ∆-normed Lie algebras formsa category with contracting Lie algebra homomorphisms as morphisms.

We willdenote this category ∆LA.9.1. Theorem.

For every ∆-normed vector space (E, p) there exists a universalarrow from this space to the forgetful functor from ∆LA to the category of ∆-normed spaces. It is an isometric embedding of (E, p) into a ∆-submultiplicateivelynormed Lie algebra lie(E).□In a particular case where ∆is a one-point set, the above construction coincideswith the construction of a free Banach-Lie algebra over a normed space consideredearlier.If ∆has countable cofinality type (in particular, is countable) then the Liealgebra lie(E) is a Fr´echet-Lie algebra.The algebra lie(E) is centerless and infinite-dimensional (unless dim E = 1).It is completely unclear whether such Fr´echet-Lie algebras are enlargable (that is,come from Fr´echet-Lie groups).

The property of being centerless gives a hope thatthe answer is “yes,” at least in some cases. However, if ∆= N and a correspond-ing sequence of seminorms, p, grows “fast enough,” there is a good evidence thatlie(E, p) can have no exponential map.9.2.Theorem.

Let (E, ∥· ∥) be a normed space.Define a ∆-norm p, where∆= N, by letting pn = n!∥· ∥, n ∈N. Suppose there exist a Fr´echet-Lie group,G, associated to the Lie algebra lie(E).

Then there is no exponential map lie(E) →G.□One can also study free locally convex Lie algebras over locally convex spaces,that is, universal arrows from an LCS E to the forgetful functor from the category oflocally convex topological Lie algebras and continuous Lie algebra homomorphismsto the category of locally convex spaces. We will denote the free localy convex Liealgebra over E by LClie(E).

If X is a Tychonoffspace, then one can consider thefree locally convex Lie algebra over X, defined either as the coomposition of thefree locally convex space L(X) and the free locally convex Lie algebra, or directlyas the universal arrow from X to the forgetful functor from the category of locallyconvex topological Lie algebras and continuous Lie algebra homomorphisms to thecategory Tych. We denote this Lie algebra by LClieX.P.

de la Harpe has kindly drawn my attention to the following problematics.Problem (Bourbaki [Bou]). Is it true that every extension of a Lie algebra g bymeans of a g-module M is trivial (in other terms, H2(g, M) = (0) for every g-module M) if and only if g is a free Lie algebra?The property H2(g, M) = (0) is readily verifiable for a free Lie algebra g, butthe validity of inverse implication is not known

UNIVERSAL ARROWS TO FORGETFUL FUNCTORS23It is not clear yet whether free locally convex Lie algebras can help in answeringthe above question (supposedly in negative), but at the very least, they enjoy asimilar property for continuous second cohomology.9.3.Theorem. Let X be a separable metrizable topological space, and let Mbe a complete normable locally convex LClieX-module.Then every locally con-vex extension of the Lie algebra LClieX by means of M is trivial.

In particular,H2c (LClieX, M) = (0).□The proof follows the argument for free Lie algebras, but the Michael SelectionTheorem (Theorem 1.4.9 in [vM]) is involved.In some cases one managed to establish the triviality of algebraic second coho-mology for locally convex (and even Banach) Lie algebras [dlH].10. SupermathematicsThe (unhappy but hardly avoidable) term “supermathematics” is used to desig-nate the mathematical background of dynamical theories with nontrivial fermionicsector in the quasi-classical limit ℏ→0.

The “supermathematics” includes super-algebra, superanalysis, supergeometry etc., all of these being obtained from their“ordinary” counterparts by incorporating odd (anticommuting) quantities [BBHR,BBHRPe, B, DeW, Man1].In one of those approaches an important role is played by the so-called groundalgebras, or algebras of supernumbers; in other approach, algebras of this typecome into being as algebras of superfunctions over purely odd supermanifolds. Asa matter of fact, those algebras turn out to be universal arrows of a special kind,and they also find an independent application in infinite-dimensional differentialgeometry.We will give necessary definitions.

The term “graded” in this paper means “Z2-graded”. A graded algebra Λ is an associative algebra over the basic field K togetherwith a fixed vector space decomposition Λ ∼= Λ0 ⊕Λ1, where Λ0 is called the evenand Λ1 the odd part (sector) of Λ, in such a way that the parity ˜x of any elementx ∈Λ0 ∪Λ1 defined by letting x ∈Λ˜x, ˜x ∈{0, 1} = Z2, meets the followingrestriction:˜xy = ˜x + ˜y, x, y ∈Λ0 ∪Λ1If in addition one hasxy = (−1)˜x˜yyx, x, y ∈Λ0 ∪Λ1then Λ is called graded commutative.10.1.Theorem [Pe16,17].

Let E be a normed space. There exists a universalarrow ∧BE from E to the forgetful functor from the category of complete submul-tiplicatively normed graded commutative algebras to the category of normed spaces.It contains B as a normed subspace of the odd part (∧BE)1 in such a way thatE ∩{1} topologically generates ∧BE and every linear operator f from E to the oddpart Λ1 of a complete normed associative unital graded commutative algebra Λ witha norm ∥f∥op ≤1 extends to an even homomorphism ˆf : ∧B E →Λ with a norm∥ˆf∥≤1□

24VLADIMIR G. PESTOVAlgebraically, ∧BE is just the exterior algebra over the space E, endowed witha relevant norm and completed after that. It enjoys one more property.

A Banach-Grassmann algebra [JP] is a complete normed associative unital graded commuta-tive algebra Λ satisfying the following two conditions.BG1 (Jadczyk-Pilch self-duality). For any r, s ∈Z2 = {0, 1} and any boundedΛ0-linear operator T : Λr →Λs there exists a unique element a ∈Λr+s such thatTx = ax whenever x ∈Λr.

In addition, ∥a∥equals the operator norm ∥T∥op of T.BG2. The algebra Λ decomposes into an l1 type sum Λ ≃K ⊕J0Λ ⊕Λ1 whereK = R or C and J0Λ is the even part of the closed ideal JΛ topologically generated bythe odd part Λ1.

In other words, for an arbitrary x ∈Λ there exist elements xB ∈K, x0S ∈J0Λ, and x1 ∈Λ1 such that x = xB+x0S +x1 and ∥x∥= ∥xB∥+∥x0S∥+∥x1∥.10.2. Theorem [Pe17].

Let E be a normed space. The following conditions areequivalent:(i) dim E = 0;(ii) ∧BE is a Banach-Grassmann algebra.□The algebra ∧Bl1 (denoted by B∞) was widely used in superanalysis [JP].The algebras of the type ∧BE appear in infinite-dimensional differential geom-etry: in [KL], Klimek and Lesniewski used them for constructing Pfaffian systemsover infinite-dimensional Banach spaces after it became clear that the earlier consid-ered Pfaffians over Hilbert spaces are insufficient for applications in mathematicalphysics.If one wishes to study algebras of superfunctions on purely odd (that is, includ-ing fermionic degrees of freedom only) infinite dimensional supermanifolds modeledover locally convex spaces, then another universal arrow comes into being.

A locallyconvex graded algebra Λ carries two structures - that of a graded algebra and of lo-cally convex space — in such a way that multiplication is continuous and both evenand odd sectors are closed subspaces of Λ. A topological algebra A is called locallymultiplicatively convex, or just locally m-convex, if its topology can be described bya family of all submultiplicative continuous seminorms.

(Equivalently: A can beembedded into the direct product of family of normable topological algebra.) [Ar,Mic] An Arens-Michael algebra [He] is a complete locally m-convex algebra.10.3.

Theorem [Pe15,16]. Let E be a locally convex space.

Then there exists auniversal arrow ∧AME from E to the forgetful functor from the category of gradedcommutative Arens-Michael algebras to the category of locally convex spaces.□The two particular cases are well-known: ∧AMRℵ0 is the DeWitt supernumberalgebra [DeW], and ∧AMRω is the nuclear (LB) algebra considered in [KoN]. (HereRℵ0 stands for the direct product of countably many copies of R, and Rω denotesthe direct limit lim−→Rn.) In addition, in the finite-dimensional case, ∧AMRq is justthe Grassmann algebra with q odd generators.Perhaps, the same sort of construction would serve as a base for study of Pfafianson infinite dimensional locally convex spaces.At present one of the most appealing unsolved problem in “supermathematics”is to give a unified treatment of all existing approaches to the notion of a superman-ifold by viewing supermanifolds over non-trivial ground algebras Λ as superbundlesover Spec Λ.Denote by G the category of finite-dimensional Grassmann algebras and unitalalgebra homomorphisms preserving the grading Let LCSGop denote the category

UNIVERSAL ARROWS TO FORGETFUL FUNCTORS25of all contravariant functors from G to the category LCS of locally convex spacesand continuous linear operators; we call the category LCSGop the category of vir-tual locally convex superspaces. Every graded locally convex space E = E0 ⊕E1canonically becomes an object of LCSGop, because it determines a functor of theform ∧(q) 7→[∧(q) ⊗E]0; we will identify this functor with E. The simplest non-trivial example of a virtual graded locally convex space is R1,1 = R1 ⊕R1.

Thecategory LCSGop is a subcategory of the category DiffLCSGop of all contravariantfunctors from G to the category DiffLCS of locally convex spaces and infinitelysmooth mappings between them.Conjecture. The set of all morphisms in the category DiffLCSGop from a purelyodd graded locally convex space E to R1,1 carries a natural structure of a gradedlocally convex algebra canonically isomorphic to the free graded commutative Arens-Michael algebra, ∧AME′β, on the strong dual space E′β.11.

C⋆algebras and noncommutative mathematicsEvery normed space E admits a universal arrow to the forgetful functor fromthe category of (commutative) C⋆-algebras and their morphisms to the categoryof normed spaces and contracting linear operators; we will denote it by C⋆(E)(C⋆com(E), in commutative case), and refer to as the free (commutative) C⋆-algebraover a normed space. The arrows in both cases are isomorphic embeddings.

Thisis simply due to the two facts: firstly, every normed space E embeds into the C⋆-algebra of continuous functions on the closed unit ball of the dual space E′ withthe weak⋆topology, and secondly, the class of (commutative) C⋆algebras is closedunder the l∞-type sum.This construction is a particular case of the Blackadar’s construction of a C⋆-algebra defined by generators and relations [Bla]. For example, the free C⋆-algebraover a set Γ of free generators [GM] is just the free C⋆-algebra in our sense over theBanach space l1(Γ).

In non-commutative topology [BOB] the C⋆-algebras C⋆(l1(Γ))(treated as objects of the opposite category) are viewed as noncommutative versionsof Tychonoffcubes Iτ, because they are couniversal objects (universal — in theopposite category).It is known that every free C⋆-algebra is residually finite-dimensional (RFD),that is, admits a family of C⋆-algebra homomorphisms to finite-dimensional C⋆-algebras separating points [GM]. The same is true for our more general objects.11.1.

Theorem. For every normed space E the C⋆-algebra C⋆(E) is residuallyfinite-dimensional.□This result seems interesting because there are few known classes of RFD C⋆-algebras [ExL].Both embeddings have been considered earlier [BlP, Ru], where the so-calledmatrix norms on E defined by those embeddings are denoted by MAX and MIN.This construction is especially important for the so-called quantized functional anal-ysis [Eff], of which the idea is that all the main functional-analytic properties andresults concerning Banach spaces can be expressed in terms of the universal arrowC⋆com(E), so their non-commutative versions stated for C⋆(E) constitute the objectof quantized (that is, noncommutative) functional analysis.In this connection, it may be useful to consider two equivalence relations onBanach spaces two such spaces E and Fbeing equivalent iffC⋆(E) ∼C⋆(F)

26VLADIMIR G. PESTOV(respectively, C⋆com(E) ∼= C⋆com(F)).If one wishes to study “quantized” theory of LCS’s then one should turn tothe similar universal arrows from a given LCS E to the forgetful functor from thecategory of the so-called pro-C⋆-algebras in the sense of N.C. Phillips [Ph] (justinverse limits of C⋆-algebras) and their morphisms to the forgetful functor to thecategory of LCS’s; there are both commutative and non-commutative versions ofthose universal arrows.Finally, we expect that a whole new class of examples of the so-called quantumalgebras in the sense of Jaffe and collaborators [JLO] can be obtained by consideringuniversal arrows from a set of data including graded normed spaces to the relevantforgetful functor.AcknowledgmentsIt is an appropriate occasion to name all Institutions that supported my in-vestigations related to universal arrows during more than 13 years; I am gratefulto all of them. Those are: Institute of Applied Mathematics and Mechanics atTomsk State University (1978-80) and Faculty of Mechanics and Mathematics ofthe same University (1983-91); Division of Mathematics of Moscow State Univer-sity (1980-83); Faculty of Mathematics of Far Eastern State University, Vladivostok(1984); Institute of Mathematics, Novosibirsk Science Centre (1988-91); Depart-ment of Mathematics, University of Genoa, and Group of Mathematical Physicsof the Italian National Research Council (1990 and 1991); Department of Mathe-matics, University of Victoria, Canada (1991-92); and Department of Mathematics,Victoria University of Wellington, New Zealand (1992—).

A 1992 research grantV212/451/RGNT/594/153 from the Internal Grant Committee of the latter Uni-versity is acknowledged.In the present area of research I was most influenced by ideas of my Ph.D.advisor Professor A.V. Arhangel’ski˘ı.

My warm thanks also go to S.M. Berger, U.Bruzzo, M.M.

Choban, R.I. Goldblatt, S.P. Gul’ko, P. de la Harpe, A.E.

Hurd, S.S.Kutateladze, T. Loring, J. Leslie, N.C. Phillips, M.G. Tkachenko, O.V.

Sipacheva,V.V. Uspenski˘ı for their help expressed in various forms.The present article is an extended version of my invited talk at the Universityof Wollongong (Australia), and I am most grateful to Professor S.A. Morris forproviding such an excellent opportunity.

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