Appl Categor Struct

DOI 10.1007/sl0485-015-9423-x V0,

Tensor Functional Topology on Woronowicz Categories

Wendy Lowen1 • Joris Mestdagh1

Received: 19 December 2015 / Accepted: 28 December 2015

© The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract Functional topology is concerned with developing topological concepts in a category endowed with certain axiomatically defined classes of morphisms (Clementino et al. 2004). In this paper, we extend functional topology to a monoidal framework, replacing categorical pullbacks by pullbacks relative to the monoidal structure (which itself replaces the product) or more generally relative to a relation on the category (Janelidze, Appl. Categ. Structures, 17(4),351-371, 2009). Our main application is to the opposite Woronowicz category of C*-algebras. In this category a natural class of proper morphisms yields the unital algebras as compact objects. When restricted to the commutative C * -algebras, we recover exactly the morphisms induced by proper continuous maps of locally compact Hausdorff spaces. We further endow this category with a factorization system and investigate the precise relation with the proper maps, building on an approach which we previously developed with the eye on the category of schemes (Lowen and Mestdagh, J. Pure Appl. Algebra 217(11), 2180-2197, 2013). We also show how our results for C*-algebras can naturally be adapted to the opposite Woronowicz category of nondegenerate algebras over a commutative ring.

Keywords Functional topology • C*-algebra • Woronowicz category • Proper morphism

To Horst Herrlich: through hugely inspiring mathematics and moonlit memories your wonderful mind shines on.

The authors acknowledge the support of the European Union for the ERC grant No 257004-HHNcdMir and the support of the Research Foundation Flanders (FWO) under Grant No G.0112.13N.

S Wendy Lowen

wendy.lowen@uantwerpen.be

Joris Mestdagh mestdaghjoris@hotmail.com

Departement Wiskunde-Informatica, Middelheimcampus, Middelheimlaan 1, 2020 Antwerp, Belgium

Published online: 29 January 2016

Springer

1 Introduction

By the famous Gelfand-Naimark theorem, the category of locally compact Hausdorff spaces with proper maps is dual to the category of commutative C*-algebras with C*-morphisms. This theorem forms the basis for the point of view of noncommutative geometry, where noncommutative C*-algebras are considered as "noncommutative spaces". An obvious shortcoming in the above result is the a priori restriction to proper maps between topologi-cal spaces. This problem is remedied by turning to the Woronowicz category of C*-algebras Wor(C*-Alg), of which the restriction to commutative C*-algebras is equivalent to the category of locally compact Hausdorff spaces with all continuous maps. The definition of the Woronowicz category is somewhat involved, and makes essential use of the construction of the multiplier algebra, which is the largest unitization of a C*-algebra and constitutes the noncommutative counterpart of the Cech-Stone compactification. This suggests that unital C*-algebras should somehow be considered as compact objects.

The main aim of this paper is to introduce and investigate notions of compactness and properness for C*-algebras in the context of so-called functional topology on the category Wor(C*-Alg)°p. In very general terms, functional topology is an approach to developing topological concepts in a category based upon certain (axiomatically defined) classes of maps. This basic idea dates back at least to the seventies with the work of Herrlich [10], Manes [18], Penon [20, 21], and in the presence of a factorization system it led to a richer theory in the work of Herrlich, Salicrup and Strecker [12], applicable in topology, group theory and order. In the 90's, on categories endowed with closure operators, a close resemblance with the topological situation was obtained by Clementino, Giuli and Tholen [4], applicable to Birkhoff closure spaces, uniform spaces, topological groups and locales. The idea of using a class of closed morphisms in order to derive both proper and separated mor-phisms from it was contained in Tholen's [22], and gave rise to the theory of functional topology as developed by Clementino, Giuli and Tholen in the presence of a factorization system [5], which is applicable for instance to approach spaces as demonstrated by Colebunders, Lowen and Wuyts [6]. An approach by Hofmann and Tholen later focussed the attention on the proper maps as primary, and is applicable to general categories of lax algebras [13].

In [16], after noticing the similarities between functional topology and the classes of proper and separated morphisms of schemes introduced by Grothendieck, the authors extended the framework from [5], making it somewhat more flexible without greatly diminishing its power, and in particular making it applicable to schemes [16]. Surprisingly, the resulting framework turns out to be well-suited to incorporate the Woronowicz category. The main new element we need is a shift from finitely complete categories to appropriate monoidal categories. Indeed the category Wor(C*-Alg)op has no (known) binary products but is endowed with a monoidal structure given by the maximal tensor product. In the classical functional topology setup, a proper class of morphisms is a class containing the isomorphisms, which is closed under compositions and pullback-stable. This last property does not seem quite natural in a monoidal context. To find the appropriate replacement, we investigate generalized limits in a more general framework: that of categories endowed with a relation R, that is a collection of compatible relations on the classes of morphisms with common domain. The notion of a relation on a category is dual to the notion of a cover relation in the sense of Janelidze [14], which we call a co-relation in the present paper. Associated R-limits originate from considering cones in which some of the involved morphisms are required to be in relation. For instance, we define the R-pullback of two morphisms

f : A —> C and g : B —> C to consist of a universal object P with morphisms P1 : P —> A and p2 : P —> B such that fpi = gp2 and (p\, p2) G R. Natural notions of R-commutative objects and R-central morphisms relative to a relation can be defined and expressed in terms of generalized limits. After establishing some basic results involving R-limits in Section 4, in Section 5 we develop some approaches to functional topology in a category with R-pullbacks and a final object. We will refer to these approaches as "tensor functional topology".

• The most basic approach takes an R-proper class F of proper morphisms as input. With respect to F one defines compact objects as objects for which the unique morphism to the final object is proper, separated morphisms as morphisms with a commutative domain and proper diagonal, and Hausdorff objects as commutative objects with proper diagonal. Starting the theory from an R-proper class rather than a closed class is preferable as for general R, we cannot naturally associate an R-proper class to a closed class due to the weaker composability properties of R-pullback diagrams.

• If C is endowed with a factorization system (E, M), one can obtain a closed class F of closed morphisms from a so-called (E, M)-closed structure (P, Fo) consisting of an auxiliary R-proper class Fo of closed immersions and an auxiliary closed class P of surjections, following [16]. I n general, F will not be R-proper. In case of general R, we may additionally require F to be an R-proper class, in which case we speak of an (E, M) -proper structure with proper morphisms F.

Many classical results from functional topology still hold true in tensor functional topology, although sometimes additional centrality conditions need to be imposed. To understand this, it is worth noting that if we restrict our category to the full subcategory of commutative objects, the relation R restricts to the standard relation for which any two morphisms with common domain are related.

In Section 6, we give an account of tensor functional topology in the opposite Woronow-icz category Wor(C*-Alg)op. The necessary background on this category is recalled in Section 2. For a morphism f : A —> B in Wor(C*-Alg)op, we write F : B —> M (A) for the associated Woronowicz-morphism. Two morphisms f : A —> B and f' : A —> B' are R-related if the associated Woronowicz-morphisms F and F' : B' —> M(A) commute. We make the following definitions for f : A —> B:

• f is in F if F(B) c A;

• f is in M if F(B) = A;

• f is in E if F is an isometry.

We show that F is an R-proper class, for which compact objects correspond precisely to unital algebras, and all commutative algebras are Hausdorff. Further, (E, M) is a factorization system on Wor(C*-Alg)op and (E n F, M) is an (E, M)-proper structure with F as its class of proper morphisms. Restricting Wor(C*-Alg)op to the commutative objects yields a category equivalent to locally compact Hausdorff spaces with continuous maps. We show that the restriction of (E, M) to this category corresponds precisely to the factorization system of dense maps and closed embeddings. Further, the class F restricts precisely to the proper continuous maps.

The application of tensor functional topology to C*-algebras raises the natural question how much of this approach goes through for ordinary algebras. As we recall in Section 3, a multiplier algebra is known to exist for nondegenerate fc-algebras over a commutative

ring k. We further introduce the nondegenerate tensor product between nondegenerate k-algebras. We obtain a Woronowicz category Wor(k-Alg) of nondegenerate k-algebras which closely resembles the category Wor(C*-Alg). In particular, its opposite category can be endowed with the commutation relation R and classes of morphisms F, M, E analogous to the higher classes. All higher statements about the C*-algebra case have precise parallels for nondegenerate k-algebras, as shown in Section 6. In fact, under mild assumptions on a symmetric monoidal closed category C, a Woronowicz category of non-degenerate semigroup objects in C can be defined, and Wor(k-Alg) can be obtained from this general construction by taking C to be the category of k-modules [17]. The development of tensor functional topology in these more general Woronowicz categories is work in progress.

2 The Woronowicz Category of C*-Algebras

The construction of the Woronowicz category for C*-algebras originates with Woronowicz [24]. We recall the details for convenience and to fix notation. More details and proofs can be found in Wegge-Olsen [23] and Lance [15]. We further prove the existence and construction of co-equalizers in the Woronowicz category (Theorem 2.14).

Definition 2.1 A C*-algebra is a (possibly noncommutative, possibly nonunital) C-algebra equipped with a norm || — || and an involution (-)* : A —> A such that

(1) (kx + /y)* = kx* + /y* for each k,/ e C and x,y e A.

(2) (xy)* = y*x* for each x, у e A.

(3) x** = x for each x e A.

(4) A is complete with respect to the norm || ||.

(5) ||xy|| < ||x||||y|| for each x, у e A.

(6) ||xx*|| = ||x||2 for each x e A.

A C*-algebra is called unital if it is unital as an algebra, i.e. if there is a unit element 1a e A such that 1ao. = a = я1а for each a e A.

The C*-morphisms are defined as follows:

Definition 2.2 Let A and B be C*-algebras and let f : A —> B be a function. We say that f is a C*-morphism if

(1) f is linear,

(2) f(xy) = f(x)f(y) for each x, y e A,

(3) f(x*) = f(x)* for each x e A.

If A and B are unital, then we say that f is unital if f(1A) = 1b.

One can show that each C*-morphism f is continuous and has norm ||f || < 1. This choice of morphisms makes the C*-algebras into a category, which we will call C*-Alg. The theorem of Gelfand-Naimark classifies the commutative C*-algebras. For a locally compact Hausdorff space X, we obtain the commutative C*-algebra Co(X, C) of continuous functions from X to C which vanish at infinity. For a compact Hausdorff space X, we obtain the unital commutative C*-algebra C(X, C) of continuous functions from X to C. In this case, we have C (X, C) = Co(X, C).

Theorem 2.3 (Gelfand-Naimark) [23] The category of locally compact Hausdorff spaces with proper maps as morphisms is equivalent to the category of commutative C*-algebras with C*-morphisms as morphisms. The (contravariant) equivalence is given by associating to each locally compact Hausdorff space X, the commutative C*-algebra Co(X, C). Furthermore, under this equivalence, the category of compact Hausdorffspaces with continuous maps as morphisms is equivalent to the category ofunital commutative C*-algebras with unital C*-morphisms as morphisms.

While the proper continuous functions between locally compact Hausdorff spaces correspond to the C*-morphisms under this equialence, the same is not true for all the continuous functions. To be able to model all continuous functions, we need the concept of the multiplier algebra.

Definition 2.4 Let A be a C*-algebra. We say that an ideal J of A is an essential ideal if every other nonzero ideal in A has nonzero intersection with A. Or equivalently, when the annihilator J± = {a G A | aJ = 0} is zero.

Definition 2.5 A unitization of a C*-algebra A is an embedding of A into a unital C*-algebra B such that A is an essential ideal of B.

Since unital C*-algebras correspond to compact Hausdorff spaces in Theorem 2.3, we see that the concept of unitization corresponds to compactification. The largest compact-ification is of course the Cech-Stone compactification. Likewise, every C*-algebra has a largest unitization, which we will call the multiplier algebra.

Theorem 2.6 [23] Let A be a C*-algebra. There exists a unique unitization M(A) of A such that if A is embedded as an ideal in a C*-algebra B, then there exists a unique mor-phism i : B —> M (A) such that i restricts to the identity on A. Moreover, i is injective if and only if A is essential in B.

We call M (A) the multiplier algebra of A.

If A = C0(X, C) for a locally compact Hausdorff space X, then the multiplier algebra M(A) is exactly C(fX, C) = Cb(X, C), the commutative C*-algebra of bounded continuous functions from X to C, where fX is the (Cech-Stone compactification of X.

The multiplier algebra is not functorial. More precisely, if f : A —> B is a C*-morphism, then this does not need to induce a morphism M(A) —> M(B).

Definition 2.7 Let A be a C*-algebra. Let H be a Hilbert space such that A is embedded as a subspace of B(H). The strict topology on A is the locally convex topology generated by the seminorms x —> ||xa\\ and x —> ||ax\\ for x G B(H) and a G A.

One can prove that the multiplier algebra M(A) is the strict completion of A.

For C*-algebras A and B, we call a C*-morphism F : A —> M(B) nondegenerate if F(A)B is dense in B. We are now able to state the following extension theorem:

Theorem 2.8 [15] Let A and B be C*-algebras and let F : A —> M(B) be a C*-morphism. The following conditions are equivalent:

(1) F is nondegenerate.

(2) F(A)B = B.

(3) BF(A) = B.

(4) F is the restriction to A of a unique unital C*-morphism M(A) —> M(B) which is strictly continuous on the unit ball.

A C*-morphism F : A —> M(B) that satisfies the conditions of the previous theorem will be called a Woronowicz-morphism from A to B. The unique unital extension of F : A —> M(B) from (4) will be denoted by F : M(A) —> M(B).

The continuous maps between locally compact Hausdorff spaces X and Y correspond exactly to the Woronowicz-morphisms from Cq(Y, C) to Cq(X, C). With this in mind, we define the category Wor(C*-Alg) to be the category of all C*-algebras with Woronowicz-morphisms. Note that the above theorem says that we can compose such morphisms, thus the Woronowicz-category is well defined. Naturally, we are more interested in the opposite category Wor(C*-Alg)op since this category can be seen as a category of "noncommutative topological spaces."

Recall that the multiplier algebra was not functorial with respect to the C*-morphisms. This is solved by the Woronowicz-morphisms: any Woronowicz-morphism from A to B gives rise to a unique unital C*-morphism M(A) —> M(B).

We would like to form tensor products of C*-algebras A and B. An obvious definition is to form the algebraic tensor product A ( B and to complete this with respect to a suitable norm. A suitable norm should satisfy ||x ( y|| < ||x||||;y|| (this is called a subcross norm). The largest possible subcross norm is called the maximal norm:

Definition 2.9 Let A and B be C*-algebras and define for each t e A ( B

H(t) = sup{a(t) | a is a C*-seminormon A ( B}.

This is called the maximal C*-norm on A ( B. The completion with respect to this norm is denoted by A B.

Definition 2.10 Let A, B and C be (possibly noncommutative, possibly nonunital) rings and let f : A —> C and g : B —> C be ring-morphisms. We say that f and g commute if for each a e A and b e B we have that f(a)g(b) = g(b) f(a).

Tensor products do not form a coproduct for C*-algebras, but we do have the following theorem:

Theorem 2.11 [23] Let fk : Ak —> Bk, k = 1, 2 be morphisms of C*-algebras, then fx ( f2 extends by continuity to a morphism f\ f2 : Ax A2 —> B\ B2.

If gk : Ak —> C, k = 1, 2 are commuting morphisms of C*-algebras, then g1 ( g2 extends by continuity to a morphism g1 g2 : A1 A2 —> C.

We collect some facts about the category Wor(C*-Alg):

Proposition 2.12 If F : A —> M(B) is a Woronowicz-morphism from A to B which constitutes an isomorphism in Wor(C*-Alg), then F(A) = B.

Proof Let G : B —> M(A) be the inverse of F. Take an a G A, then we have by nondegeneracy of G that there exist bi G B and ai G A such that a = ^ G(bi)ai. But then F(a) = ^ biF(ai). But since B is an ideal of M(B), this implies that F(a) G B. This shows that F(A) C B. Likewise, we have that G(B) C A. Now take b G B, then b = F(G(b)), and thus b g F(A). □

Proposition 2.13 [23] Any surjection of C*-algebras p : A —> B induces a nondegener-ate morphism P : A —> M(B).

Theorem 2.14 The co-equalizer in the category Wor(C*-Alg) exists. In particular, the co-equalizer of two nondegenerate C*-morphisms F,G : A —> M(B) is given by B/J, where J is the two-sided ideal in B generated by F(a)b — G(a)b and bF(a) — bG(a) for each a G A and b G B .

Proof Let p : B —> B/J be the canonical surjection. This surjection extends to a non-degerate morphism P : B —> M(B/J). We prove that PF = PG. Let (bi)iGI be an approximate unit in B, i.e. it is a net in B which converges strictly to the unit in M(B). By strict continuity, we know that (P(bi))iGI converges to the unit in M(B/J) (see [23]). Thus follows that F(a)bi converges strictly to F(a) and that G(a)bi converges strictly to G(a). By definition of C as a quotient space, we have that P(F(a)bi) = P(G(a)bi) and since P(F(a)bi) converges to P(F(a)) and P(G(a)bi) converges to P(G(a)), we have that P(G(a)) = P(G(a)). _ _

Let H : B —> M(C) be a Woronowicz morphism from B to C such that HF = HG. Then H(J) = 0 and thus there is a unique morphism K : B/J —> M(C) such that H = KP. This is clearly nondegenerate since H is. □

Proposition 2.15 Let G : A —> M(C) and G : B —> M(C) be two Woronowicz-morphisms. If F and G commute, then so do their extensions F and G.

Proof Take x g M(A) and y G M(B). Then there exist nets (a{)iGI in A and (bj)jgj in B such that ai —> x strictly and bi —> y strictly. But then F(ai) converges strictly to F(x) and G(bj) converges strictly to G(y). Then F(a{)G(bj) converges strictly to F(x)G(y) and G(bj)F(ai) converges strictly to G(y)F(x). Since F(a{)G(bj) = G(bj)F(a{), it follows that T(x)G(y) = G(y)F(x). □

Proposition 2.16 [15] Let M : A —> M(C) be an injective and nondegenerate C*-morphism. Then M extends uniquely to an isometric, unital C*-morphism M : M(A) —> M(C). Furthermore, the image of M is given by the idealizer of M(A), i.e.

M(M(A)) = {x G M(C) I xM(A) C M(A) and M(A)x C M(A)}.

3 The Woronowicz Category of Associative Algebras

In this section we generalize the Woronowicz category to nondegenerate associative algebras over a commutative ground ring k with unit. The basic notions necessary for this generalization can be found in De Commer, Van Daele [7]. In [17], we show that under mild assumptions on a symmetric monoidal closed category C, a Woronowicz category of non-degenerate semigroup objects in C can be defined. The Woronowicz category of this section can be obtained from this general construction by taking C to be the category of k-modules. However, the general categorical proofs turn out to be much more involved. In this section we provide a self-contained treatment of the results for associative k-algebras.

Definition 3.1 A k-algebra A is nondegenerate if the following hold for all a G A:

(1) If ab = 0 for each b G A, then a = 0.

(2) If ba = 0 for each b G A, then a = 0.

(3) We have that A = AA := aia'i | at, a[ G A}.

The category of all nondegenerate k-algebras with the usual k-algebra morphisms is denoted by к — Alg.

Definition 3.2 The multiplier algebra of a nondegenerate k-algebra A is the algebra M (A) consisting of couples (k, p), where k and p are maps A —> A such that

p(a)b = ak(b) for all a,b G A.

This k-algebra is a k-unital algebra equipped with the following operations:

(1) For all (к, p), (к', p) G M(A) and r, s G A, we set

r(k, p) + s(k', p') = (rk + sk', rp + sp').

(2) For all (p, k), (p', k') G M(A), we set

(k,p) ■ (k',p') = (k о k',p' о p).

(3) The unit is given by (IdA, IdA).

Lemma 3.3 If A is a nondegenerate k-algebra and if (k, p) G M (A), then both k and p are k-linear. Furthermore, it holds for each a, b G A that

k(ab) = k(a)b and p(ab) = ap(b).

Proof Let a,b, c G A be arbitrary, and let r, s G k, then

ck(ra + sb) = p(c)(ra + sb)

= rp(c)a + sp(c)b = c(rk(a) + sk(b))

Since c is arbitrary and A is nondegenerate, it follows that k(ra + sb) = rk(a) + sk(b). The other statements in the lemma have a similar proof. □

There is a natural k-algebra morphism i : A —> M (A) : a —> (ka, pa) where ka(b) = ab and pa(b) = ba.

Lemma 3.4 Let A be a nondegenerate k-algebra. Then the map i : A —> M(A) is injective.

Proof Assume that for an a G A, we have that i(a) = 0. This means that ka = 0 and pa = 0. Hence for each b G A, we have that ab = 0 and ba = 0. The nondegeneracy of A then implies that a = 0. Thus i is injective. □

Lemma 3.5 Let A be a nondegenerate k-algebra, then we have for each (p,k) G M(A) that

(k, p) ■ i(a) = i(k(a)) and i(a) ■ (k, p) = i(p(a)). Thus i(A) is an ideal of M(A). Hence if A is unital, then i is an isomorphism.

Proof We have that

(k, p) ■ i(a) = (kka, PaP).

For b G A, we have that

(kka)(b) = k(ab) = k(a)b. Thus kka = kk(a). We also have

(PaP)(b) = p(b)a = ak(b). Thus pap = pk(b). The rest of the lemma is similar. □

Corollary 3.6 Let A be a nondegenerate k-algebra, and let (k, p) G M(A). If (k, p) ■ i(a) = 0 for each a G A, then (k, p) = 0.

In what follows, we will usually not explicitly write i. Thus we view A as a subset of M(A).

Definition 3.7 Let A and B be nondegenerate k-algebras. We say that an algebra morphism F : A —> M(B) is nondegenerate if B = F(A)B and B = BF(A). A Woronowicz-morphism between A and B is then defined as a nondegenerate algebra morphism F : A M(B).

Proposition 3.8 A Woronowicz-morphism F : A —> M(B) can be extended to a unique unital algebra morphism F : M(A) —> M(B).

Proof Let (k, p) G M(A). Every b G B can be written as b = ^i F(a{)bi. Now we define k'(b) = ^ F(k(ai))bi. Using nondegeneracy of A and F, one easily checks that this is well-defined. Similarly, one defines p'(b) = ^2 biF(p(a{)). The required extension is F : M(A) M(B) : (k, p) (k',p').

Assume that F : M(A) —> M(B) is another extension of F. Then we have for each x G M(A) and b'F(a') G B that a'x G A. Hence

b'F(a')(F(x) - F(x)) = 0 Thus for each b G B we have that b(F(x) - F(x)) = 0. It follows that F(x) = F(x). □

This proposition has as a corollary that we can compose nondegenerate maps. Indeed, if F : A —> M(B) and G : B —> M(C) are nondegenerate, then G extends to a morphism G : M(B) —> M(C), and the composition GF makes sense.

Lemma 3.9 The composition GF is nondegenerate.

Proof Take c e C, then by nondegeneracy of G, we can write c = i G(bt)ct, where bi e B and ci e C. Now, since F is nondegenerate, we can write bj = F(at,j)bf,j, where a e A and b' e B. Thus c = ^t j G(F(aij))F(bij)ci and G(bi,j)ci e C. Thus we have shown that G(F(A))C = C. Similarly, it follows that CG(F(A)) = C.

We can now form the category of all k-algebras with Woronowicz morphisms. We will denote this category as

Wor(k-Alg)

Proposition 3.10 Let F : A —> M(B) be an injective, nondegenerate map, then its extension F : M(A) —> M(B) is injective too.

Proof Assume that x e M(A) satisfies F(x) = 0. Then for each a e A, we have that F(ax) = F(a)F(x) = 0. Since f is injective on A and since ax e A, it follows that ax = 0. This holds for each a e A; it follows that x = 0. □

Proposition 3.11 For every surjective algebra morphism g : A —> B, we have that G := ig : A —> M(B) is nondegenerate and that this extends to a morphism G : M(A) —> M(B).

Proof Take b G B, then there exists an a G A such that g(a) = b. We can write a = ^ a[a'!. Hence b = g(a[)g{a'!). Since g(A) ç B, it follows that g(a^), g(a'!) G B. Hence b G g(A)B and b G Bg(A). □

Proposition 3.12 If F : A —> M(B) is a Woronowicz-morphism which constitutes an isomorphism in Wor(k-Alg), then F(A) = B.

Proof Let G : B —> M(A) be the inverse of f. Take an a e A; then we have by nondegeneracy of G that there exists bi e B and an ai e A such that a = G(bi)ai. But then F(a) = biF(ai). But since B is an ideal of M(B), this implies that F(a) e B. This shows that F(A) C B. Likewise, we have that G(B) C A. Now take b e B, then b = F(G(b)), and thus b e F(A). □

The quotients of nondegenerate algebras do not need to be nondegenerate. But we do have the following

Proposition 3.13 Let A be a nondegenerate algebra and let I be an ideal of A. We let F = {a G A | ab G I and ba G I for all b G B}. Then

(1) F is an ideal and A/F is nondegenerate.

(2) If F : A —> M(B) is a nondegenerate morphism between nondegenerate algebras such that F(I) = 0. Then it follows that F(F) = 0 and F induces a nondegenerate morphism A/F —> M(B).

Proof (1) That F is an ideal is easily checked. Assume that p(x)p(y) = 0 for all y G A. Then holds that xy G F for all y G A. It follows that for each y' G A that xyy' G I. But since A is nondegenerate, we know that every element of A has the form ^t yiy't. So for every a G A, we have that xa G I. Thus follows that x G F and thus p(x) = 0. Since A = AA, it follows at once that A/J = (A/J)(A/J).

(2) Take x G F Then holds for each a G A that ax G I and xa G I. Take b G B, then by nondegeneracy, we can write b = ^i F(a{)bi. But then

F(x)b = J2 F(x)F(ai)bi F(xai)bi = 0.

It follows that F(x) = 0. The remainder of the proof is now clear. □

Theorem 3.14 The co-equalizer in the category Wor(k-Alg) exists. In particular, the co-equalizer of two nondegenerate morphisms F, G : A —> M(B) is given by B/F, where I is the two-sided ideal in B generated by F(a)b - G(a)b and bF(a) - bG(a) for each a G A and b G B.

Proof Since B is nondegenerate, it follows that B/F is nondegenerate. Let p : B —> B/F be the canonical surjection. This surjection is nondegenerate and thus extends to a nondegenerate surjection P : M(B) —> M(B/F). We prove that PF = PG. Indeed, take P(b) G B/F, then

[P(F(a)) - P(G(b))](P(b)) = P(F(a)b - G(a)b) = 0.

Hence ~P(F(a)) = P(G(a)) for each a G A. _ _

Let H : B —> M(C) be a nondegenerate morphism such that HF = HG. Then H(F) = 0 and thus there is a unique morphism K : B/F —> M(C) such that H = KP. This is clearly nondegenerate since H is. □

Proposition 3.15 Let F : A —> M(C) and G : B —> M(C) be two Woronowicz-morphisms. If F and G commute, so do their extensions F and G.

Proof Take x e M(A) and y e B. Take c e C, then we can write c = ^i ciF(ai). Note that aix e A. Then

cF(x)G(y) = ^ ctF(atx)G(y) i

= y^ CiG(y)F(aix) i

= £ CiG(y)F(ai)F(x) i

= £ CiF(ai)F(y)F(x) i

= cG(y)F(x).

It follows that F(x)G (y) = G(y)F(x). Now let x e M(A) and y e M(B). Similarly, ilt follows that F(x)G(y) = G(y)F(x). □

Definition 3.16 For an associative fc-algebra A and for an ideal I of A, we say that I is nondegenerate if for each x e A the following hold:

(1) If for each y e A we have that yx e I, then x e I.

(2) If for each y e A we have that xy e I, then x e I.

The ideal A is always nondegenerate in A. Furthermore, an algebra is nondegenerate if and only if A = AA and {0} is a nondegenerate ideal of A.

Proposition 3.17 The intersection of nondegenerate ideals is nondegenerate.

In particular, every ideal I can be extended to a smallest nondegenerate ideal which contains it. We will denote this ideal by I. We give a more explicit description of {0}.

Proposition 3.18 Let A be an associative k-algebra. We have that a e {0} if and only if there exists n > 0 such that for all bi,...,bn e A we have bib2...bna = 0 and abib2...bn = 0.

Proposition 3.19 If A and B are nondegenerate k-algebras, then A (g> B/{0} is a nondegenerate k-algebra.

For nondegenerate fc-algebras A and B, we put A ( B = A ( B/{0} and we call it the nondegenerate tensor product.

Proposition 3.20 If fc is afield, then A ( B = A ( B.

Proof Assume for a 0 b we have that aa' 0 bb' = 0 for all a' G A and b' G B. This implies that there exists finitely supported maps : B —> R such that

bb'=Y,ka> b (f)f,

ka>,V (f )aa' = 0 for all f G B If ka'b (f) = 0 for all f G B, then follows that bb' = 0. Otherwise, it follows by taking inverses that aa' = 0.

Thus if there exists some b' such that bb' = 0, then it must hold that aa' = 0 for all a' G A. Thus by nondegeneracy of A, we have that a = 0 and thus a 0 b = 0.

Otherwise, it must hold that bb' = 0 for all b' G B. But then b = 0 and hence a 0 b = 0. □

Proposition 3.21 For nondegenerate algebras A and B, there exist commuting, nondegen-erate maps

Ja : A M(A) M(A0B)

Jb : B M(B) M(A0B).

Proof Take (k, p) G M(A), let p : A 0 B —> A0B be the canonical quotient. Take p(a0b) G A0B, then we set k'(p(a0b)) = p(k(a)0b) and p'(p(a®b)) = p(p(a) ®b). This is well-defined: if a 0 b G {0} and if I is any nondegenerate ideal of A 0 B, then a 0 b G I. Thus (x 0 y)(k(a) 0 b) = (xk(a)) 0 (yb) = (p(x)a) 0 (yb) G I. By nondegeneracy of I follows that k(a) 0 b G I. Since this is true for all nondegenerate ideals of A 0 B, it follows that k(a) 0 b G {0}. Similarly, we have that p' is well-defined. So we have an element (k', p') G M(A0B). This shows that we have a morphism

M(A) M(A 0 B) : (k, p) (k', p').

Similarly, we also have a morphism M(B) —> M(A 0 B). These morphisms are clearly commuting. Note that for (ka, pa) G M(A), we have k'a(p(a' 0 b')) = p(aa' 0 b') and k'a (p(a' 0 b')) = p(a'a 0 b'). We check nondegeneracy. Take p(a 0 b) G A0B. Then we can write a = ^2i aia'. But then

p(a 0 b) = J2 p(aia' 0 b) = k'a. (p(a' 0 b) G Ja(A)(A0B). ii

4 Generalized Colimits

It is well known that in categories without (useful) coproducts, a monoidal structure may sometimes serve as an alternative. More generally, if a category is endowed with compatible relations Rc on the sets of morphisms with common domain C, there results a natural notion of R-coproduct. The link between such relations on the one hand, and monoidal

structures on the other hand, is studied by Janelidze [14]. In Section 4.1, we describe R-coproducts in the Woronowicz categories from the previous sections. In Section 4.2, we introduce more general R-colimits. Under an additional assumption on R, we prove some properties involving combinations of R-colimits. For instance, we show that in general R-pushouts cannot be composed, but they can be composed under an additional centrality requirement for one of the involved morphisms. In the sequel, we will make fundamental use of the dual notions, which are discussed in Section 4.3.

4.1 Co-Relations and Generalized Coproducts

Recall that for a category C and an object C e C, the slice category C/C is the category of all morphisms with codomain C.

Definition 4.1 [14] A co-relation R on a category C consists of the datum of a relation Rc on Ob (C/C) for every object C e C, such that the following properties are satisfied:

(1) If f : A —> C and g : B —> C are Rc -related and if h : D —> A is an arbitrary morphism, then fh and g are Rc-related.

(2) If f : A —> C and g : B —> C are Rc-related and if h : C —> D is an arbitrary morphism, then hf and hg are Rd -related.

For simplicity, we usually denote Rc simply by R for all objects C. In [14], a co-relation R on a category is called a cover relation and the author investigates cover relations associated to the natural order on subobjects (whence the terminology), as well as cover relations associated to monoidal structures. In this paper we are mainly interested in (generalizations of) the latter type, where in particular R is symmetric. Note that in general, for a co-relation R on C, the opposite relations R^ for C e C do not define a co-relation due to the asymmetric nature of condition (1). If they do define a co-relation, we denote it by Rop and we call R a bi-co-relation (bicover relation in [14]).

Example 4.2 (1) If C is an arbitrary category, then we can say that f : A —> C and g : B —> C are always R-related. We will call this co-relation standard.

(2) Let F : A —> M(C) and G : B —> M(C) be morphisms in Wor(C*-Alg). We say that F and G are R-related if they commute.

(3) Let Ring be the category of all (possibly noncommutative) unital rings with as mor-phisms the usual ring homomorphisms. Two morphisms f and g are R-related if they commute.

(4) Let G : A —> M(C) and G : B —> M(C) be in Wor(fc-Alg). We say that F and G are R-related if they commute.

With these examples in mind, we make the following definition:

Definition 4.3 Let C be a category with a co-relation R on C.

(1) A morphism f : B —> A in C is R-central if (f, 1a) e R and R-op-central if

(1A,f) e R.

(2) An object A e C is R-commutative if (1a, 1a) e R.

The central morphisms in C constitute a left ideal in C, that is if f : A —> B is central and h : A' —> A is arbitrary, then fh is central. The commutative objects in C form a

full subcategory Com(C) C C. The co-relation R on C gives rise to an obvious restricted co-relation on any subcategory. In particular, the restricted co-relation on Com(C) is the standard relation on Com(C).

A co-relation R on a category can be used to generalize the notions of coproduct and pushout:

Definition 4.4 Let A and B be objects of a category C. The R-coproduct of A and B is an object Q together with R-related morphisms qA : A —> Q and qB : B —> Q such that for every two R-related morphisms fA : A —> C and fB : B —> C, there is a unique morphism f : Q —> C such that fA = fqA and fB = fqB.

We denote the R-coproduct of A and B by A ]_[R B. Clearly, if it exists, an R-coproduct is unique up to isomorphism. In [14], Janelidze characterizes the co-relations R for which the associated R-coproduct defines a monoidal structure on C. Note that unless R is symmetric, we do not necessarily have A ]_[R B = B ]_[R A.

We can give a similar definition for the R-pushout:

Definition 4.5 Let f : C —> A and g : C —> B be morphisms in a category C. The R-pushout of f and g is an object Q together with R-related morphisms qA : A —> Q and qB : B —> Q such that qAf = qBg and such that for every two R-related morphisms fA : A —> C and fB : B —> C with fAf = fBg, there is a unique morphism h : Q —> C such that fA = hqA and fB = hqB.

We denote the R-pushout of f and g as above by A\\CB. Again, if it exists, an R-pushout is unique up to isomorphism. We call qB the R-pushout of f by g, and by a pushout of f we mean a pushout qB of f by some morphism g.

If R is the standard co-relation, then the R-coproduct and R-pushout are simply the usual coproduct and pushout.

Theorem 4.6 In the category Wor(C*-Alg), R-coproducts exist and are given by the maximal tensor product.

Proof Let A and B be C*-algebras. We can form the maximal tensor product A 0B. It is shown in [23] that

A 0„B c M(A) 0„ M(B) c M(A 0„ B).

Thus the canonical maps qA : A —> M(A) 0M(B) : a —> a 0/x 1 and qB : B —> M(A) 0^ M(B) : b —> 1 0^ b extend to maps QA : A —> M(A 0/x B) and QB : B —> M(A 0^ B). These maps are clearly commuting. Furthermore, they are strictly continuous since

\\qA(a){a'0^b')\\ = \\(aa') 0^ b'\\ < \\aa'\\\\b'\\.

Now let Fa : A —> M(C) and Fb : B —> M(C) be commuting nondegenerate morphisms. Then we define F : A 0 B —> M (C) by f(a0 b) = fA (a)fB (b). This is clearly the unique map for which Fa = FQa and Fb = FQb. We must prove that F is non-degenerate. For this, take c G C. Then we can write c = ^i FA(ai)ci and we can write

ci = £j FB(bi,j)ci,j. Thus ci = Y.i,j FA(ai)FB(bi,j)ci,j G f(A 0 B)C. □

Corollary 4.7 The category Wor(C*-Alg) is naturally a monoidal category with as monoidalproduct the R-coproduct and with C as unit.

Proof It is well-known that the category of C*-algebras with usual C*-morphisms is monoidal with as monoidal product the maximal tensor product. Thus for each three C*-algebras A, B and C we have a suitable isomorphism ty : A (B C) ^ (A B) C. This induces a nondegenerate morphism since ty is surjective. Hence, we have an isomorphism in Wor(C*-Alg). □

Theorem 4.8 In the category Wor(fc-Alg), R-coproducts exist and are given by the nondegenerate tensor product (.

Proof Let A and B be fc-algebras. We can form the tensor product A(B. We have shown that we have maps

Ja : A M(A) M(A ( B)

Jb : B M(B) M(A ( B) which are nondegenerate and commuting.

Now let Fa : A —> M(C) and Fb : B —> M(C) be commuting nondegenerate morphisms. Then we define H : A ( B —> M(C) by H(a ( b) = FA(a)FB(b). We show that H(I) = 0, which proves that h descends to a map F : A(B —> M(C). So take a ( b e 1. Then there is some fc > 0 such that for each yx,..., yt e A ( B holds that yi...yfc(a ( b) = 0. Now take c e C, by nondegeneracy of Fa and Fb and by using that Fa and FB are commuting, we can write c as

c = cFA(ail)FB(bi1 )...FA(aifc)FB(bifc).

i1,...,ifc

cFA(a)FB(b) = c ^ fa^j...aifca)FB(bi1 ...b^b)

i\,...,ifc

= c J2 H((aix ( bix)...H(aifc ( bifc)(a ( b)) ii,...,ifc = 0.

Since c is arbitrary, it follows that FA(a)FB(b) = 0, as desired.

The map F is clearly the unique map for which Fa = FQa and Fb = FQb. We must prove that F is nondegenerate. For this, take c e C. Then we can write c = ^i Fa (ai)ci and wecanwriteci = Y,j FB(bi,j)ci,j.Thusci = J2,j FA(ai)FB(bi,j)ci,j e H(A(B)C. □

Corollary 4.9 The category Wor (fc-Alg) is naturally a monoidal category with as monoidal product ( and with as unit fc.

Proof We show that the tensor product is associative as the rest of the statement is obvious. Let A, B,C be nondegenerate fc-algebras and let ty : A ( (B ( C) ^ (A ( B) ( C be the canonical isomorphism such that ty(a ( (b ( c)) = (a ( b) ( c. This induces a canonical isomorphism ty : A((B(C) ^ (A(B)(C. Since ty is an isomorphism, it is nondegenerate and thus it implements an isomorphism in Wor (fc-Alg). □

4.2 Generalized Colimits

The following notion generalizes R-coproducts and R-pushouts:

Definition 4.10 Let F : I —> C be a functor, let R be a co-relation on C as before, and let r

be a relation on Ob(I). An (r, R)-cocone on F consists of a cocone (C, (pi : F(i) —> C)i)

such that irj implies piRpj. An (r, R)-colimit on F is a universal cocone on F.

Obviously, if it exists, an (r, R)-colimit of F is unique.

Example 4.11 (1) If r = 0 and R is the standard co-relation on C, then (r, R)-colimits are categorical colimits in C.

(2) Let I = (a b) be the "coproduct category". If r = {(a, b)}, then an (r, R)-colimit is an R-coproduct. If r = {(a, b), (b, a)}, we call an (r, R)-colimit a two-sided R-coproduct. If R is symmetric, an R-coproduct is automatically two-sided.

(3) Let I = (a c —> b) be the "pushout category". If r = {(a, b)}, then an (r,R)-colimit is an R-pushout. If r = {(a, b), (b, a)}, we call an (r,R)-colimit a two-sided R-pushout. If R is symmetric, an R-pushout is automatically two-sided.

(4) Let I = (a) be the "object category" with ara. The diagram determined by the object A has (r, R)-colimit (A, 1a : A —> A) if and only if A is commutative with respect to R. For C = Ring the category of unital rings with the commutation co-relation R from Example 4.2 (3), the (r, R)-colimit of the diagram determined by the ring A is the canonical morphism A —> A/[A, A] where [A, A] is the ideal generated by the commutators aa' — a'a for a, a' G A.

(5) Let I = (b —> a) be the "morphism category" with r = {(b, a)} (resp. r = {(a, b)}). The diagram determined by the morphism f : B —> A has (r, R)-colimit (A, 1a : A —> A, f : B —> A) if and only if f is central (resp. op-central) with respect to R. For C = Ring the category of unital rings with the commutation co-relation R from Example 4.2 (3), the (r, R)-colimit of the diagram determined by the ring map f : B —> A is the canonical morphism A —> A/[f(B), A] where [f(B), A] is the ideal generated by the commutators f(b)a — af (b) for a G A and b G B.

Proposition 4.12 Suppose C has R-coproducts as well as categorical coequalizers. Then C

has R-pushouts.

Proof For morphisms f : C —> A and g : C —> B, consider the R-coproduct (Q, sa : A —> Q, sb : B —> Q) and the categorical equalizer q : Q —> E of SAf and SBg. By Definition 4.1 (2), qsA and qsB are R-related. It is easily seen that they make E into an R-pushout of f and g. □

In order to proceed, it is useful to consider how a co-relation R interacts with certain classes of morphisms. Consider the following weakening of the notion of a pretopology on a category C: a cover system T on C consists of the datum, for every object C G C, of a collection T(C) of families of morphisms (fi : Ci —> C)i. Dually, an co-cover system S on C consists of the datum, for every object C G C, of a collection S(C) of families of morphisms (gi : C —> Ci)i. A class of morphisms T in C gives rise to a

cover system T with T(C) = {/ : D —> C | / G T} and a co-cover system T with T(C) = {/ : C D | / g T}.

Definition 4.13 Let R be a co-relation on a category C.

(1) Let T be a cover system on C. We say that R is T-generated if the following condition holds for morphisms / : A —> C and g : B —> C in C and a collection of morphisms (fi : Ai —> A)i in T(A): if for all i we have f/iRg, then we have /Rg.

If R is a bi-co-relation, we call R bi-T-generated if both R and R°P are T-generated.

(2) Let S be an co-cover system on C. We say that R is T-cogenerated if the following condition holds for morphisms / : A —> C and g : B —> C in C and a collection of morphisms (gi : C —> Ci)i in T(A): if for all i we have gi/Rgig, then we have /Rg.

Consider the classes Epi of epimorphisms and Mono of monomorphisms in C, the cover system Epi of jointly epimorphic families and the co-cover system Mono of jointly monomorphic families. Note that the co-relations listed in Example 4.2 are bi-Ep;-generated and Mono-op-generated (and hence in particular bi-Epi-generated and Mono-co-generated).

Lemma 4.14 Suppose C is endowed with a bi-co-relation R.

(1) Suppose Rop is Epi-generated. Suppose / : B —> A has an (r, R)-colimit q : A —> A' as in Example 4.11 (5). Then q is an epimorphism and q/ : B —> A' is central.

(2) Suppose R is bi-Epi-generated. Suppose A has an (r, R)-colimit q : A —> A' as in Example 4.11 (4). Then q is an epimorphism and A' is commutative.

Under the assumptions of Lemma 4.14, we call q : A —> A' in (1) the centralizator of / : B —> A and q : A —> A' the commutativizator of A.

Proof That q is an epimorphism immediately follows from the universal property of the colimits. In case (1), we consider the epimorphism q. By the assumption on Rop, from q/Rq we deduce q/R 1a . In case (2), we twice consider the epimorphism q : A —> A'. By the assumptions upon R and Rop, from qRq we deduce 1a'r1a'. c

Proposition 4.15 Suppose C is endowed with a bi-Epi-generated bi-co-relation R. Consider morphisms / : C —> A and g : C —> B with two-sided R-coproduct (Q,sa : A —> Q, sb : B —> Q) as in Example 4.11 (2).

(1) sa/ = SBg is central and op-central.

(2) If A and B are commutative, then so is Q.

Proof (1) We show centrality, that is sAfRlQ. Since {sa, sb } is jointly epimorphic, it suffices that SAfRsA and SAfRsB .The second relation follows from saRsb , and the first one can be rewritten as sBgRsA which follows from sbRsa.

(2) Using twice the family (sa, s в}, 1gR1g is equivalent to the following four relations: saRsa, sbRsb, saRsb, sbRsa- The first two follow from commutativity of A and of B, the last two follow from the definition of twosided R-pushout. □

The following example shows that it is not always true that the composition of .R-pushout diagrams is an R-pushout diagram, or that the R-pushout of an isomorphism remains an isomorphism.

Example 4.16 Take the category of unital algebras over R in which f : A —> C and g : B —> C are R-related if and only if they commute. For central morphisms A —> B and A —> C, it is easily checked that the R-pushout is given by B ®a C. Now consider the following diagram, where H is the quaternion algebra:

> (H О H)/I

The left diagram is clearly an R-pushout diagram. The right diagram is by definition an R-pushout diagram, where I is the ideal generated by h ® 1 — 1 ® h. The composite diagram would only be a pushout diagram if (H ® H)/I = H. This is not the case since otherwise we would have p = q = 1h, but the map 1h : H —> H is not central.

In general, we have the following:

Proposition 4.17 Suppose R is a bi-Epi-generated bi-co-relation on C. Consider the following composition of diagrams in which y is R-central:

(1) If both squares are two-sided R-pushouts, then so is the composed diagram.

(2) If the left square and the composed diagram are two-sided R-pushouts, then so is the right square.

Proof (1) Obviously, we have y'ß'Ra" since y'Ra". Consider morphisms f : A —> E and g : D —> E with fRg. It follows that fRgy, so from the left R-pushout we obtain a

unique morphism x : P —> E with f = xft' and gy = xa. If we can show that xRg, then we obtain the required unique morphism x' : Q —> E. Since {ft', a'} is jointly epimorphic, it suffices that xft'Rg and xaRg. The first relation is fRg which is given. For the second relation, centrality of y gives yRId, whence gyRg which can be rewritten as xaRg as desired.

(2) If we can show that y'Ra", we can finish the proof in the usual fashion. Along the lines of part (1), we use the jointly epimorphic {ft', a'} to show this.

Example 4.18 In the category of unital fc-algebras over a commutative ground ring, if in the notations of Proposition 4.17 a, ft and y are central, part (1) of the result corresponds to the familiar change of rings formula (A (c B) (b D = A (c D.

Next we give a construction of R-pushouts using centralizations and R-pushouts of central morphisms.

Proposition 4.19 Suppose R is a bi-Epi-generated bi-co-relation on C. Consider mor-phisms a : C —> A and ft : C —> B. Suppose the centralizators qA : A —> A' of a and qB : B —> B ' of ft exist. Suppose further that (Q, sa' : A' —> Q, sb ' : B ' —> Q) is the two-sided R-pushout of qAa and qBft. Then (Q, sa'qA, sb'qB) is the two-sided R-pushout of a and ft.

Proof It suffices to check the universal property. Consider f : A —> D and g : B —> D with fRg, gRf and fa = gft. This implies faRg, in other words gftRg, and gftRf, in other words faRf. From the universal properties of the centralizators, we thus obtain f' : A' —> D with f 'qA = f and g' : B' —> D with g'qB = g. If we can show f 'Rg' and g'Rf', we obtain the required morphism Q —> D. But since both qA and qB are epimorphisms, it suffices that f 'qARg'qB and g'qBRf 'qA which is part of our assumption of f and g. □

4.3 Relations and Generalized Pullbacks

We can easily dualize the notions of co-relation R, R-coproduct and R-pushout. Since we will make frequent use of these dual notions, we state them explicitely to fix terminology. For a category C and an object C G C, the slice category C/C is the category of all morphisms with domain C.

Definition 4.20 A relation R on a category C consists of the datum of a relation Rc on Ob (C/C) for every object C G C, such that the following properties are satisfied:

(1) If f : C —> A and g : C —> B are Rc -related and if h : A —> D is an arbitrary morphism, then hf and g are Rc -related.

(2) If f : C —> A and g : C —> B are Rc-related and if h : B —> C is an arbitrary morphism, then fh and gh are Rb-related.

The relation on C for which every two morphisms with the same codomain are related is called the standard relation.

Definition 4.21 Let C be a category with a relation R on C.

(1) A morphism f : A —> B in C is R-central if (f, 1a) is in R and R-op-central if (1A,f) is in R.

(2) An object A G C is R-commutative if (1a, 1a) G R.

The R-central morphisms in C constitute a right ideal in C. The R-commutative objects in C form a full subcategory Com (C ) ç C. The relation R on C gives rise to an obvious restricted relation on any subcategory. In particular, the restricted relation on Com(C) is the standard relation on Com(C).

Definition 4.22 Let A and B be elements of a category C with relation R. The R-product of A and B is an object P together with R-related morphisms pa : P —> A and pb : P —> B such that for every two R-related morphisms qA : Q —> A and qB : Q —> B, there is a unique morphism f : Q —> B such that qA = PAf and qB = pb f.

We denote the R-product of A and B by A xR B.

Definition 4.23 Let f : A —> C and g : B —> C be morphisms in a category C with relation R. The R-pullback of f and g is an object P together with R-related morphisms PA : P —> A and pb : P —> B such that fpA = gPB and such that for every two R-related morphisms qA : D —> A and qB : D —> B with fqA = gqB we have that there is a unique morphism h such that qA = PAh and qB = PBh.

We denote the R-pullback of f and g as above by AxRB. We call pb the R-pullback of f along g, and by an R-pullback of f we mean an R-pullback pb of f along some morphism g.

Just like R-pushouts, in general R-pullbacks do not behave well with respect to compositions and isomorphisms.

The following observation will be useful later on:

Lemma 4.24 Consider a monomorphism m : Y —> Z and an R-central morphism g X —> Y. The following is the R-pullback of m along mg:

x--—>z

The following terminology may be somewhat confusing, but we prefer to use the words "generated" and "cogenerated" in analogy with the classical meaning of an object C being generated by morphisms landing in C resp. cogenerated by morphisms starting in C.

Definition 4.25 Let R be a relation on a category C.

(1) Let T be a cover system on C. We say that R is T-generated if the following condition holds for morphisms f : C —> A and g : C —> B in C and a collection of morphisms (f : Ci —> C)i in T(C): if for all i we have ffiRgfi, then we have fRg.

(2) Let S be an co-cover system on C. We say that R is T-cogenerated if the following condition holds for morphisms f : C —> A and g : C —> B in C and a collection of morphisms (gi : A —> Ai)i in T(A): if for all i we have gifRg, then we have fRg.

If R is a bi-relation, then R is called bi-S-cogenerated if both R and Rop are S-cogenerated.

The following result generalizes well-known stability properties for categorical pullbacks:

Proposition 4.26 Consider the commutative diagram

x---s>z

and suppose it is either the R-pullback of f and m, or the R-pullback of m and f, or the two-sided R-pullback of f and m.

(1) If m is a monomorphism, then so is m'.

(2) Suppose (E, M) is a factorization system on C and R is E-generated. If m G M, then m' G M.

Proof Consider the following commutative diagram

in which the right square is the R-pullback of f and m, m G M and e G E. Since e L m there is a unique morphism h : X' —> Y with mh = fv and he = gu. In order to apply

the R-pullback property, we need vRh. We have he = gu and ve = m'u, from m'Rg we deduce m'uRgu in other words veRhe. Since R is E-generated, we have vRh so we get a unique map h' : X' —> P with m'h' = v and gh' = h. Now m'h'e = ve = m'u is equivalent to h'e = u since M C Mono. This finishes the proof of the first statement. The proofs of the other statements are similar. □

5 Tensor Functional Topology

The basic idea of functional topology is that a certain amount of "topology" can be developed in a category based upon axiomatically defined classes of morphisms. After this idea originated in the seventies in work of Herrlich, Manes and Penon [10], [18], [21] [20], since the nineties the subject received a lot of attention in work of Herrlich, Salicrup, Strecker, Clementino, Giuli, Tholen, Hofmann and others with applications to Birkhoff closure spaces, uniform spaces, topological groups, locales, approach spaces, lax algebras and schemes [12], [4], [22] [5], [6], [13], [16]. In this section, we adapt some of these approaches to a monoidal context, or, more generally, to the context of a category endowed with a relation R in the sense of Definition 4.20. We mainly focuss our attention on the following two approaches:

• a minimal approach inspired by [13], taking an R-proper class of so-called proper morphisms as primary (Section 5.1). This allows for the definition of compact objects (Section 5.4), of commutative Hausdorff objects (Section 5.4) and of separated morphisms with commutative domain (Section 5.2). Many familiar relations between these notions hold true, sometimes under suitable centrality conditions. This approach can be refined by adding a second class Fo of closed immersions to the data (Section 5.3).

• an approach building on [5], [16], in the presence of a factorization system (E, M). This involves the definition of a closed class F of closed morphisms based upon an auxiliary R-proper class Fo of closed immersions and an auxiliary closed class P of surjections, together constituting a co-called (E, M)-closed structure (P, Fo). If F is moreover an R-proper class, we call (P, Fo) an (E, M) -proper structure with proper morphisms F (Section 5.5).

If R is such that R-pullback diagrams can naturally be composed, and R-pullbacks of identity morphims remain identity morphisms, then the results from [16] integrally go through. In this section however, we concentrate on the results which still hold true more generally. Unlike in the classical case, there is no standard way of associating an R-proper class to a closed class, so the most natural thing to do in this setup is start at once with the class of morphisms one would like to view as "proper", and show by hand that it is an R-proper class. Even in the second approach, there is no general way of checking whether the closed class F we obtain from (E, M) and (P, Fo) is actually R-proper (that is we have an (E, M)-proper structure). On the other hand, if we start from a closed class F and we have a factorization system (E, M) with M C F, then under the additional conditions that R is E-generated and F is stable under cancellation of post-composed M-morphisms, there is a canonical way of associating the closed structure (E n F, M) to it (see Propositions 5.18 and 5.19), which is at the other extreme from the case P = E corresponding to the setup in [5].

From now on, we will always work in a category C with a fixed relation R.

5.1 Proper Morphisms

Let C be a category with R-pullbacks. We make use of the standard classes Mor of all morphisms, Iso of isomorphisms, Mono of monomorphisms and Epi of epimorphisms.

In general, one is interested in classes F of morphisms satisfying some of the following stability properties:

(1) Iso c F;

(2) F is closed under composition;

(3) F is R-pullback stable, that is every R-pullback of f G F along an arbitrary morphism is again in F.

If F satisfies (1) and (2), we call F a closed class. If F satisfies (1), (2) and (3), it is called an R-proper class.

Definition 5.1 Let F be a class of morphisms. We say that a morphism g is F-proper if every R-pullback of g exists and is in F.

We thus obtain the class F — Prop of F-proper morphisms.

Lemma 5.2 Let F be a class of morphisms.

(1) If F c G then F — Prop c G — Prop.

(2) If Fo c F with Fo R-pullback-stable, then each morphism in Fo is F-proper.

Unlike in the familiar situation for the standard relation, in general F — Prop fails to be a proper class. We have:

Lemma 5.3 Suppose R is symmetric and Mono-cogenerated. Let F be a class of morphisms. Consider morphisms f : X —> Y and g : Y —> Z. If f, g G F — Prop and f is central, then gf G F — Prop (and gf is central).

Due to the failure of F — Prop to be proper in general, it is more natural to start at once from a proper class F considered as "proper morphisms", as is done in [13]. Obviously, a proper class F satisfies F = F — Prop.

5.2 Separated Morphisms

Let C be a category with R-pullbacks. For a morphism g : X —> Y, we consider the following R-pullback diagram:

X--->Y

X x$X-

If X is R-commutative, we obtain a unique diagonal morphism

= (1x, 1x) : X XxRX

from the universal property of an R-pullback. We can now introduce separated morphisms:

Definition 5.4 Let F be a class of morphisms. Let g : X —> Y be a morphism such that X is a commutative object.

(1) We say that g is F-separated if the diagonal Ag e F.

(2) We say that g is F-perfect if it is F-proper and F-separated.

We thus obtain the corresponding classes F — Sep of F-separated morphisms and F — Perf of F-perfect morphisms. A class F is called separating if every F-separated morphism g satisfies Ag e F — Prop. Obviously, every proper class is separating.

Lemma 5.5 Let F be a class of morphisms.

1. If F c G then F — Sep c G — Sep.

2. Suppose Iso c F. Every monomorphism m : X —> Y with X commutative is F-separated.

Proof (2) For a monomorphism m : X —> Y with X commutative, the diagram

X-=—

is an R-pullback by Lemma 4.24, whence Am = 1x : X —> X is in F. □

The following generalizes [16, Lemma 2.6]:

Lemma 5.6 Suppose R is symmetric and Mono-cogenerated. Let F be a proper class of morphisms. Consider morphisms f : X —> Y and g : Y —> Z. If f is central, g e F — Sep and gf e F, then f e F.

Proof Since f and Ag are central, the proof from [16, Lemma 2.6] goes through. □

5.3 Closed and Proper Pairs

Definition 5.7 Consider a proper class of monomorphisms F0 C Mono and a closed class F with F0 C F. The pair (F0, F) is called a closed pair, the morphisms in F are called closed morphisms and the morphisms in F0 are called closed immersions. Is F is moreover a proper class, then (F0, F) is called a proper pair and the morphisms in F are called proper morphisms.

A closed pair is called separating if for every F-separated morphism g : X —> Y, we have Ag e F0.

Proposition 5.8 Assume that C is a category with R-pullbacks. If H is a proper class, then (H n Mono, H) is a separating proper pair.

Proof By Proposition 4.26 (1), Mono is a proper class and H n Mono is a proper class of monomorphisms. The resulting proper pair is separating since the diagonal of any morphism is a monomorphism. □

5.4 Compact and Hausdorff Objects

Suppose C has a final object 1. The unique morphism from an object X to 1 is denoted by !x : X 1.

We will define compact and separated objects at once with respect to a proper class, thus avoiding some of the problems that arise if one starts from a closed class.

Definition 5.9 Let H be a proper class, morphisms of which are called proper morphisms.

(1) An object X G C is H-compact if the morphism !x : X —> 1 is proper.

(2) A commutative object X G C is H-Hausdorff if !x : X —> 1 is H-separated.

Proposition 5.10 Suppose F is a closed class for which H = F — Prop is a proper class.

(1) An object X G C is H-compact if and only if for every Y G C we have that the second projection p2 : X xR Y —> Y is in F.

(2) If F is separating, then a commutative object X G C is H-Hausdorff if and only if !x : X —> 1 is F-separated.

In the category Top with F the usual closed maps and H = F — Prop the usual proper maps, the H-compact and H-Hausdorff objects are the usual compact and Hausdorff spaces.

We now give some stability properties of compact objects. Proposition 5.11 Let H be a proper class.

(1) If f : X —> Y is proper and Y is H-compact, then X is H-compact too.

(2) If X and Y are H-compact, then so is X xR Y.

Proof (1) is clear. (2) For X and Y are compact, consider p\ : X xR Y —> X and p2 : X xR Y —> Y. Since !x : X —> 1 is in H, so is its pullback p2. Now also !y : Y —> 1 is in H, hence so is the composition XxRy = Yp2 : X xR Y —> 1. □

The following corollary expresses the well-known topological statement that a closed subspace of a compact space is again compact:

Proposition 5.12 Suppose (Fo, F) is a closed pair for which H = F — Prop is a proper class. If f : X —> Y is in Fo and Y is H-compact, then X is H-compact too.

Finally, we can recover the famous topological statement that a continuous morphism from a compact to a Hausdorff space is proper.

Proposition 5.13 Let H be a proper class. Consider a central morphism f : X —> Y with Y commutative. If X is H-compact and Y is H-Hausdorff, then f is proper.

Proof This follows from Lemma 5.6 by taking g =!y. □

5.5 (E, M)-Closed Structures

Let C be a category with R-pullbacks endowed with a factorization system (E, M) with M C Mono. From now on, we denote the (E, M)-factorization of a morphism f : X —> Y by

>f(X)—>Y.

Definition 5.14 An (E, M)-closed structure (P, F0) (or simply closed structure if (E, M) is understood) consists of:

(a) An R-proper class F0 Ç M of closed immersions.

(b) A closed class P of surjections.

With respect to a closed structure, a morphism f : X —> Y is called closed if and only if for every m : X' —> X in F0, we have e(fm) G P and /(fm) G F0. The class of closed morphisms is denoted by F. The closed structure is called separating if the following condition is satisfied.

(c) For any morphism f : A —> B with A commutative, it holds that if the diagonal A f of f is in F, then it is in F0.

The closed structure is called a proper structure if the class F is R-proper, and in this case the closed morphisms are also called proper morphisms.

The following is an adaptation of [16, Proposition 3.13]:

Proposition 5.15 Let (P, F0) be an (E, M)-closed structure. We have:

(1) F is closed under compositions, thus (F0, F) is a closed pair.

(2) F0 = F n M.

(3) If we have mg G F for a monomorphism m and an R-central morphism g, then also g G F.

If (P, F0) is a separating (E, M)-closed structure, then (F, F0) is a separating closed pair.

Remarks 5.16 (1) Suppose we take P = E. Then a morphism f : X —> Y is closed if and only if for m : X' —> X in F0 we have /(fm) G F0. Thus, this is precisely the situation of [5].

(2) Suppose we take F0 = M. Then a morphisms f : X —> Y is closed if and only if for m : X' —> X in F0 we have e(fm) G P.

The following is an adaptation of [16, Proposition 3.15]:

Proposition 5.17 Let (E, F0) be an (E, M)-closed structure. Suppose in an R-pullback

X--->Y

of m G F0 and f G E, we have f ' G E. If gf G F and f G E, then g G F.

Proof Take f : X —> Y and g : Y —> Z morphisms. Let gf G F with f G E. Let m : M —> Y be an arbitrary morphism in F0. We need to show that n(gm) G F0. For this, we take the R-pullback of m and f. We have the following diagram:

/J,(gm)

Since F0 is stable under R-pullbacks, we get that m' G F0. Since f G E, the assumptions of the proposition ensure that f ' G E. Thus ¡i(gfm') = ¡i(gm). But since gf is closed, this is an element of F0. □

In some situations we have M = F0. The following two propositions deal with this case:

Proposition 5.18 Let (P, M) be an (E, M)-closed structure and let F be the class of closed morphisms with respect to this closed structure.

(1) If we have mg G F for m G M, then also g G F.

(2) F is closed under R-pullbacks along morphisms in M.

Proof (1) Consider m : Y —> Z in M and f : X —> Y with mg G F. Consider m' : X' —> X in M. We have fx(mgm') = m^(gm') and e(mgm') = e(gm'). Since mg G F, the latter is in P.

(2) Consider the pullback square

with f G F, m G M. Then mf ' = fm' G F, whence by (1), f ' G F. Conversely, we have:

Proposition 5.19 Let (E, M) be a factorization system and suppose R is E-generated. Let F be a closed class with M Ç F and assume that mg G F for m G M implies g G F. Then (E n F, M) is an (E, M)-closed pair, and F is its class of closed morphisms.

Proof Obviously E n F is a closed class, and by Proposition 4.26, M is R-proper. It remains to show that F is the associated class of closed morphisms. Assume that f G F and m G M, then it obviously holds that /(fm) G M and e(fm) G E. But since / (fm)e(fm) = fm G F, it follows by the hypothesis that e(fm) G F.

Conversely, assume that f : A —> B is closed with respect to the (E, M)-closed structure (E n F, M). Since 1a G M, it follows that f = f 1a = e(f 1a) G F. We also have /(f 1A) G M Ç F. Thus it follows that f G F by composition. □

6 Tensor Functional Topology in Woronowicz Categories

In this section, we show that the opposite Woronowicz categories of both associative algebras (Section 6.2) and C*-algebras (Section 6.3) can naturally be endowed with proper structures. In each case, this entails three main steps:

(1) The definition of an R-proper class F of proper morphisms;

(2) The definition of a factorization system (E, M);

(3) The verification that the conditions of Proposition 5.19 are fulfilled and thus (E n F, M) is an (E, M)-closed structure.

In each case, compact objects correspond precisely to unital algebras, and all commutative algebras are Hausdorff.

In the C*-algebra case, restricting the opposite Woronowicz category to the commutative objects yields a category equivalent to locally compact Hausdorff spaces with continuous maps. We show that the restriction of (E, M) to this category coincides precisely with the factorization system of dense maps and closed embeddings. Further, the class F restricts precisely to the proper continuous maps.

6.1 Topological Situations

In the category Top of topological spaces and continuous maps, let F be the class of the usual closed maps. The standard factorization system (E0, M0) has E0 given by the surjec-tions and M0 given by the embeddings. It is well known that taking F0 to be the closed

embeddings and putting P0 = E yields a separating closed structure describing the closed maps F [5].

In [16], we alternatively consider the factorization system (E, M) with E given by the dense maps and M given by the closed immersions. Put Fo = M. Clearly, we are in the situation of Proposition 5.19 and P = E n F corresponds to the usual class of surjections. We thus obtain a separating closed structure (P, M) describing the closed maps F.

Let CHd C Top be the full subcategory of compact Hausdorff spaces. The factorization systems (E0, M0) and (E, M) both restrict to CHd and coincide on this category, corresponding to the fact that every morphism is closed.

Let LcHd C Top be the full subcategory of locally compact Hausdorff spaces. In this case only the factorization system (E, M) restricts to LcHd, and taking F0 = M and P the surjections again yields a separating closed structure describing the closed maps in LcHd.

6.2 Associative Algebras

Let k be a commutative ring. In this section we develop functional topology on the opposite Woronowicz category Wor(k-Alg)op from Section 3. For a morphism f : A —> B in Wor(k-Alg)op, we write F : B —> M(A) for the associated Woronowicz-morphism. We make the following definitions for f : A —> B:

• f is in F if F(B) C A;

• f is in M if F(B) = A;

• f is in E if F is injective.

Lemma 6.1 F is an R-proper class.

Proof It suffices to check that F is stable under R-pullbacks. Consider F : B —> M (A)

and G : B —> M(C) arbitrary such that F(B) C A. Let us introduce some notation:

• We let q: A ( C —> A(C be the canonical quotient map.

• We let I be the ideal of A(C generated by F(b)a(c — a ( G(b)c and aF(b) (c — a ( cG(b), for arbitrary elements a G A, b G B and c G C. We let p : A(C —> (A(C)I1 be the quotient map.

• We let F' : C —> M(C) —> M(A®C/T) be defined as F'(c) = (k', p'), where k'(p(q(a' ( c'))) = p(q(a' ( cc')) and p'(p(q(a' ( c'))) = p(q(a' ( c'c)).

• We let G' : A —> M(C) —> M(A(ClT) be defined G(a) = (k',p'), where k'(p(q(a' ( c'))) = p(q(aa' ( c')) and p'(p(q(a' ( c'))) = p(q(a'a ( c')).

The following is then the R-pullback diagram:

-»■M(A)

M(C)-> M{{A®C)/i)

We need to prove that F'(c) is in (C ( A)/J. But since G is nondegenerate, we have that G(B)C = C. Hence we can write c = ^n G(bn)cn. Thus

FJ(c) = J2 F (G(bn)cn).

Note that F(bn) e A by hypothesis on F. Thus by definition of F', we have

F(G(bn)cn) = (X', p')

X'(p(q(b'( c'))) = p(q(b'® G(bn)cnc')) = p(q(F(bn)b'( cnc')) = XF(bn)®cn(p(q(b' ( c'))). Hence X' = XF(bn)(cn, and similarly p' = PF(bn)(cn. This is thus in (B(C)/I. □

Lemma 6.2 The classes (E, M) form a factorization system.

Proof (1) E and M are closed under composition with isomorphisms. This is obvious.

(2) Every morphism f decomposes as f = me, with m e M and e e E.

In Wor(k--Alg), for every morphism f : A —> M(B), we can decompose it as A A/Ker(f) M(B). This induces a suitable factorization on the dual category.

(3) Every e e E is orthogonal to every m e M. Consider a commutative diagram in Wor(k-Alg):

M(B)<t

Since e(D) = C, we find for each x G C that there exists an y G D such that m(y) = x. Then we define d(x) = v(y). This is a well-defined morphism since e is injective. It is obvious that d is nondegenerate since v is nondegenerate. □

Proposition 6.3 (E n F, M) is a separating (E, M)-proper structure on Wor(k-Alg)op with F as proper maps. With respect to the proper class F:

(1) An object A is compact if and only if it is a unital k-algebra.

(2) An object A is Hausdorff if and only if it is a commutative k-algebra.

(3) All morphisms with commutative domain are separated.

Proof By Lemma 6.2, (E, M) is a factorization system and by Lemma 6.1, F is a proper class. We verify that we can apply Proposition 5.19. For this, we must first verify that R is

E-generated. For this we take nondegenerate morphisms F : A ^ M (C) and G : B ^ M(C), and we take a nondegenerate injection F' : C ^ M(D). Assume that F'FRF'G for each i, this means that for each a G A and b G B, we have F (F(a)) commutes with F (G(b)). Since F' is an injection, this implies directly that F(a) commutes with G(b) and thus that FRG.

Next, we must verify the other condition of Proposition 5.19 that mg G F and m G M implies that g G F. Dually, this comes down to taking nondegenerate morphisms M : A ^ M(B) and G : B ^ M(C) such that GM(A) C C and M(A) = B. To verify that G(B) C C, take some b G B. Then there is some a G A such that M(a) = b. Then G(b) = G(b) = G(M(a)) G C. This verifies the conditions of Proposition 5.19.

(1) The initial object of the category Wor (k-Alg)op is k. A Woronowicz-morphism from k to A is then a nondegenerate morphism F : k —> M(A). Demanding that this morphism is proper is exactly asking that F(k) C A. This is true if and only if A contains a unit.

(2) Let A be a commutative object, that is a commutative k-algebra. The diagonal morphism is represented by D : A(A —> M(A) such that D(a ( b) = ab. Since the algebra is nondegenerate, we obtain that D(A ( A) = A.

(3) Let A be a commutative object, that is a commutative k-algebra and let G : B ^ M(A) be the dual of a morphism in Wor(k-Alg)op. The diagonal is given by D : A(bA ^ M(A) such that D(a ( b) = ab G A. This diagonal thus obviously satisfies the condition D(A(bA) C A. And thus the dual of the diagonal lies in F.

6.3 C*-Algebras

In this section we develop functional topology on the opposite Woronowicz category Wor(C*-Alg)op from Section 3, and we discuss the relation with the situation in LcHd. For a morphism f : A —> B in Wor(C*-Alg)op, we write F : B —> M (A) for the associated Woronowicz-morphism. We make the following definitions for f : A —> B:

• f is in F if F(B) C A;

• f is in M = F0 if F(B) = A;

• f is in E if F is an isometry.

The following is proven along the lines of Proposition 6.3 (see also [19]):

Proposition 6.4 (E n F, M) is a separating (E, M)-proper structure on Wor(C*-Alg)op with F as proper maps. With respect to the proper class F:

(1) An object A is compact if and only if it is a unital C*-algebra.

(2) An object A is Hausdorff if and only if it is a commutative C*-algebra.

(3) All morphisms with commutative domain are separated.

To end this section, we investigate the natural fully faithful functor $ : LcHd —> Wor(C*-Alg)op : X i—> C0(X) which sends a continuous map f : X —> Y to the Woronowicz-morphism $(f) = F : Co(Y) Cb(X) : g g ◦ f.

The image of this functor is equivalent to the category Com(Wor(C*-Alg)op) of R-commutative objects. Recall from Section 6.1 that LcHd is endowed with the factorization system of dense maps and closed embeddings, and taking the closed immersions equal to the closed embeddings yields a closed structure describing the usual closed maps.

Theorem 6.5 [9] Let f : X —> Y be a continuous map between locally compact Hausdorff spaces with associated Woronowicz-morphism F = &(f).

(1) f is proper if and only if F e F, that is F(C0(Y)) c C0X).

(2) f is a closed embedding if and only if F e M, that is F(C0(Y)) = C0X).

(3) f is dense if and only if F e E.

Proof (1) Assume that f is proper and let g e C>(Y). For each e > 0, there exists a compact set K c Y such that ||g|y\kII<xj < e. Since f is proper, we have that K' = f -1(K) is a compact set such that || (g o f) |x\k' II to < e. Thus g o f vanishes at infinity. For the converse implication, assume for each g e C0Y) that g o f e C0X). Let K be a compact subset of Y. By local compactness of Y, we know that there exists some compact set K' such that K lies in the interior of K'. Furthermore, K is the intersection of such sets. Now let g be a continuous function such that g is supported in K' and such that gK' (K) = 1. Since g is compactly supported and thus lies in C>(Y), we see that g o f lies in C0X). Thus there exists a compact set S c X such that |g o f | > 1/2 only on S .If x e f-1 (K), then gK' (f(x)) e K and thus gK (f(x)) = 1. Thus x e S. Hence we see that f-1 (K) c S. Thus f-1(K) is a closed subset of a compact set and is thus closed.

(2) Assume that f is a closed embedding, then f is certainly proper and thus it follows that F(C0(Y)) c C0X). To prove the other inclusion, assume that g e C0X). Then by the Tietze extension theorem (applied on the Alexandroff compactification of X), we see that g extends to a function g' e C>(Y). For this function, we clearly have

F(g') = g.

For the converse implication, we already know by the previous point that f is proper. It suffices to show that f is injective. So let x = y in X such that f(x) = f(y) in Y. We can exhibit a function g e C0X) such that g(x) = g(y). There exists a function g' e C0(Y) such that g' o f = g. But then g(x) = g'(f(x)) = g'(f(y)) = g(y), a contradiction.

(3) Assume that f is dense. It suffices to prove that F is injective. Thus take g, g' e C0 (Y) such that g o f = g' o f .By density of f, it follows that g = g'.

Conversely, assume that F is an isometry. Assume that X is not dense. By local compactness, there then exists a compact subset K of Y with nonempty interior. Let g be a nonzero function supported in K. Then F(g) = 0; while by injectivity of F it follows that g = 0, a contradiction.

Remark 6.6 From (2) and (3) of the above theorem, it formally follows that F e F implies that f is closed (see for instance [16, Proposition 3.48]). In (1), a much more precise result is obtained, namely that F captures precisely the usual proper maps between locally compact Hausdorff spaces when restricted to the commutative objects.

One could remark that our system of functional topology for C*-algebras has M the closed embeddings and not general embeddings. We now describe more general classes

which would also encompass usual embeddings. Sadly, we do not yet know whether these classes give rise to a factorization system.

For a morphism f : A ^ B in the category Wor(C*-Alg)op, we write F : B ^ M(A) for the associated Woronowicz-morphism. We make the following definitions for f : A ^ B:

• f is in E' if f G E and if for each pure state t : B ^ C there exists a pure state t' : M(A) ^ C such that t' o F = t.

• We define the following equivalence relation: we say for x, y G B that x ~f y if for each pure state t : B ^ C for which there does not exist a pure state t ' : M(A) ^ C with t' o F = t, we have that t(x) = T(y).

• We define f G O if for each a G A, there exists some b G B such that F(b) = a and such that b ~ f 0.

• Finally, we define f G M' if its (E, M) factorization f = me satisfies e G O.

This does yield the appropriate notions in the commutative case. So let f : X ^ Y be a continuous map between locally compact Hausdorff spaces, then we get the following Woronowicz-morphism F : C0Y) ^ Cb(X) : g ^ g o f .In this case, we get

Theorem 6.7 Under these conditions, we have:

(1) f is surjective if and only if f is in E';

(2) f is a open embedding if and only if f is in O;

(3) f is an embedding if and only if f is in M'.

Proof (1) This follows since by Gelfand duality we have that any pure state t : C0Y ^ C has the form evy, the evaluation in y.

(2) By Gelfand duality the pure state t' : C0X) ^ C corresponds to evaluations evx with x G X. Thus t' o F corresponds to the evaluation evf (x). Thus the pure states t : C0(Y) ^ C which are not of the form t ' o F correspond to evaluations evy with y G f(X). Hence, we see that for g, g' G C0(Y), we have that g ~f g' if and only if g(y) = g'(y) for each y G f(X). Thus g ~ f 0 if and only if g vanishes outside

Now let f be an open embedding and let g G C0X). We can extend g by 0 on entire Y. Assume that this is not continuous, then there exists some y G dX and some open subset U around y such that |g| > 1/2 on U n X. But since g vanishes at infinity in X, we see that U n X must be contained in a compact subset K of X. This is also a compact subset of Y, and thus y G K. This means that some boundary point of U lies in K C X, this is a contradiction since X was assumed open.

Conversely, let y be a boundary point of f(X) and assume that y G f(X). Then there exists some g G C0X) such that g(y) = 1. By hypothesis, for g there exists some g' G C0Y) such that g' = g o f and such that g' vanishes outside f(X). This is in contradiction with demanding that g(y) = 1 for the boundary point y. We finally prove that f is injective. To show this, assume that x = y in X such that f(x) = f(y). Take g G C0X) such that g(x) = g(y). There exists a g' G C0Y) such that g = g'f and then g(x) = g'(f(x)) = g'(f(y)) = g(y), a contradiction. Thus f must have been injective.

(3) This follows since any locally compact subspace of a locally compact space is an open subspace of a closed subspace. □

Acknowledgments The authors wish to thank Venkatesh Chandrasekaran for making suggestions on the English language used in this article.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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