The unitary symmetric monoidal model category of small C*-categories

In the same way that C * -algebras provide an axiomatization of precisely the normclosed and * -closed subalgebras A ⊆ L(H) of bounded operators on Hilbert space, C * -categories axiomatize the norm-closed and * -closed subcategories of the category of all Hilbert spaces and bounded operators between them. An important abstract example is the collection of all * -homomorphisms between any two fixed C * -algebras (or, more generally, of all * -functors between two C * -categories) A and B, which forms a C * -category C * (A, B) in a natural way. Another example is the C * -category of Hilbert modules over a fixed C * -algebra. C * -categories occur naturally in the detailed study of analytic assembly maps and they provide functorial versions of various constructions in coarse geometry (see the numerous references provided in [15]). Most notably, perhaps, C * -categories (with additional structure) feature prominently in Doplicher-Roberts duality of compact groups [4]. References to other domains of applications are given in [6,19]. At the end of this introduction, and in view of our own 'homotopical' results, we shall advocate an entire new course of application. The abstract theory of C * -categories was initiated in the article [6] by Ghez, Lima and Roberts, who were mainly concerned with W*-categories (generalizing von Neumann algebras and comprizing the representation C * -categories C * (A, Hilb)). It was later picked up by Mitchener [15], who described useful basic constructions such as the minimal tensor product ⊗ min and the reduced and maximal groupoid C * -categories C * r G and C * max G associated to every discrete groupoid G; by Kandelaki [11], who has introduced multiplier C * -categories and has used them to characterize C * -categories of countably generated Hilbert modules; and by Vasselli [18,19], who also studies multiplier C * -categories as well as bundles of C * -categories in order to extend Doplicher-Roberts duality to groupoids. All three latter authors have worked with non-unital C * -categories, which appear as the kernels of * -functors. In the present article, we extend the basic theory by generalizing some other constructions from the world of algebras to that of categories. For reasons that will become apparent, we limit ourselves to unital C * -categories (i.e., those with identity arrows for all objects) and units preserving functors. We begin with some recollections (Section 1), and proceed (Section 2) with a concise, but precise, treatment of so-called "universal constructions", which include as special cases all small colimits in the category of small C * -categories and Mitchener's maximal groupoid C * -categories. Another, apparently new, example is provided by the maximal tensor product A ⊗ max B of two C * -categories. Its theoretical importance is testified by the following result (see Theorem 3.15): Theorem. The maximal tensor product ⊗ max defines a closed symmetric monoidal structure on the category C * 1 cat of small (unital) C * -categories and (units preserving) * -functors, whose internal Hom objects are precisely the C * (A, B). We observe (Theorem 3.21): Theorem. The maximal groupoid C * -category is a symmetric monoidal functor ) from discrete groupoids to C * -categories. Our main results are to be found in Section 3, where we uncover some remarkable features of C * -categories. We collect them here in the following omnibus theorem, comprising Theorem 4.2, Proposition 4.15 and Theorems 4.20 and 4.21. Theorem. The category C * 1 cat of small C * -categories and * -functors admits the structure of a cofibrantly generated simplicial model category, where: 1. Weak equivalences are the unitary equivalences, namely those * -functors F : A → B that are equivalences of the underlying categories, such that the isomorphisms F F -1 ≃ id B and F -1 F ≃ id A can be chosen with unitary components. (In fact, as it turns out, the latter property is automatic; see Lemma 4.6.) 2. The cofibrations are the * -functors that are injective on objects. 3. The fibrations are the * -functors F : A → B that allow the lifting of unitary isomorphisms F x ≃ y. Moreover, the above unitary model structure is compatible with the maximal tensor product ⊗ max , that is, together they endow C * 1 cat with the structure of a symmetric monoidal model category. In fact, the previous theorem shows that C * 1 cat is a symmetric sSet-algebra in the sense of Hovey [8]. We should note that the simplicial structure is defined via the nice formulas A ⊗ K := A ⊗ max πK , A K := C * (πK, A) and Map(A, B) := νC * (A, B) (for A, B ∈ C * 1 cat and K ∈ sSet), where πK = C * max (ΠK) is the groupoid C * -category of the fundamental groupoid of K, and ν denotes the simplicial nerve of the subcategory of unitary isomorphisms. The whole structure is intimately connected to the canonical model on the category of small categories (or small groupoids) of homotopists' folklore, where weak equivalences are the categorical equivalences in the usual sense. In the last subsection of our article we explain how these models are related to each other. We now briefly suggest why the operator algebraists should care. The hard-won experience of representation theorists and algebraic geometers has shown that a good way of studying certain invariants of rings, dg-algebras, or schemes, such as K-theory and Hochschild and cyclic (co)cohomology, is to proceed as follows (see [12]): First, substitute your object (ring, scheme, . . . ) with a suitable small dgcategory of representations. The setting of dg-categories is convenient in part because it is closed under many constructions, such as taking tensor products, functor categories, localizations . . . (only the latter works well for, say, triangulated categories). Second, factor your invariant through a suitable localization of the category of small dg-categories, by inverting classes of morphisms (quasi-equivalences, Morita equivalences) that induce isomorphisms of the invariants in question. In order to retain control on the result, one should realize this localization as the homotopy category of a suitable model structure on dg-categories. Another reason dg-categories are convenient is that they allow such models. The resulting localization is now a unified convenient setting where the powerful methods of modern homotopy theory can be applied to the collective study of the invariants. We propose that a similar strategy could be useful for enhancing the study of invariants of C * -algebras (groupoids, . . . ) such as (equivariant and bivariant) K-theory. We have provided here the first pieces of the puzzle: the category of small C * -categories is closed under numerous constructions, and carries a nice model structure for what is perhaps the strongest and most natural notion of equivalence after that of isomorphism, namely, unitary equivalence. The next logical step should be to investigate Morita(-Rieffel) equivalence by providing a suitably localized model structure, and to identify convenient small C * -categories of representations (Hilbert modules, . . . ) that should stand for the C * -algebra. If this vision can be carried out to some extent, we prophesy bountiful applications. Conventions. The base field will be denoted by F, and it is either the field R of real numbers or C of complex numbers. In this article, all categories (included C *categories) have identity arrows 1 x for all objects x, and all functors (included *functors) are required to preserve the identities (but cf. Remark 3.1). An F-category is a category enriched over F-vector spaces. Concretely, an Fcategory consists of a category A whose Hom sets A(x, y) have the structure of Fvector spaces, and whose composition law consists of linear maps A(y, z) ⊗ F A(x, y) → A(x, z). A * -category A is an F-category equipped with an involution, by which we mean an involutive antilinear contravariant endofunctor which is the identity on objects. In detail, an involution consists of a collection of maps A(x, y) → A(y, x) for all objects x, y ∈ ob A, each denoted by a → a * , satisfying the identities A * -functor F : A → B between * -categories is an F-linear functor commuting with the involution: F (a * ) = F (a) * . In the following we shall occasionally work in the purely algebraic category of small * -categories and * -functors. A (semi-)normed category is an F-category whose Hom spaces A(x, y) are (semi-) normed in such a way that the composition is submultiplicative: for any two composable arrows b ∈ A(y, z) and a ∈ A(x, y). A normed category is complete, or is a Banach category, if each Hom space is complete. Every semi-normed category A can be completed to a Banach category, by first killing the null ideal {a | a = 0} ⊆ A and then completing each quotient Hom space. We now come to our main object of study: Definition 2.1 (C * -category). A pre-C * -category is a normed * -category A satisfying the two additional axioms: (iv) C * -identity: a * a = a 2 for all arrows a ∈ A. (v) Positivity: For every arrow a ∈ A(x, y), the element a * a of the endomorphism algebra A(x, x) is positive (i.e., its A C * -category is a complete pre-C * -category. We shall denote by C * 1 cat the category of small C * -categories and * -functors between them. Remark 2.2. Axiom (v) is not particularly elegant. We refer to [15, §2] for a discussion of alternative axiomatizations of C * -categories in the real and complex cases, as well as for a simple example showing that (v) is necessary (i.e., it does not follow from the other axioms). In the complex case, we may substitute (v) with the following axiom: In both the real and complex cases, we may simultaneously substitute (v) with (v) ′ , and (iv) with: (iv) ′ C * -inequality: a 2 a * a + b * b for all parallel arrows a, b ∈ A(x, y). In any case, the ultimate justification for the notion of C As for C * -algebras, we say that an arrow a ∈ A(x, y) in a C * -category is unitary if its adjoint is its inverse: a * a = 1 x ∈ A(x, x) and aa * = 1 y ∈ A(y, y). The present article could be construed as an investigation of the following notion: Definition 2.4 (Unitary equivalence). A * -functor F : A → B is a unitary equivalence if there exist a * -functor G : B → A and isomorphisms u : GF ≃ id A and v : F G ≃ id B such that the components u x ∈ A(GF x, x) and v y ∈ B(F Gy, y) are unitary elements for all x ∈ ob A and y ∈ ob B. Unitary equivalences are simply called equivalences in [15]. Remark 2.5. If A and B are W*-categories then according to [6, after Def. 6.3] every * -functor A → B which is an equivalence is also a unitary equivalence, as can be seen by objectwise applying the polarization identity. Although the latter is not available in general C * -categories (or even C * -algebras), the conclusion is actually true for general C * -categories, as we have learned from [10, §3.1]. As this fact had caused us some confusion in the past we now record it carefully in the next proposition. Proposition 2.6. If two objects of a C * -category A are isomorphic, then they are also isomorphic via a unitary isomorphism. Indeed, if a : x → y is the given isomorphism, a unitary isomorphism u : x → y is obtained by the formula u := a(a * a) -1/2 . Proof. (Cf. [2,Prop. 4.6.4].) If a ∈ A(x, y) is an isomorphism, it follows from 1 * x = 1 x and 1 * y = 1 y that a * is also invertible with inverse (a -1 ) * . Thus a * a is an invertible element of the endomorphism C * -algebra A(x, x), and is moreover a positive element by axiom (v) of C * -categories. By functional calculus there exists a self-adjoint r := (a * a) -1/2 ∈ A(x, x) commuting with a * a and satisfying r 2 = (a * a) -1 . Setting u := ar, we can now compute u * u = r * a * ar = r 2 a * a = 1 x and uu * = ar 2 a * = a(a * a) -1 a * = aa -1 (a * ) -1 a * = 1 y , proving the claim. It follows in particular that a unitary equivalence between two C * -categories is precisely the same thing as a * -functor which induces an equivalence of the underlying ordinary categories (see Lemma 4.6 if necessary). Many pleasant features of C * -algebras generalize almost effortlessly to C * -categories (see [6,15]). We now recall those that we shall use in the following, often without mention. First of all, every * -functor F : A → B between C * -categories is automatically norm decreasing: a A for all a ∈ A, so in particular it is continuous on every Hom space. If F is faithful (that is, injective on arrows) then it is automatically isometric. By combining these two facts, we see that every * -category possesses at most one C * -norm (i.e., one that turns it into a C *category). Similarly, we see that an invertible * -functor F : A → B (an isomorphism in C * 1 cat) is isometric and thus identifies all the structure of the C * -categories A and B, norms included. More generally, every unitary equivalence F is also isometric, since a F a GF a = u * y • a • u x = a (for any a ∈ A(x, y), where G is a * -functor quasi-inverse to F and u : GF ≃ id is unitary). In the next section we shall give a unified treatment of colimits in C * 1 cat together with other universal constructions. The case of limits is much easier: . It is now straightforward to check that the inclusion * -functor E ֒→ A is the equalizer of F and G. The product of a set {A i } i of C * -categories has also an easy construction: let P be the category with objects ob P := i ob A i and with morphism sets As with C * -algebras, one checks easily that the coordinatewise operations and the norm • ∞ equip P with the structure of a pre-C * -category. Moreover, each P (x, y) is complete, so P is a C * -category. This P , together with the canonical projections P → A i , provides the product of the A i in C * 1 cat. Finally, one can always prove properties of C * -categories by way of the basic representation theorem: Proposition 2.8. Every small C * -category is isomorphic to a concrete C * -category. Or equivalently, every small C * -category A has a faithful representation A → Hilb. Proof. See [6, Prop. 1.9], [14,Thm. 3.48] and [15,Thm. 6.12] for more details. Let A and B be two C * -categories, with A small. We recall from [6] the definition of the C * -category C * (A, B) (denoted (A, B) in loc. cit.). Its object set consists of all * -functors F : A → B. For any two * -functors F, F ′ ∈ ob C * (A, B), the set of natural transformations α : F → F ′ between them forms an F-vector space. Setting we define the Hom spaces of C * (A, B) to be the subspaces of bounded natural transformations. Then the pointwise algebraic operations Remark 2.9. Not all morphisms of * -functors are bounded. For instance, let A be the C * -algebra with ob A := N and with Hom spaces A(n, n) := F and A(n, m) := 0 for n = m (this is the coproduct It is natural to ask whether there exists a monoidal structure on C * 1 cat for which C * (-, -) is the internal Hom. The answer is Yes, and will be provided by the maximal tensor product in Theorem 3.15. In this section we construct colimits in C * 1 cat, groupoid C * -algebras C * max G (see also their generalizations C * ism (C) in Definition 4.22) and maximal tensor products A ⊗ max B. These are all special cases of "universal constructions", whose analog for C * -algebras is well-known and widely used. For future use, we provide here a treatment of such general universal constructions for C * -categories. This also suits the spirit of the article, which is to show that C * -categories offer a powerful and flexible alternative setting to that of C * -algebras. As for C * -algebras or pro-C * -algebras [16], it is possible to construct a C * -category by generators and relations, provided the set of relations is well-behaved. Here in addition we have to specify a set of objects, and for this end it is natural to employ the notion of quiver. A quiver is a labeled oriented graph, possibly with loops and multiple edges; in other words, a quiver Q consists of a set ob Q of vertices, here called objects, and of a set Q of arrows, together with source and target functions s, t : Q → ob Q. We write q ∈ Q(x, y) to indicate that the arrow q has source s(q) = x and target t(q) = y. A morphism of quivers ρ : Q → Q ′ assigns to every object x ∈ ob Q an object ρ(x) ∈ ob Q ′ , and to every arrow q ∈ Q(x, y) an arrow ρ(q) ∈ Q ′ (ρ(x), ρ(y)). Thus categories are quivers with the extra structure given by composition, and functors are quiver morphisms that preserve composition. We also want to specify relations that should hold between the arrows, when interpreted in a C * -category. Given a quiver Q, a relation for Q is simply a statement about arrows of Q which makes sense for elements of a C * -category. A quiver with relations (Q, R) is a quiver Q together with a set R of relations for Q. A representation of the quiver with relations (Q, R) is a quiver morphism ρ : Q → A into (the underlying quiver of) a C * -category A, such that the arrows ρ(q) satisfy in A all the relation of R. (For example, a relation r ∈ R for Q may read " q In particular, it must make sense.) Remark 3.1. The only relations that will be used in this article are statements involving algebraic combinations of arrows and norms thereof. However, since there is no extra effort involved, it seems worth it to give here the general treatment of universal constructions in view of future applications, where more sophisticated operations and conditions could be involved. We are thinking for instance of analytic functions such as exponentiation, or continuity conditions when dealing with topological spaces (cf. [16] and [9]). We also note at this point that the definitions and results of this section all have evident non-unital versions. But for simplicity and focus we shall stick with unital C * -categories. The next definition, as well as the proof of Theorem 3. (1) The unique quiver morphism ρ : Q → 0 to the final C * -category (= the zero C * -algebra) is a representation of (Q, R). ( which contains the image under ρ of all objects and arrows of Q, then the restriction ρ ′ : Q → B is also a representation of (Q, R). ( (4) For every arrow q ∈ Q there exists a constant c(q) such that ρ(q) c(q) for all representations ρ of (Q, R). (5) If {ρ i : Q → A i } i is a nonempty small set of representations of (Q, R), then the quiver morphism (ρ i ) i : Q → i A i into the product C * -category mapping q ∈ Q(x, y) to (ρ i (q)) i (which is always well defined if (4) holds, cf. the proof of Lemma 2.7) is again a representation of (Q, R). Theorem 3.3. Given any small (i.e., the arrow and object sets are small ) admissible quiver with relations (Q, R), there exist a small Construction 3.4. Given any quiver Q, the free * -category over Q is the * -category F(Q) with the same set of objects ob Q and whose morphisms are recursively constructed by adding formal units, compositions, adjoints and finite linear combinations of arrows of Q, and then by modding out the algebraic relations which make the units units, the Hom sets F-linear spaces, composition bilinear and * an involution. This construction provides a left adjoint for the forgetful functor from * -categories to quivers, whose unit is the evident inclusion Q → F(Q). Indeed, every quiver morphism ρ : Q → A to a * -category extends to a unique * -functor F(Q) → A, by setting ρ(1 x ) := 1 ρ(x) and for each of the three moves in the recursive definition of the arrows of F(Q). Terminology 3.5. We say that a representation ρ : Moreover, in this case A is isomorphic to the completion of the quotient * -category F(Q)/ ker(π) with respect to the norm induced from A. But Q is small, so there is only a set of possible quotients F(Q)/I (where I indicates some F-linear categorical ideal compatible with the involution), and for each of them there is only a set of possible norms ν. Using condition (3), we conclude that every dense representation is isomorphic to a representation of the form Q → F(Q)/I ν just described, of which there is only a set. Proof of Theorem 3.3. Given an admissible quiver with relations (Q, R), choose a small full set of representatives {ρ i : Q → A i } i∈I for the isomorphism classes of its dense representations, as in Lemma 3.6. By condition (1), I is non-empty. By condition (4) the induced quiver morphism (ρ i ) i : Q → i∈I A i into the product is welldefined, and by condition ( 5) it is again a representation of (Q, R). Let be the restriction of this representation to the smallest sub-C * -category of i∈I A i still containing the image of (ρ i ) i . By condition (2), ρ (Q,R) is a representation of (Q, R), which moreover is dense by construction. Let us verify that ρ (Q,R) satisfies the universal property of the theorem. Let by taking A ′ to be the sub-C * -category of A generated by ρ(Q) and because of (2)), followed by the faithful inclusion ι : A ′ → A. By the definition of I, there exist an i 0 ∈ I and an isomorphism of C * -categories ϕ : and any * -functor U(Q, R) → A satisfying this equality must agree with ρ on Q, and therefore is equal to ρ by the density of ρ (Q,R) . Example 3.7 (Algebraic relations). Let Q be any small quiver and let R be any set of algebraic relations on the arrows of Q. That is, R consists of equations between elements of the free * -category F(Q). Then (Q, R) is admissible and the universal C * -category U(Q, R) exists, as soon as condition (4) holds. Indeed (1), ( 2) and (3) are obvious, and ( 5) is satisfied because the * -algebraic operations in a product of C * -categories are defined coordinatewise. In particular, if B is any * -category we can consider the pair (Q(B), R(B)), consisting of the underlying quiver Q(B) and the set R(B) of all algebraic relations of B. Then the enveloping C * -category Note that, by the representation theorem for C * -categories (Prop. 2.8), in order to compute • ∞ it suffices to consider * -functors B → Hilb. Also, the canonical * -functor ρ B is faithful if and only if there exists a faithful * -functor into any C *category, if and only if there exists a faithful * -functor B → Hilb. We now give a modest example of a universal C * -category that is not defined by algebraic relations. It will be needed to prove Proposition 4.15. Example 3.8. Let Q = {a : 0 → 1} be the quiver with one single arrow between two distinct objects, and consider the single relation R = { a 1}. It is immediately verified that (Q, R) is admissible, and we shall denote the resulting universal C *category by 1. Note that 1 has the property that, for any C * -category A, * -functors F : 1 → A correspond bijectively to arrows a ∈ A of norm at most equal to one. Since there exist operators of norm one, it follows that indeed a = 1 in 1. For similar reasons, we see that a * a = 1 0 and aa * = 1 1 . Our next application is to show the cocompleteness of C * 1 cat. Lemma 3.9. The category of small * -categories and * -functors between them has all small colimits. Proof. It is enough to construct coproducts and coequalizers. Coproducts i B i are easy: just take the disjoint union i B i on objects, with Hom spaces ( i B i )(x, y) := B j (x, y) for x, y ∈ B j and ( i B i )(x, y) := 0 for x ∈ B j , y ∈ B k and j = k. Then i B i inherits a unique composition and involution from those of the B i 's, so that the canonical inclusions B j → i B i are * -functors. Coequalizers are slightly trickier. Let F 1 , F 2 : B → C be any two parallel * -functors between * -categories, and denote by Q the forgetful functor to quivers. Clearly if we quotient the quiver QC by the relations we obtain a quiver D which is the coequalizer, as a quiver, of QF 1 and QF 2 . Then the coequalizer of F 1 and F 2 is the small * -category F(D)/I, the quotient of the free * -category on D by the * -closed ideal generated by all the relations of C. Proposition 3.10. The category C * 1 cat has all small colimits. Proof. Let X : I → C * 1 cat be a small diagram. Let colim V X be the colimit of the diagram V X, where V is the forgetful functor to the category of small * -categories and * -functors. This exists by Lemma 3.9. Now, by Example 3.7 we only have to check (1), the boundedness of the supremum norm over all representations, in order for the enveloping C * -category colim V X → U(colim V X) to exist. It is then clear that this construction, together with the composite * -functors X(i) → colim V X → U(colim V X), enjoys the universal property of colim X in C * 1 cat. Let ρ : colim V X → A be any * -functor to some C * -category. Let f be an arrow of colim V X. By construction (see the proof of Lemma 3.9), f is represented by some algebraic combination of arrows f k of the categories X(i), i ∈ ob I, where, as before, "algebraic" means that only finite linear combinations, compositions and adjoints are allowed. Call this combination C(f 1 , . . . , f n ). Then the following holds, where C indicates the obvious corresponding algebraic combination of the norms. The first inequality is due to the triangle inequality and sub-multiplicativity of the norm and to the isometricity of the involution in A. The second one holds because each * -functor X(i) → colim V X ρ → A starts and ends at C * -categories and is therefore automatically norm-reducing. In particular, the bound c(f ) < ∞ does not depend on the representation ρ. Hence colim V X satisfies condition (1), and we are done. Remark 3.11. While the inclusion C * 1 alg ֒→ C * 1 cat, of (unital) C * -algebras into C * -categories is easily seen to preserve all limits, it does not preserve colimits for the obvious reasons, e.g.: 1 cat has two objects and therefore is not even an algebra. Given two * -categories A and B, their algebraic tensor product A ⊗ F B is simply their tensor product as F-categories As in the case of C *algebras, if A and B are C * -categories there are in general different ways to make the algebraic tensor product into a C * -category. One possibility is to complete with respect to the spatial norm: Proposition 2.8 provides faithful representations ρ : A → Hilb and σ : B → Hilb, which we may combine to form a representation ρ ⊗ σ : A ⊗ F B → Hilb by sending a ⊗ b : (x, y) → (x ′ , y ′ ) to ρ(a) ⊗ σ(b) ∈ L(ρ(x) ⊗ σ(y), ρ(x ′ ) ⊗ σ(y ′ )) and extending linearly. We can thus define a norm f min := (ρ ⊗ σ)(f ) Hilb on the algebraic tensor product A ⊗ F B, which turns out to be independent of the choices of ρ and σ. This • min is a C * -norm, and the corresponding completion A ⊗ min B of A ⊗ F B is called the minimal tensor product of A and B (see [15]). On the other hand, it is also possible to generalize the maximal tensor product of C * -algebras: Proposition 3.12. Let A, B be two small C * -categories. The supremum norm (1) on the * -category A ⊗ F B is bounded, therefore the universal enveloping C * -category of A ⊗ F B exists. Denote it by A ⊗ max B and call it the maximal tensor product of A and B. The canonical * -functor from A ⊗ F B into it is faithful, and the construction specializes to the usual maximal tensor product of (unital ) C * -algebras. Proof. Let us check that f ∞ is finite for each arrow f ∈ A ⊗ F B. For every object x ∈ ob A, we can define a * -functor J x : B → A ⊗ F B by sending b : y → y ′ to 1 x ⊗ b : (x, y) → (x, y ′ ). In the same way we can define a * -functor I y : A → A ⊗ F B for every choice of y ∈ ob B. Let F be any representation of A ⊗ F B. Then the compositions F I y and F J x are * -functors between C * -categories, and therefore they are norm-decreasing. Using this fact and the commutative triangles ai⊗1y / / (x ′ , y) Presumably, we should expect nuclearity to play a fundamental role in the general theory of C * -categories, similar to the role of nuclearity in the general theory of C *algebras -but we have not explored this line of thought yet. Lemma 3.14. Let C be any C * -category, and let A, B be small C * -categories. Then the usual exponential law for (ordinary) categories induces an isomorphism Proof. It is an exercise (and a basic results of enriched category theory) to verify that the exponential law for categories upgrades to F-categories, in the form of an F-linear isomorphism (Hom F denotes the F-category of F-linear functors and natural transformations between them). Under Φ, an F-functor F : A ⊗ F B → C is sent to the F-functor ΦF : A → Hom F (B, C) defined as follows. On objects, ΦF sends x ∈ ob A to F (1 x ⊗ -) : B → C. An arrow a : x → x ′ is sent to the natural transformation ΦF (a) = F (a ⊗ -): The inverse Ψ of Φ takes a functor G : A → Hom F (B, C) and assigns to it the unique F-functor ΨG : Since this is all we need for the next theorem, we leave to the reader the straightforward verification that Φ and Ψ extend to an isomorphism of C * -categories as claimed (actually, this will also be a formal consequence of Theorem 3.15). Theorem 3.15. The maximal tensor product ⊗ max endows C * 1 cat with the structure of a closed symmetric monoidal category, with unit object F and internal Hom's the C * -categories C * (A, B). Proof. Using the universal property of the maximal tensor product, it is straightforward routine to verify that it defines a functor -⊗ max -: , and that the structural associativity, left and right identity, and symmetry isomorphisms of the symmetric monoidal structure ⊗ F of small F-categories induce similar isomorphisms for ⊗ max , which again satisfy the axioms for a symmetric monoidal category with unit object F. The internal Hom construction induces via composition of * -functors a functor C * (-, -) : , and the bijection of Lemma 3.14, which is readily seen to be natural in A, B, C ∈ C * 1 cat, shows that the monoidal structure is closed with internal Hom given by C * (-, -). In order to define the simplicial structure on C * 1 cat we shall need the maximal (or "full") groupoid C * -category C * max G associated to a small discrete groupoid. We recall its construction from [15,Def. 5.10], but here we shall rather emphasize the universal property it enjoys. With this perspective, we can show that it transforms products of groupoids into maximal tensor products: max G mapping all arrows of G to unitary elements of a C * -category. Clearly this defines a functor C * max : Gpd → C * 1 cat on the category of small groupoids and functors between them, which sends equivalences of groupoids to unitary equivalences of C * -categories. To see that the universal notion just described actually exists, we can realize it as the enveloping C * -category (see Example 3.7) of the * -category FG with (This FG is the groupoid category of [15,Def. 5.4].) Thus it suffices to verify that f ∞ = sup ρ ρ(f ) , with supremum taken over all * -functors ρ : FG → Hilb, is finite for every arrow f ∈ FG. Indeed we have because unitaries have norm one or zero (the latter happens if the unitary's domain and codomain have zero endomorphism algebras). Remark 3.17. The completion of FG with respect to another, more concretely defined norm • r , produces the reduced groupoid C * -category C * r G, which generalizes the reduced C * -algebra of a group (see [15,Def. 5.8]). Since the canonical * -functor FG → C * r G is by construction a faithful representation, we conclude that the canonical *functor FG → C * max G is also faithful. For any C * -category A, denote by uni A its subcategory of unitary elements. It follows immediately from Definition 3.16 that composition with the faithful functor ρ for any small groupoid G and any, possibly large, C * -category A. In particular: Proof. The bijection on objects is (4). Now let F, F ′ : C * max G → A be any two *functors, and let α : F → F ′ be a unitary isomorphism. That is, for every x ∈ ob(C * max G) = ob G we have a unitary arrow α x : F (x) → F ′ (x) in A, and for every arrow f ∈ (C * max G)(x, x ′ ) the square is commutative. In particular, it commutes for every f ∈ G(x, x ′ ), so α can also be seen as an (iso)morphism F ρ → F ′ ρ in the category Hom(G, uni A). Conversely, assume that we have a collection (α x : F x → F ′ x) x∈ob G of unitaries in A rendering the above squares commutative for all f ∈ G(x, x ′ ). Then by the linearity of composition in A, the squares commute for all f ∈ FG(x, x ′ ), and by continuity of composition they commute for all f in the completion Lemma 3.20. Let G 1 and G 2 be two small groupoids and A be any C * -category. There is an isomorphism of groupoids Proof. We get the following isomorphisms of categories , as in Lemma 3.19; the standard closed monoidal structure of Gpd; the universal property of C * max G 2 (after which one applies Hom(G 1 , -)); the universal property of C * max G 1 ; and finally the exponential law of Lemma 3.14, to which one applies uni. By looking closely at the constructions of these isomorphisms, we see that the above composition is the result of extending functors and restricting * -functors along the canonical maps, as claimed. induced by the universal property of with the two canonical maps in Lemma 3.20. From the lemma it follows in particular that C * 1 cat(-, A) applied to (5) yields a bijection for every small C * -category A. We conclude by the Yoneda lemma ([13, III.2, p. 62]) that ( 5) is an isomorphism in C * 1 cat. Moreover: Theorem 3.21. The natural isomorphism (5) and the canonical identification F ≃ F1 = C * max (1) make C * max into a (strong) symmetric monoidal functor from the symmetric monoidal category of groupoids, (Gpd, ×, 1), to the symmetric monoidal category of C * -categories, (C * 1 cat, ⊗ max , F). Proof. The commutativity of the coherence diagrams involving (5), F ≃ C * max (1), and the structural isomorphisms of the two symmetric monoidal categories (see [13,VI]), follows immediately from the uniqueness of the arrows induced by the universal properties of C * max and ⊗ max . We shall now prove the main theorem, that there exists a well-endowed model structure on the category of small unital C * -categories, whose weak equivalences are precisely the unitary equivalences (Def. 2.4). This section owes much to Charles Rezk's neat presentation [17] of the canonical (or "folk") model structure on the category of small categories. Our reference for the theory of model categories will be [8]; for a pleasant gentle introduction we refer to the expository article [5]. Definition 4.1. Let F : A → B be a * -functor between unital C * -categories. We call F a cofibration if the map on objects ob F : ob A → ob B is injective. We call F a fibration if for every object y ∈ ob B and every unitary isomorphism v : F x → y there exists a unitary u : x → x ′ in A such that F u = v (and therefore F x ′ = y). Although the next lemma is obvious (in view of Proposition 2.6), we spell out the argument again as this will later help us clarify some constructions. Lemma 4.6. Let F : A → B be a * -functor between C * -categories. The following are equivalent: (i) F is a unitary equivalence. (ii) F is fully faithful (as a functor of the underlying categories of A and B) and also unitarily essentially surjective, i.e., for every y ∈ B there is a unitary arrow y ≃ F x in B for some x ∈ ob A. (iii) F is an (ordinary) equivalence of the underlying categories of A and B. (iv) F is fully faithful and essentially surjective, in the usual sense. Proof. The implications (i) ⇒ (ii), (ii) ⇒ (iv), and (iii) ⇒ (iv) are seen immediately, and (iv) ⇒ (ii) follows from Proposition 2.6. The implication (iv) ⇒ (iii) is an exercise application of the axiom of choice, and (ii) ⇒ (i) is quite similar, as we now verify in detail. Assume that F : A → B is fully faithful and unitarily essentially surjective. By the latter condition, we are free to choose for each y ∈ ob B an object Gy ∈ ob A and a unitary isomorphism v y : F Gy → y. Let b ∈ B(y, y ′ ). Since F is bijective on Hom spaces, we may set Gb := F -1 (v * y ′ bv y ) ∈ A(Gy, Gy ′ ). One checks immediately that the assignments x → Gx and b → Gb define a functor G : B → A. Moreover, G is a * -functor, as one sees by applying F -1 to the equality ), and the unitaries (v y ) y∈ob B define an isomorphism v : F G → id B . Finally, note that for every x ∈ ob A we have chosen an object GF x ∈ ob A and a unitary element v F x ∈ B(F GF x, F x). It is straightforward to verify that the elements u x := F -1 (v F x ) ∈ A(GF x, x) are unitary and form an isomorphism u : GF → id A . Therefore F is a unitary equivalence with quasi-inverse G. Remark 4.7. When proving the implication (ii)⇒(i) of Lemma 4.6, note that if F happens to be injective on objects, for every y = F (x) ∈ ob B lying in the image of F we may well choose G(y) := x, and similarly we may choose the unitary v y : F Gy → y to be the identity of y. The resulting quasi-inverse G : B → A has then the additional property that GF = id A . Proof. Assume that F is a trivial fibration. Being a unitary equivalence, it is fully faithful and unitarily essentially surjective (Lemma 4.6). Hence, for every object y ∈ ob B there are an x ∈ ob A and a unitary v : y ≃ F x. Since F is also a fibration, we may lift v to a unitary u : x ′ ≃ x in A with F u = v. In particular F x ′ = y, showing that F is surjective on objects. Conversely, assume that F is fully faithful and surjective on objects. Then F is unitarily essentially surjective (because identity maps are unitaries), and therefore it is a unitary equivalence by Lemma 4.6. Also, given a unitary v : F x → y there is an x ′ with F x ′ = y, and F -being a * -functor -induces a bijection F : A(x, x ′ ) ≃ B(F x, y) of unitary elements. Hence F is a fibration too. We now prove the five axioms MC1-MC5 of a model categories. MC1 (Limits and colimits axiom). We have already established in Lemma 2.7 that C * 1 cat is complete, and in Proposition 3.10 that it is cocomplete, so the first axiom holds. MC2 (2-out-of-3 axiom). We must verify that unitary equivalences have the 2out-of-3 property, i.e., that whenever two * -functors F, G are composable and two out of {F, G, F G} are in Weq, then so is the third. This follows by combining the 2-out-of-3 property of the usual equivalences of categories with that of unitary arrows of C * -categories. We leave the easy exercise to the reader. Then there exists a lifting L : B → C making the two triangles commute. Proof. (i) Let F ∈ Weq ∩ Cof. By Lemma 4.6 and Remark 4.7, there exists a quasiinverse F ′ : B → A such that F ′ F = id A , and such that the other unitary v : F F ′ → id B is the identity for every object in the image of ob F : Let us first define the lifting L on objects. For every x ∈ ob B, let Note that y x is such that Gy x = V F F ′ x, by the commutativity of the square. Since G is a fibration and V v x : Gy x = V F F ′ x → V x is a unitary in D, we can choose an object Lx ∈ ob C and a unitary w x : Moreover, for x = F z in the image of ob F , we may certainly (and will) choose We should now (and will) define L on morphisms b : x → x ′ by the formula: x , so that the following square commutes. 6), ( 7), (8) and F ′ F = id A , it is now immediate to verify that L : B → C is a * -functor such that GL = V and LF = U . (ii) This time, we can choose the map ob L by using the injectivity of ob F and the surjectivity of ob G (Corollary 4.8). We take care to set Lx := U z whenever x = F z is in the image of F . Since G is also fully faithful, it provides isomorphisms G : C(Lx, Lx ′ ) ≃ D(GLx, GLx ′ ) = D(V x, V x ′ ) for all x, x ′ ∈ ob C. Composing their inverses with V defines the unique * -functor L : C → D with the chosen object-map and such that GL = V and LF = U . Proof. Our functorial factorizations will have the classical and familiar form: The two midway objects are defined by the pullback, resp. pushout, squares This is a C * -category with the evident operations, and it comes equipped with the two projections ẽv 0 : à → A and F : à → C * (I, B) (the latter sending (x, u, y) to the unique * -functor F (x, u, y) : I → B which assigns the unitary u : F x ≃ y to the generator 0 → 1, and sending a : (x, u, y) → (x ′ , u ′ , y ′ ) to the morphism F a : F (x, u, y) → F (x ′ , u ′ , y ′ ) with components ( F a) 0 := F a : F x → F x ′ at 0 and ( F a) 1 := u ′ • F a • u * : y → y ′ at 1). It is now straightforward to check that Ã, together with its two projections, satisfies the universal property of the pullback C * (I, B) × B A. With this picture, it is easy to see that I : A → à is given by x → (x, 1 x , F x). Thus it is clearly fully-faithful and also unitarily essentially surjective, since for every (x, u, y) ∈ ob à the identity 1 x defines a unitary isomorphism 1 x : (x, u, y) ≃ (x, 1 F x , F x) = Ix; therefore I is a unitary equivalence by Lemma 4.6 (a quasi-inverse is given by ẽv 0 : à → A). (ii) F has the right lifting property with respect to V : F ⊔ F → 1 if and only if it is full. (iii) F has the right lifting property with respect to W : P → 1 if and only if it is faithful. Proof. This is a straightforward translation of the three lifting properties, using both the universal properties of F, Proof. Recall that the object-set of a colimit of C * -categories is the colimit of the object-sets. Thus if A ∈ {∅, F, F ⊔ F}, we see that every * -functor from A into a filtering colimit must factor through some stage (in the case A = F ⊔ F, we need the colimit to be filtering to ensure that the two objects determining the * -functor out of F ⊔ F will eventually land in the same C * -category). Thus in particular ∅, F and F ⊔ F are finite objects of C * 1 cat. Now let us consider ) α<λ be a sequence of * -functors indexed by an uncountable limit ordinal λ, and let F : P → colim α<λ B α =: B be a * -functor into the colimit of the sequence in C * 1 cat. Let a, a ′ : 0 → 1 be the two generating arrows of P . By the construction of colimits as a completion (Prop. 3.10), the arrow F a is the norm colimit in B(F 0, F 1) of a countable sequence (b n ) n∈N with b n ∈ B αn ; since λ is an uncountable limit ordinal, F a must be the image of some b ∈ B β for some β < λ. Similarly, there exists a b ′ ∈ B β ′ , for some β ′ < λ, mapping to F a ′ ∈ B. Moreover, reasoning with objects as above we see that b and b ′ must become parallel arrows at some stage γ < λ. Thus there exists a γ < λ and there exist arrows f, f ′ ∈ B γ (y 0 , y 1 ) (for some y 0 , y 1 ∈ ob B γ ) mapping to F a and F a ′ respectively. Therefore F : P → B factors through the unique * -functor G : P → B γ determined by Ga = f and Ga ′ = f ′ . This shows that P is ℵ 1small in C * 1 cat. By Lemma 4.14, the domains of the generating maps I and J are small in C * 1 cat, thus ensuring that the (weaker) smallness condition required by the definition of cofibrantly generated model categories is satisfied. Adding up all the lemmas, we conclude: Proof. It suffices to verify that π preserves cofibrations and that it sends the generating trivial cofibrations of sSet to trivial cofibrations. If f : K → L is a cofibration (i.e., a dimensionwise injective map), then in particular it is injective in dimension zero and therefore π(f ) is injective on objects, i.e., it is a cofibration of C * 1 cat. Now let ι n,k : Λ k [n] → ∆[n] (for n 1, 0 k n) be a generating trivial cofibration of sSet. Precisely as in [17,Thm. 6.1], one can verify that, for n > 1, the functor Π(ι n,k is an isomorphism of fundamental groupoids, and that Π(ι 1,k ) is the inclusion 1 → I of the trivial group as one end of the interval groupoid (Def. 4.4). Hence each π(ι n,k ) = C * max Π(ι n,k ) is an isomorphism of C -categories for n > 1, and π(ι 1,k ) = (0 : F → I) is the generating trivial cofibration of C * 1 cat. In particular, these are all trivial cofibrations as required. 1 cat is a simplicial symmetric monoidal left Quillen functor. Thus, in the language of [8], the unitary model category C * 1 cat is a symmetric sSet-algebra. Proof. The functor Π : sSet → Gpd commutes with finite products, thus defining a symmetric monoidal functor (sSet, ×, pt) → (Gpd, ×, 1); by Theorem 3.21, C * max is also symmetric monoidal, and therefore so is their composition π. We have seen (Prop. 4.18) that π is a left Quillen functor, and by Definition 4.19 it transports the simplicial structure of sSet to that of C * 1 cat. There is a commutative tetrahedron of model categories where every edge is a Quillen adjoint pair: (for each pair (L, R), the arrow shows the direction of the left adjoint L). triangle on the left hand side is and comprises the usual model category of simplicial sets, together with the so-called canonical, or folk, models on categories and groupoids. For both Cat and Gpd, the weak equivalences are the equivalences in the usual sense of category theory, the cofibrations are the functors that are injective on objects, and the fibrations are the functors F : C → D enjoying the lifting property for isomorphisms: if u : F x ≃ y is an isomorphism in D, then there exists an isomorphism v in C such that F v = u (cf. [17] and [1, §5]). It should now be obvious that the unitary model for C * 1 cat has been adapted from the latter ones. All labeled Quillen pairs on the diagram should be clear. Just recall that Π denotes the fundamental groupoid, left adjoint to the simplicial nerve N . With inc we have denoted the inclusion of groupoids into categories, whose right adjoint iso maps a category to its subcategory of isomorphisms. Observe that all model categories are symmetric monoidal (C * 1 cat for ⊗ max , the other three for the categorical product), and that all left Quillen functors are symmetric monoidal. We now explain the remaining, still unlabeled, Quillen pair Cat ⇆ C * 1 cat, which provides a direct connection between categories and C * -categories. * * -categories with one single object can be identified with (unital) C * * * -algebra. All groupoids in this article are discrete.