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Comparison of the geometric bar and W-constructions

We show that, for a simplicial group K,the realization of the W-construction of K is naturally homeomorphic to the universal bundle of its geometric realization. The argument involves certain recursive descriptions of the W-construction and classifying bundle and relies on the facts that the realization functor carries an action of a simplicial group to a geometric action of its realization and preserves reduced cones and colimits


Introduction
Let K be a simplicial group; its realization |K| is a topological group suitably interpreted when K is not countable. The W -construction W K → W K yields a functorial universal simplicial principal K-bundle, and the classifying bundle construction E|K| → B|K| of its geometric realization |K| yields a functorial universal principal |K|-bundle. The realization of the W -construction also yields a universal principal |K|-bundle |W K| → |W K|, by virtue of the general realization result in [26]. In this note we identify the the classifying bundle E|K| → B|K| with the realization of the W -construction. A cryptic remark about the possible coincidence of the two constructions may be found in the introduction to Steenrod's paper [25] but to our knowledge this has never been made explicit in the literature.
Spaces are assumed compactly generated, and all constructions on spaces are assumed carried out in the compactly generated category. It is in this sense that the realization |K| is always a topological group; in general, the multiplication map will be continuous only in the compactly generated refinement of the product topology on |K| × |K|. For countable K, there is no difference, though. Here is our main result.
Theorem. There is a canonical homeomorphism of principal |K|-bundles between the realization |W K| → |W K| of the W -construction and the classifying bundle E|K| → B|K| which is natural in K.
The map from |W K| to E|K| could be viewed as a kind of perturbed geometric Alexander-Whitney map while the map in the other direction is a kind of perturbed geometric shuffle map (often referred to as Eilenberg-Zilber map) but this analogy should not be taken too far.
The classifying space B|K| is the realization of the nerve N K of K as a bisimplicial set. The latter is homeomorphic to the realization of its diagonal DN K since this is known to be true for an arbitrary bisimplicial set [20]. The diagonal DN K, in turn, does not coincide with the reduced W -construction W K, though, but after realization the two are homeomorphic. We shall spell out the precise relationships in Section 4 below.
Eilenberg-Mac Lane introduced the bar and W -constructions in [6] and showed that, for any (connected) simplicial algebra A, there is a "reduction" of the realization |W A| of the reduced W -construction of A onto the (reduced normalized) bar construction B|A| of the normalized chain algebra |A| of A and raised the question whether this reduction is in fact part of a contraction. By means of homological perturbation theory, in his "Diplomarbeit" [27] supervised by the second named author, Wong answered this question by establishing such a contraction. Wong's basic tool is the "perturbation lemma" exploited in [8]; see [10] for details and history.
Our result, apart from being interesting in its own right, provides a step towards a rigorous understanding of lattice gauge theory. See [11] for details. Using the notation K Y for Kan group [12] of a reduced simplicial set Y , at this stage, we only spell out the following consequence, relevant for what is said in the references just cited.
Corollary. For a reduced simplicial set Y , there is a canonical map from its realization |Y | to the classifying space B|K Y | of the realization of K Y which is natural in Y and a homotopy equivalence.
The proof of our main result involves certain recursive descriptions of the Wconstruction and classifying bundle and relies on the facts that the realization functor carries an action of a simplicial group to a geometric action of its realization and preserves reduced cones and colimits. It would be interesting to extend the results of the present paper to simplicial groupoids, so that a result of the kind given in the Corollary would follow for an arbitrary connected simplicial set, with the Kan group replaced by the Kan groupoid [5]. We hope to return to this issue elsewhere. The argument we shall give here involves suitable (reduced) cone constructions which no longer serve for the desired purpose for the more general groupoid case.
We are indebted to Jim Stasheff and to the referee for a number of most helpful comments.
1. The classifying space of a topological group Let G be a topological group. Its nerve N G [2], [3], [21] is the simplicial space having in degree k ≥ 0 the constituent N G k = G ×k , with the standard simplicial operations. The usual lean realization BG = |N G| of N G is a classifying space for G, cf. [13], [21], [24]; there is an analoguous construction of contractible total space EG together with a free G-action and projection ξ onto BG, and this projection is locally trivial provided (G, e) is an NDR (neighborhood deformation retract) [25]. We note, for completeness, that the fat realization ||N G|| yields Milnor's classifying space [16], and the projection from the corresponding total space to ||N G|| is always locally trivial whether or not (G, e) is an NDR. Below (G, e) will always be CW-pair and hence an NDR, cf. e. g. the discussion in the appendix to [22], and we shall exclusively deal with the lean realization BG = |N G|. To reproduce a description thereof, and to introduce notation, write ∆ for the category of finite ordered sets [q] = (0, 1, . . . , q), q ≥ 0, and monotone maps. We recall the standard coface and codegeneracy operators , (0, 1, . . . , j − 1, j, . . . , q + 1) → (0, 1, . . . , j, j, . . . , q), respectively. As usual, for a simplicial object, the corresponding face and degeneracy operators will be written d j and s j . The assignment to [q] of the standard simplex ∇[q] = ∆ q yields a cosimplicial space ∇; here we wish to distinguish clearly in notation between the cosimplicial space ∇ and the category ∆. The lean geometric realization |N G| is the coend N G ⊗ ∆ ∇, cf. e. g. [14] for details on this notion. Exploiting this observation, Mac Lane observed in [13] that |N G| coincides with the classifying space for G constructed by Stasheff [23] and Milgram [15]; see also Section 1 of Stasheff's survey paper [24] and Segal's paper [21]. Mac Lane actually worked with a variant of the category ∆ which enabled him to handle simultaneously the total space EG and the base BG.
Steenrod [25] has given a recursive description of |N G| which we shall subsequently use. For ease of exposition, following [1], we reproduce it briefly in somewhat more categorical language. This will occupy the rest of this Section. For a space X endowed with a G-action φ: X ×G → X we write η = η G X : X → X ×G for the unit given by η(x) = (x, e). For an arbitrary space Y , right translation of G induces an obvious free G-action µ on Y × G. In categorical language [14], the functor ×G and natural transformations µ and η constitute a monad (×G, µ, η) and a G-action on a space X is an algebra structure on X over this monad. Sometimes we shall refer to an action of a topological group on a space as a geometric action.
Let D be any space and E a subspace endowed with a G-action φ: E × G → E; the inclusion of E into D is written β. Recall that the enlargement D ⊇ D of the G-action is characterized by the property: if Y is any G-space, and f any map from D to Y whose restriction to E is a G-mapping, then there exists a unique G-mapping f from D to Y extending f . The space D then fits into a push out diagram When D is based and E is a based subspace, the products E × G and D × G inherit an obvious base point, and the square (1.1) is one in the category of based spaces whence, in particular, the enlargement D inherits a base point. This notion of enlargement of G-action is functorial in the appropriate sense. See [25] for details. This kind of universal construction is available whenever one is given an algebra structure over a monad preserving push out diagrams.
The unit interval I = [0, 1] is a topological monoid under ordinary multiplication having 1 as its unit, and hence we can talk about an I-action X × I → X on a space X. Such an I-action is plainly a special kind of homotopy which, for t = 1, is the identity. In the above categorical spirit, the interval I gives rise to a monad (×I, µ, η) and an I-action on a space X is an algebra structure on X over this monad.
The base point of I is defined to be 0. Following [25], for a based space (X, x 0 ), we shall refer to an I-action ψ: X × I → X as a contraction of X (to the base point x 0 ∈ X) provided ψ sends the base point (x 0 , 0) of X × I to x 0 and factors through the reduced cone or smash product for all x ∈ X, t ∈ I; the reduced cone will be endowed with the obvious base point, the image of X × {0} ∪ {x 0 } × I in CX. Whenever we say "contraction", we mean "contraction to a pre-assigned base point". Abusing notation, the corresponding map from CX to X will as well be denoted by ψ and referred to as a contraction.
Moreover we write η = η C X for the map, the corresponding unit, which embeds X into CX by sending a point x of X to (x, 1) ∈ CX. The right action of I on X × I induces a contraction µ C X : CCX → CX of CX. Again we can express this in categorical language: the functor C and natural transformations µ and η constitute a monad and a contraction of a based space X is an algebra structure on X over this monad. Sometimes we shall refer to a contraction of a space as a geometric contraction.
Let (E, x 0 ) be any based space and (D, x 0 ) a based subspace endowed with a contraction ψ: CD → D; the inclusion of D into E is written α. The enlargement (E, x 0 ) ⊇ (E, x 0 ) of the contraction is characterized by the property: if f is any map from E to a space Y having a contraction to some point y 0 whose restriction to D is an I-mapping, then there exists a unique I-mapping f from E to Y extending f . The space E then fits into a push out diagram which provides a construction for E. Moreover, the composite of the unit η: E → CE with the map from CE to E in (1.4) embeds E into E and the right action of I on E × I induces a contraction of CE which, in turn, induces a contraction This notion of enlargement of contraction is functorial in the appropriate sense. See [25] for details. Alternating the above constructions, in [25], Steenrod defines based spaces and injections of based spaces by induction on n together with contractions ψ n : CD n → D n (Steenrod writes these contractions as I-actions D n × I → D n ) and G-actions φ n : E n × G → E n in the following way: Let D 0 consist of the single point e with the obvious contraction. Let E 0 = G, the right action being right translation. Now define (D 1 , e) to be the enlargement to (E 0 , e), (E 0 , e), of the contraction of (D 0 , e); then D 1 is just the reduced cone on E 0 . Define E 1 to be the enlargement to D 1 , D 1 , of the G-action on E 0 . In general, D n is the enlargement to (E n−1 , e), (E n−1 , e), of the contraction ψ n−1 of (D n−1 , e) so that D n fits into a push out square the requisite injection β n−1 : E n−1 → D n is the map denoted above by β, cf. (1.5); and the requisite contraction ψ n : CD n − → D n of (D n , e) or, equivalently, I-action ψ n : D n × I − → D n , is the map denoted above by ψ, cf. (1.6). Likewise, E n is the enlargement to D n , D n , of the G-action φ n−1 on E n−1 , so that E n fits into a push out square the requisite G-action φ n : E n × G − → E n and injection α n : D n → E n are the action denoted above by φ, cf. (1.2), and the map denoted above by α, cf. (1.3), respectively. The union endowed with the weak topology, inherits a G-action φ: E G ×G → E G and contraction ψ: CE G → E G from the φ n 's and ψ n 's, respectively. The G-action is free, and the orbit space BG = E G G equals the lean geometric realization |N G| of the nerve of G. This is Steenrod's result in [25].

The recursive description of the W -construction
In [1], the first named author observed that the W -construction admits a recursive description of formally the same kind as (1.7) above, except that it is carried out in the category of based simplicial sets. This is among the key points of the paper. We shall explain it now.
Let K be a simplicial group. Let e denote the trivial simplicial group viewed at the same time as the simplicial point. For a simplicial set X endowed with a K-action φ: X × K → X we write η = η K X : X → X × K for the unit of the action; in each degree, it is given by η(x) = (x, e). Given an arbitrary simplicial set Y , right translation of K induces an obvious action µ of K on Y × K. Much as before, in categorical language, the functor ×K and natural transformations µ and η constitute a monad (×K, µ, η) in the category of simplicial sets and a K-action on a simplicial set X is an algebra structure on X over this monad. Moreover realization preserves monad and algebra structures. In other words: the realization of a K-action φ: X × K → X on a simplicial set X is a geometric action |φ|: |X| × |K| → |X| in the usual sense. Notice this involves the standard homeomorphism [17] between the realization |X × K| of the simplicial set X × K and the product |X| × |K| of the realizations (with the compactly generated topology). The homeomorphism between |X × K| and |X| × |K| is of course natural and relies on the fact that, for an arbitrary bisimplicial set, the realization of the diagonal is homeomorphic to the realization as a bisimplicial set, cf. [20] (Lemma on p. 86). Note, however, that the product |X| × |K| yields a realization of X × K only after subdivision of the product CW-decomposition of |X| × |K|, cf. [19] (Satz 5 p. 388). This reflects, of course, the decomposition coming into play in the Eilenberg-Zilber map.
Recall that in the category of simplicial sets there are two natural (reduced) cone constructions. The first one is defined by the simplicial smash product with the standard simplicial model ∆ [1] of the unit interval. We shall say more about this in Section 4 below. The recursive description of the W -construction crucially involves the second somewhat more economical cone construction which relies on the observation that an (n + 1)-simplex serves as a cone on an n-simplex. We reproduce this cone construction briefly; it differs from the one given in [4] (p. 113) by the order of face and degeneracy operators; our convention is forced here by our description of the W -construction with structure group acting from the right, cf. what is said in (2.6) below.
Let X be a simplicial set. For j ≥ 0, we shall need countably many disjoint copies of each X j which we describe in the following way: For j ≥ 0, consider the cartesian product X j × N with the natural numbers N. Let o be a point which we formally assign dimension -1 and, given i ∈ N, write X −1 (i) = {(o, i)} so that each X −1 (i) consists of a single element; next, for j ≥ 0, let X j (i) = X j × {i}. The unreduced simplicial cone CX on X is given by with face and degeneracy operators given by the formulas Notice that in these formulas n − i = dim x; in particular, Let now (X, * ) be a based simplicial set. The unreduced simplicial cone C{ * } of the simplicial point { * } is the simplicial interval, and the reduced simplicial cone CX is simply the quotient CX = CX C{ * }.
For each n ≥ 0, its constituent (CX) n arises from the union X n (0) ∪ · · · ∪ X 0 (n) by identifying all ( * , i) to a single point written * , the base point of CX. The nondegenerate simplices of CX different from the base point look like (x, 0) and (x, 1) where x runs through non-degenerate simplices of X. We write η = η C X : X → CX for the unit induced by the assignment to x ∈ X n of (x, 0) ∈ X n (0). A (simplicial) contraction is, then, a morphism ψ: CX → X of based simplicial sets satisfying The cone CX itself admits the obvious contraction A contraction ψ is called conical provided We note that a geometric contraction in the sense of Section 1 above, being defined as an action of the associative monoid I, automatically satisfies the usual associativity law for an action. Under the present circumstances, the property of being conical corresponds to this associativity property. The contraction µ C X of CX is conical; in categorical terms, the triple (C, µ, η) is a monad in the category of simplicial sets, and a conical contraction is an algebra structure in the category of simplicial sets over this monad.
We now have the machinery in place to reproduce the first named author's recursive description the W -construction: Define based simplicial sets and injections of based simplicial sets by induction on n together with conical contractions ψ n : CD n → D n and K-actions φ n : E n × K → E n on each E n from the right in the following way: Let D 0 = e, with the obvious conical contraction ψ 0 , let E 0 = K, viewed as a based simplicial set in the obvious way, the right action φ 0 being translation, and let α 0 be the obvious morphism of based simplicial sets from D 0 to E 0 . For n ≥ 1, define (D n , e) to be the enlargement to (E n−1 , e) of the contraction ψ n−1 : CD n−1 → D n−1 , that is, D n is characterized by the requirement that the diagram be a push out square of (based) simplicial sets; the composite of the unit η from E n−1 to CE n−1 with the morphism CE n−1 − → D n of simplicial sets in (2.2) yields the requisite injection β n−1 : E n−1 → D n , and the contraction ψ n−1 and the conical contraction of CE n−1 induce a conical contraction ψ n : CD n − → D n . Likewise, E n is the enlargement to D n of the K-action φ n−1 on E n−1 , that is, E n is characterized by a push out square of based simplicial sets of the kind the requisite K-action φ n : E n ×K − → E n is induced by φ n−1 and the obvious K-action on D n × K, and the requisite injection α n : D n → E n is the composite of the unit with the morphism D n × K − → E n of simplicial sets in (2.3). The limit inherits a K-action φ: W K × K → W K and conical contraction ψ: CW K → W K from the φ n 's and ψ n 's, respectively. The K-action is free, and the projection map to the quotient W K = W K K yields the universal simplicial K-bundle or W -construction of K, cf. [1], with action of K from the right. For intelligibility, we explain some of the requisite details: A straightforward induction establishes the following descriptions of the simplicial sets D k and E k : . , i s , e, i s+1 , . . . ) ∼ (. . . , i s + i s+1 , . . . ), (. . . , k s , 0, k s+1 , . . . ) ∼ (. . . , k s k s+1 , . . . ).
Thus, for n ≥ 0, From this, adding the requisite neutral elements wherever appropriate, we deduce the following more common explicit description: For n ≥ 0, with face and degeneracy operators given by the formulas further, (W K) 0 = {e} and, for n ≥ 1 with face and degeneracy operators given by the formulas Remark 2.6. Here preferred treatment is given to the last face operator, as is done in [9] and [12]. This turns out to be the appropriate thing to do for principal bundles with structure group acting on the total space from the right and simplifies comparison with the bar construction. See for example what is said on p. 75 of [9]. The formulas (2.4) and (2.5) arise from those given in (A.14) of [9] for a simplicial algebra by the obvious translation to the corresponding formulas for a simplicial monoid; they differ from those in [4] (pp. 136 and 161) where the constructions are carried out with structure group acting from the left.

The proof of the Theorem
The realization of a conical contraction ψ: CX → X of a based simplicial set (X, x 0 ) is a geometric contraction |ψ|: C|X| → |X| in the sense reproduced in Section 1 above. In fact, the association (|x|(t 0 , . . . , t n ), t) −→ (|(x, 1)|(tt 0 , . . . , tt n , 1 − t), x ∈ X n , n ≥ 0, yields a homeomorphism from the reduced cone C|X| on the realization |X| to the realization |CX| of the cone and, furthermore, the realizations of the unit η and C-algebra structure µ C X : CCX → CX yield the geometric unit |X| → C|X| and geometric C-algebra structure µ C |X| : CC|X| → C|X|, that is, the realization preserves monad-and C-algebra structures.
The proof of the Theorem is now merely an elaboration of the observation that the realization functor | · | carries an action of a simplicial group to a geometric action of its realization, preserves reduced cones and, having a right adjoint (the singular complex functor), also preserves colimits. In fact, denote the corresponding sequence (1.7) of based topological spaces for the realization |K| by and, likewise, write for the corresponding sequence (2.1) in the category of based simplicial sets. Realization carries the sequence (3.2) to the sequence are commutative; (2) each τ j identifies the realization |ψ j K|: |CD j K| − → |D j K| of the conical contraction ψ j K: CD j K − → D j K of simplicial sets with the geometric contraction ψ j |K|: CD j |K| − → D j |K|; (3) each ρ j identifies the realization |φ j K|: Consider the realization of (2.2); it is a push out square of topological spaces. Hence the maps τ n−1 and ρ n−1 induce a map τ n from D n |K| to |D n K|, necessarily a homeomorphism, so that C|τ n−1 |, |τ n−1 |, C|ρ n−1 | and |τ n | yield a homeomorphism of squares between the realization of (2.2) and (1.8). Moreover, the homeomorphism τ n identifies the realization |ψ n K|: |CD n K| − → |D n K| of the conical contraction ψ n K: CD n K − → D n K of simplicial sets with the contraction ψ n |K|: CD n |K| − → D n |K|. Likewise the maps ρ n−1 and τ n induce a map ρ n from E n |K| to |E n |K|, necessarily a |K|-equivariant homeomorphism, so that |ρ n−1 × Id K |, |ρ n−1 |, |τ n × Id K | and |ρ n | yield a homeomorphism of squares between the realization of (2.3) and (1.9). Moreover, the homeomorphism ρ n is |K|-equivariant and identifies the realization |φ n K|: |(E n K) × K| − → |E n K| of the simplicial K-action φ n K: (E n K) × K − → E n K with the topological |K|-action φ n |K|: (E n |K|) × |K| − → E n |K|. The requisite diagrams (3.4) for j = n are manifestly commutative. This completes the inductive step.
The limit ρ = lim ρ n = lim τ n : E|K| − → |W K| is a |K|-equivariant homeomorphism; it identifies the principal |K|-bundles E|K| → B|K| and |W K| → |W K| as asserted, is plainly natural in K and, in particular, induces a natural homeomorphism from B|K| to |W K|. This proves the Theorem.

The other cone construction
The classifying space B|K| is the realization of the nerve N K of K as a bisimplicial set. On the other hand, the diagonal DN K is a simplicial set which does not coincide with the reduced W -construction W K, but its realization is homeomorphic to the realization of the nerve N K of K as a bisimplicial set since this is known to be true for an arbitrary bisimplicial set [20]. The purpose of this Section is to clarify the relationships between the various spaces and constructions.
As already pointed out, the construction (2.1) can be carried out with the simplicial smash product (·) ∧ ∆ [1] instead of the reduced cone: The simplicial interval ∆[1] carries a (unique) structure of a simplicial monoid having (1) as its unit, and hence we can talk about an action X × ∆[1] → X of ∆[1] on a simplicial set X; such an action is a special kind of simplicial homotopy which "ends" at the identity morphism of X. The fact that the naive notion of homotopy of morphisms of simplicial sets is not an equivalence relation is not of significance here. Much as before, the simplicial interval ∆ [1] gives rise to a monad (×∆ [1], µ, η) in the category of simplicial sets and an action of ∆[1] on a simplicial set X is an algebra structure on X over this monad.
The base point of ∆ [1] is defined to be (0). For a based simplicial set (X, x 0 ), we shall refer to an action ψ: X × ∆[1] → X as a ∆[1]-contraction of X provided ψ sends the base point (x 0 , 0) of X × ∆[1] to x 0 and factors through the simplicial smash product The latter is viewed endowed with the obvious base point, the image of X × {0} ∪ {x 0 } × ∆ [1] in X ∧ ∆ [1]. Abusing notation, the corresponding map from X ∧ ∆[1] to X will as well be denoted by ψ and referred to as of X ∧∆ [1]. In categorical language, the functor (·)∧∆ [1] and natural transformations µ and η constitute a monad in the category of simplicial sets, and a ∆[1]-contraction of a based simplicial set X is an algebra structure on X over this monad.
Formally carrying out the construction (2.1) with the simplicial smash product (·) ∧ ∆[1] instead of the reduced cone yields based simplicial sets and injections of based simplicial sets together with morphisms ψ ′ n : D ′ n ∧ ∆[1] → D ′ n of simplicial sets having certain properties and free K-actions φ ′ n : E ′ n × K → E ′ n . Its limit n inherits a morphism ψ ′ : D ∧ ∆[1] → D of simplicial sets and a free K-action φ ′ : D ×K → D. To explain the significance thereof, recall that the nerve construction yields a simplicial object in the category of principal simplicial K-bundles which is natural for morphisms of simplicial groups. Here EN K and N K inherit structures of bisimplicial sets, one from the nerve construction and the other one from the simplicial structure of K, and the projection from EN K to N K is a morphism of bisimplicial sets; further, for each simplicial degree q ≥ 0 coming from the nerve construction, (4.2) amounts to a principal K-bundle while for each simplicial degree p ≥ 0 of K = {K p } itself, (4.2) comes down to the universal simplicial principal K p -bundle  Proof. The classifying space B|K| is the realization of N K as a bisimplicial set, and the same kind of remark applies to E|K| and the projection to B|K|. The already cited fact that, for an arbitrary bisimplicial set, the realization of the diagonal is homeomorphic to the realization as a bisimplicial set [20] implies the following statement.

4.4.
There is a canonical |K|-equivariant homeomorphism between |DEN K| and E|K| and hence a canonical homeomorphism between |DN K| and B|K|. These homeomorphisms are natural in K.
We conclude from this that the statement of the Theorem (in the Introduction) is formally equivalent to the statement of (4.3). In fact, the Theorem identifies the realization the of W -construction with the realization of the nerve as a bisimplicial set whereas (4.3) identifies the realization of the W -construction with the realization of the diagonal of the nerve. Remark 1. While the statement of (4.4) is obtained for free, the identifications just mentioned, in turn, are not obtained for free, as we have shown in this paper.
Remark 2. For a based simplicial set (X, * ), the realization |CX| of the cone CX is naturally homeomorphic to the realization |X ∧ ∆[1]| of X ∧ ∆ [1]. In fact, a suitable subdivision of |CX| yields a realization of X ∧ ∆ [1]. It is tempting trying to construct a homeomorphism between |DEN K| and |W K| in a combinatorial way by inductively constructing the requisite maps between the realizations of the constituents of (4.1) and of the corresponding terms in (2.1) but we did not succeed in so doing. The problem is that the realization of the simplicial monoid ∆ [1] does not yield the geometric monoid structure on the interval I coming into play in Section 1 above whence the realization of an action X × ∆[1] → X of ∆[1] on a simplicial set X is not an I-action on the realization of X in the sense of Section 1. Rather, the realization of the simplicial monoid structure on ∆[1] yields the function from I × I to I which sends (a, b) to min(a, b). A suitable homeomorphism identifies this monoid structure with the one considered in Section 1 above. For example, as pointed out by the referee, one could take the function which assigns (a max(a, b), b max(a, b)) ∈ I 2 to (a, b) ∈ I 2 . Further, the monoid structure arising from the function min also gives rise to a monad in the category of spaces and with reference to it, the construction (1.7) can still be carried out; formally the same argument as that for the proof of our main result then identifies the limit (say) LK of the resulting sequence of spaces with the realization |DN K| of DN K and, by virtue of (4.4), LK is naturally homeomorphic to B|K|. However we do not see how this homeomorphism may be obtained directly since we are unable to identify the monad in the category of spaces arising from the unit interval having the usual multiplication as monoid structure with the other monad arising from the function min as monoid structure.