Secondary theories for simplicial manifolds and classifying spaces

We define secondary theories and characteristic classes for simplicial smooth manifolds generalizing Karoubi's multiplicative K-theory and multiplicative cohomology groups for smooth manifolds. As a special case we get versions of the groups of differential characters of Cheeger and Simons for simplicial smooth manifolds. Special examples include classifying spaces of Lie groups and Lie groupoids.


Introduction
We introduce and analyze secondary theories and characteristic classes for bundles with connections on simplicial smooth manifolds. Classical Cheeger-Simons differential characters for simplicial smooth manifolds with respect to Deligne's 'filtration bête' [4] of the associated de Rham complex were first introduced by Dupont-Hain-Zucker [7] in order to study the relation between the Cheeger-Chern-Simons invariants of vector bundles with connections on smooth algebraic varieties and the corresponding characteristic classes in Deligne-Beilinson cohomology.
In the case of a smooth manifold Dupont, Hain and Zucker showed that the group of Cheeger-Simons differential characters is isomorphic to the cohomology group of the cone of the natural map from Deligne's 'filtration bête' on the de Rham complex of the manifold to the complex of smooth singular cochains.
In a series of fundamental papers Karoubi [14,15] introduced multiplicative K-theory and multiplicative cohomology groups, defined for any filtration of the de Rham complex of a smooth manifold. By taking the filtration to be the 'filtration bête' it follows that Karoubi's multiplicative cohomology groups are generalizations of the classical Cheeger-Simons differential characters in appropriate degrees.
The first author in [11] studied the relationship between differential characters and multiplicative cohomology further. He gave a definition of differential characters associated to an arbitrary filtration of the de Rham complex, which in the case of the 'filtration bête' reduces again to the classical case of Cheeger-Simons. The advantage is that this more general definition allows for the definition of an explicit map at the levels of cocycles between Karoubi's multiplicative cohomology groups and Cheeger-Simons differential characters. It turns out that Karoubi's multiplicative cohomology groups are the natural gadgets for systematically constructing and studying secondary characteristic classes.
Following a similar route in this article we generalize Karoubi's multiplicative cohomology groups and the groups of Cheeger-Simons differential characters even further to simplicial smooth manifolds and arbitrary filtrations of the associated simplicial de Rham complex and study their relations. This allows for a wider range of applications, for example to classifying spaces of Lie groups and Lie groupoids.
The outline of the paper is as follows: After introducing the main background of simplicial de Rham and Chern-Weil theory, mainly following Dupont [5,6] we introduce multiplicative cohomology groups and groups of differential characters for arbitrary filtrations of the simplicial de Rham complex. We discuss briefly some examples like classifying spaces of Lie groups and Lie groupoids. After introducing the concept of multiplicative bundles and multiplicative K-theory on smooth simplicial manifolds, we construct characteristic classes of elements in the multiplicative K-theory with values in multiplicative cohomology and in the groups of differential characters.
In a sequel to this paper we will use this approach to construct and study in a unifying way secondary theories and characteristic classes for smooth manifolds, foliations, orbifolds, differentiable stacks etc. basically for everything to which one can associate a groupoid whose nerve gives rise to a simplicial smooth manifold. Differential characters for orbifolds were already introduced by Lupercio and Uribe using closely the approach of Hopkins and Singer [13]. Chern-Weil theory for general etale groupoids was systematically analyzed by Crainic and Moerdijk [3] using a very elegant approach toČech-de Rham theory, which especially applies well to leaf spaces of foliated manifolds. Working instead in the algebraic geometrical context using de Rham theory for simplicial schemes a similar machinery allows for defining secondary characteristic classes for Deligne-Mumford stacks, most prominently for the moduli stack of families of algebraic curves. Especially multiplicative cohomology with respect to the Hodge or

Elements of simplicial de Rham and Chern-Weil theory
We recall the ingredients of simplicial de Rham and Chern-Weil theory as can be found in Dupont [5,6] or Dupont-Hain-Zucker [7].
A simplicial smooth manifold X • is a simplicial object in the category of C ∞ -manifolds. In other words a simplicial smooth manifold is a functor We can think of X • as a collection X • = {X n } of smooth manifolds X n for n ≥ 0 together with smooth face and degeneracy maps ε i : X n → X n−1 , η i : X n → X n+1 for 0 ≤ i ≤ n such that the usual simplicial identities hold. These maps are functorially associated to the inclusion and projection maps For the differential geometric constructions on X • as introduced below, the degeneracy maps play no role and everything can be defined for so-called strict simplicial or ∆-manifolds [7].
The fat realization of a simplicial space X • is the quotient space where the equivalence relation is generated by There are two versions of the de Rham complex on a simplicial manifold X • (see Dupont [5,6]).

The de Rham complex of compatible forms
in Ω k dR (∆ n−1 × X n ) for all 0 ≤ i ≤ n and all n ≥ 1. Let Ω k dR (X • ) be the set of all simplicial smooth complex k-forms on X • . The exterior differential on Ω k dR (∆ n × X n ) induces an exterior differential d on Ω k dR (X • ). We denote by (Ω * dR (X • ), d) the de Rham complex of compatible forms.
Note that (Ω * dR (X • ), d) is the total complex of a double complex (Ω * , * dR (X • ), d , d ) with where Ω r,s dR (X • ) is the vector space of (r + s)-forms, which when restricted to ∆ n × X n are locally of the form where (t 0 , . . . , t n ) are barycentric coordinates of ∆ n and the {x j } are local coordinates of X n . Furthermore the differentials d and d are the exterior differentials on ∆ n and X n respectively.
We remark that ω = {ω (n) } defines a smooth k-form on n≥0 (∆ n × X n ) and the compatible condition is the necessary and sufficient condition to define a form on the fat realization X • of X • in view of the generating equivalence relation for defining the quotient space X • . The simplicial de Rham complex is the set of smooth complex s-forms on the smooth manifold X r . Furthermore the differential dR (X • ) is the exterior differential on Ω * dR (X r ) and the differential δ :

The singular cochain complex
Given a commutative ring R and a simplicial smooth manifold X • we can also associate a singular cochain complex (S * (X • ; R), ∂). It is defined as a double complex (S * , * (X • ; R), ∂ , ∂ ) with is the set of singular cochains of degree s on the smooth manifold X r .
There is an integration map I : A r,s dR (X • ) → S r,s (X • ; C) which gives a morphism of double complexes and Dupont's general version of the de Rham theorem (see [6, Proposition 6.1] for details) shows that this integration map induces natural isomorphisms Stoke's theorem gives that there is also a morphism of complexes A result of Dupont [5, Theorem 2.3 with Corollary 2.8] gives that this morphism is in fact a quasi-isomorphism, that is,

The singular cochain complex of compatible cochains
Let R be a commutative ring. A compatible singular cochain c on X • is a sequence {c (n) } of cochains c (n) ∈ S k (∆ n × X n ; R) satisfying the compatibility condition in S k (∆ n−1 × X n ) for all 0 ≤ i ≤ n and all n ≥ 1. Let C k (X • ; R) be the set of all compatible singular cochains on X • and (C * (X • ; R), d) be the singular cochain complex of compatible cochains.
It follows that the natural inclusion of cochain complexes is a quasi-isomorphism (see Dupont-Hain-Zucker [7]).
Integrating forms preserves the compatibility conditions and we therefore get an induced map of complexes [7] I : A * dR (X • ) I / / S * (X • ; C) and which is again a quasi-isomorphism, that is, we have . We will use these compatible de Rham and cochain complexes for the definition of multiplicative cohomology and differential characters of X • in Section 2.
We recall finally the basic aspects of Chern-Weil theory in the simplicial context as developed by Dupont [6], and by Dupont, Hain and Zucker [7].

Principal bundles
Let G be a Lie group. A principal G-bundle over a simplicial smooth manifold X • is given by a simplicial smooth manifold P • and a morphism π • : P • → X • of simplicial smooth manifolds, such that (i) for each n the map π p : P n → X n is a principal G-bundle over X n (ii) for each morphism f : [m] → [n] of the simplex category ∆ the induced map f * : P n → P m is a morphism of G-bundles, that is, we have a commutative diagram

Connections and curvature on principal bundles
A connection θ on a principal G-bundle π • : P • → X • over a simplicial manifold X • is a G-invariant 1-form (in the de Rham complex of compatible forms) taking values in the Lie algebra g of G, on which G acts via the adjoint representation, such that for each n the restriction is a connection on the bundle π n : ∆ n × P n → ∆ n × X n . So θ = {θ (n) } can as well be interpreted as a sequence of g-valued compatible 1-forms.
The curvature Ω of the connection form θ is the differential form We have the following general theorem concerning the Chern-Weil map of a simplicial smooth manifold.
is a closed form and descends to a closed form in Ω * dR (X • ) and its cohomology class represents the image of the class Φ ∈ H * (BG; C) under the Chern-Weil map With an abuse of notation, in the sequel we will denote also by Φ(θ) the form in Ω * dR (X • ). In order to classify differential geometric invariants on simplicial smooth manifolds it is useful to extend the constructions outlined above to the category of bisimplicial smooth manifolds. This is straightforward and we will only briefly describe the constructions (compare also Dupont-Hain-Zucker [7] and Dupont-Just [8]).
A bisimplicial smooth manifold X •• is a simplicial object in the category of simplicial smooth manifolds. In other words a bisimplicial smooth manifold is a functor We can think of X •• as a collection X •• = {X m,n } of smooth manifolds X m,n for m, n ≥ 0 together with smooth horizontal and vertical face and degeneracy maps ε i : X m,n → X m−1,n , ε j : X m,n → X m,n−1 η i : X m,n → X m+1,n , η j : X m,n → X m,n+1 for 0 ≤ i ≤ m and 0 ≤ j ≤ n, where the horizontal and vertical maps commute and the usual simplicial identities hold horizontally and vertically.
The fat realization of a bisimplicial space X •• is the quotient space where the equivalence relation is generated by for any (t, s, x) ∈ ∆ m × ∆ n−1 × X m,n .
In a similar manner as for simplicial smooth manifolds, we can associate two de Rham complexes and a singular cochain complex for bisimplicial smooth manifolds.

The de Rham complex of compatible forms
satisfying the compatibility conditions , the de Rham complex of compatible forms on X •• . We note also that we can view the complex (Ω * dR (X •• ), d) as a triple complex is the complex vector space of (r +s+t)-forms, which when restricted to ∆ m × ∆ n × X m,n are locally of the form with (t 0 , . . . , t m ) and (s 0 , . . . , s n ) the barycentric coordinates of ∆ m and ∆ n respectively and the {x k } are local coordinates of X m,n .

The simplicial de Rham complex
Again we also have the simplicial de Rham complex (A * (X •• ), δ) of X •• given as the total complex of the triple complex

The singular cochain complex
For a commutative ring R, we similarly define the singular cochain complex (S * (X •• ; R), ∂) of X •• given as the total complex of the triple complex (S * , * , * (X and ∂ = ∂ + ∂ + ∂ .
Using iteratively the arguments as in the case for simplicial smooth manifolds, we can finally also derive a de Rham theorem relating the cohomology of all the complexes defined with the cohomology of the realization of X •• , that is, we have natural isomorphisms . We remark that we can also define again the singular cochain complex of compatible forms C * (X •• ; R) in a same way as for X • using two compatibility conditions instead. Again we have quasi-isomorphisms as in the simplicial case between the various complexes.
Finally we can extend the elements of simplicial Chern-Weil theory to bisimplicial smooth manifolds, especially we remark that we can define principal G-bundles for the action of a Lie group G and a connection θ on π •• which is again a 1-form . The curvature Ω of the connection form ∇ is again the differential form Again a version of Dupont's theorem (Theorem 1.1) holds in the context of bisimplicial manifolds. When defining characteristic classes we will need that given any connection on a principal bundle, we can construct a connection on (a model of) the universal bundle that pulls back to the given one. For the convenience of the reader, we recall the theorem stating this fact and outline its proof, which for GL n (C)-principal bundles is [7, Proposition 6.15].
Theorem 1.2 Let G be a Lie group, X • a simplicial smooth manifold and π • : . Then there exists a bisimplicial smooth manifold B •• of the homotopy type of the classifying space BG and a G- Proof We define the bisimplicial manifold U •• as follows: with face maps and degeneracy maps The fat realization U •• of this simplicial manifold is contractible, that is, homotopy equivalent to a point (see Segal [19]). Now the free G-action on P • induces a free G-action on U •• . We define the classifying bisimplicial smooth manifold as the quotient We get a principal G-bundle U •• → B •• , the universal principle G-bundle and B •• is homotopy equivalent to the classifying space BG of G.
We define now the connection where (t 0 , . . . , t p ) are the barycentric coordinates of ∆ p and pr j : The canonical isomorphism of simplicial manifolds P and induces a map ψ : such that (Ψ, ψ) pulls back the principal G-bundle P • over X • and the connection θ as stated in the theorem.
2 Multiplicative cohomology and differential characters We will now define general versions of Karoubi's multiplicative cohomology and Cheeger-Simons differential characters for smooth simplicial manifolds with respect to any given filtration of the simplicial de Rham complex. As a special case with respect to the 'filtration bête' we will recover the group of Cheeger-Simons differential characters for smooth simplicial manifolds as introduced by Dupont, Hain and Zucker [7].
In general, for a given complex C * let σ ≥p C * denote the filtration via truncation in degrees below p and similarly σ <p C * denotes truncation of C * in degrees greater or equal p. Let us first consider the special case of Deligne's 'filtration bête' [4] for the simplicial de Rham complex Ω * dR (X • ) of a simplicial manifold X • . The 'filtration bête' σ = {σ ≥p Ω * dR (X • )} is given as truncation in degrees below p We define the group of Cheeger-Simons differential characters as follows: Definition 2.1 (See Dupont-Hain-Zucker [7].) Let X • be a simplicial smooth manifold and Λ be a subgroup of C. The group of (mod Λ) differential characters of degree k of X • is given bŷ Now let F = {F r Ω * dR (X • )} be any given filtration of the simplicial de Rham complex. We define the multiplicative cohomology groups of X • with respect to F as follows: Definition 2.2 Let X • be a simplicial smooth manifold, Λ be a subgroup of C and F = {F r Ω * dR (X • )} be a filtration of Ω * dR (X • ). The multiplicative cohomology groups of X • associated to the filtration F are given by In order to be be able to introduce secondary characteristic classes for connections whose curvature and characteristic forms lie in a filtration of the simplicial de Rham complex we introduce a more general version of differential characters associated to any given filtration. For smooth manifolds these invariants were studied systematically by the first author in [11].
. The groups of differential characters (mod Λ) of degree k of X • associated to the filtration F are given bŷ The truncation in degrees below k of a complex which is already truncated in degrees below k leaves it unchanged, hence if F is Deligne's 'filtration bête' of Ω * (X • ), we recover the ordinary groups of differential characters of X • as in Definition 2.1.
We have the following main theorem generalizing [11, Theorem 2.3].
Theorem 2.4 Let X • be a simplicial smooth manifold, Λ be a subgroup of C and There exists a surjective map Ξ :Ĥ 2r−n−1 r (X • ; C/Λ; F) → MH 2r n (X • ; Λ; F) whose kernel is the group of forms in F r Ω 2r−n−1 dR (X • ) modulo those forms that are closed and whose complex cohomology class is the image of a class in H * (X • ; Λ).
Proof Let A(F r ) and B(F r ) denote the cone complexes used in the definition of the groups of differential characters and multiplicative cohomology associated to the filtration F , that is, and we get a short exact sequence of complexes where σ <k denotes truncation in degrees greater or equal to k. The statement follows now from the long exact sequence in cohomology associated to this short exact sequence of complexes, because for k = 2r − n the cohomology group is trivial.
We can identify the classical Cheeger-Simons differential characters with multiplicative cohomology groups as follows Corollary 2.5 Let X • be a simplicial smooth manifold and Λ be a subgroup of C.
There is an isomorphismĤ Proof This is a direct consequence of Theorem 2.4 in the case when n = r and the filtration F is Deligne's 'filtration bête' using the definitions and the quasi-isomorphism of complexes mentioned in the proof of Theorem 2.4. above In Karoubi's original approach [15,16] towards multiplicative cohomology and differential characters for a smooth manifold M the complex of modified singular cochains is used instead. However this complex is chain homotopy equivalent to the usual complex of (smooth) singular cochains S * (M; Z) of M . Again, also in the more general case of a simplicial smooth manifold X • we can define the complex of modified compatible cochainsC * (X • ; Z) as and proceed as in [16] or [11] for the definition of multiplicative cohomology. But we can then show that the resulting complex using modified cochains is quasi-isomorphic to the compatible cochain complex C * (X • ; Z) used here and the resulting cohomology groups are isomorphic to the ones defined above.
As in the manifold case, it can be shown that the multiplicative cohomology groups fit in the following long exact sequence (compare [16]) The groups of differential characters fit also in short exact sequences analogous to the ones in Cheeger-Simons [2], which are again a special case of the one above.

Remark 2.6
There are several equivalent conventions for the cone of a map of complexes f * : A * → B * . Throughout this paper we will use the following: cone(f * ) n = A n ⊕B n−1 with differential given by d(a, b) = (d A a, (−1) n+1 f n (a)+d B b), where a ∈ A n , b ∈ B n−1 and d A , d B are the differentials in the complexes A * , B * respectively.
We will discuss some applications to specific examples of simplicial smooth manifolds. In order to deal with them in a unified way, we briefly recall the notion of a nerve for a topological category (see Segal [19] or Dupont [5,6]).
Let C be a topological category, that is, a small category such that the set of objects is the subset of composable strings of morphisms The face maps ε i : N (C) n → N (C) n−1 are given as The degeneracy maps η i : N (C) n → N (C) n−1 are given as The nerve N is a functor from the category of topological categories and continuous functors to the category of simplicial spaces.

Classifying spaces of Lie groups
Let G be a Lie group viewed as a topological category with one object, that is, Furthermore letḠ be the topological category defined as There is an obvious functor . . , g n ) = (g 0 g −1 1 , . . . , g n−1 g −1 n ) between simplicial smooth manifolds and applying the fat realization functor gives the universal principal G-bundle γ G : EG → BG.
Using the simplicial smooth manifold N (G) • we can now define As in the general case we get the identification from Corollary 2.5 in the case of the 'filtration bête' σĤ r−1 (BG, C/Λ) ∼ = MH 2r r (BG, Λ, σ).
for the classical Cheeger-Simons differential characters. These invariants were studied in the case G = GL n (C) already by Dupont, Hain and Zucker [7].
We can generalize this situation much further in the following way.

Classifying spaces of Lie groupoids
Let G : X 1 → X 0 be a Lie groupoid, that is, both the set of objects X 0 and the set of morphisms X 1 are C ∞ -manifolds and all structure maps are smooth and the source and target maps are both smooth submersions.
As in the example above we can apply the nerve functor to the category G and we get again a simplicial smooth manifold Let BG be the classifying space of G , that is, the fat realization of the nerve BG = N (G • ) . We define

Actions of Lie groups on smooth manifolds
Let X be a C ∞ -manifold and G a Lie group which acts smoothly from the left on X .
We have a Lie groupoid G : G × X → X with source map s : G × X → X, s(g, x) = x, target map t : G × X → X, t(g, x) = gx and composition map This Lie groupoid was studied in detail by Getzler [12] in order to define an equivariant version of the classical Chern character. Applying the nerve functor again gives a simplicial manifold, the homotopy quotient, which allows us to define equivariant versions of the multiplicative cohomology invariants Definition 2.9 Let G be a Lie group, acting smoothly on a smooth manifold X , Λ a subgroup of C and let The equivariant multiplicative cohomology groups of X associated to the filtration F are defined as MH 2r,n G (X, Λ, F) = MH 2r n (N (G × X → X) • , Λ, F) and the group of equivariant differential characters aŝ We will study secondary theories for classifying spaces of Lie groupoids and Lie groups in more detail in further papers in view of applications to foliations, differentiable orbifolds and differentiable stacks. Equivariant differential characters for orbifolds of type [M/G] for a smooth manifold M with smooth action of a Lie group G with finite stabilizers were constructed and studied systematically by Lupercio and Uribe [18]. Their approach follows closely the modified definition of Cheeger-Simons cohomology due to Hopkins and Singer [13]. It would be interesting to study the relation of these invariants with the ones defined here, especially for different filtrations of the de Rham complex. Chern-Weil theory for principal G-bundles over a Lie groupoid was systematically analyzed by Laurent-Gengoux, Tu and Xu [17]. This framework can be applied to differentiable stacks using the general de Rham cohomology of differentiable stacks as developed by Behrend [1]. The framework developed here allows the definition of multiplicative cohomology groups and groups of differential characters for arbitrary filtrations of the de Rham complex of a differentiable stack, which will be the topic of a sequel to this paper.

Multiplicative bundles and multiplicative K-theory
Let G be a Lie group and θ 0 , . . . , θ q be connections on the principal G-bundle π • : P • → X • , that is, θ j ∈ Ω 1 dR (P • ; g) such that for all p and all 0 ≤ j ≤ q that is, the restrictions θ (p) j are connections on the bundle ∆ p × P p → ∆ p × X p .
Fix q and let ∆ q be the standard simplex in R q+1 parameterized by coordinates (s 0 , . . . , s q ).
Lemma 3.1 The form q j=0 θ j s j defines a (partial) connection on the pullback bundle Proof For each m the sum ( q j=0 θ j s j ) (m) = q j=0 θ (m) j s j is a connection on the bundle We have to verify that the compatibility conditions hold. The strict simplicial structure on X • × ∆ q is given by the maps ε i = ε i × id ∆ q for all i, where ε i is the map given by the strict simplicial structure on X • . We have . Now, since the θ j satisfy the compatibility conditions we have As before we have which proves the lemma.
Given an invariant polynomial Φ of degree k, we denote bỹ Θ q (Φ; θ 0 , . . . , θ q ) ∈ Ω 2k dR (X • × ∆ q ) the characteristic form (on X • ) associated to Φ for the (curvature of the) connection q j=0 θ j s j . When Φ is understood, we will omit it from the notation for the above form. The closed formΘ q (θ 0 , . . . , θ q ) is a family of compatible closed forms We define a form Θ q (θ 0 , . . . , θ q ) ∈ Ω 2k−q dR (X • ) by that is, Θ q (θ 0 , . . . , θ q ) is the family of forms These forms satisfy the compatibility conditions since the diagram commutes and the formsΘ (m) q (θ 0 , . . . , θ q ) ∈ Ω 2k dR (∆ m × X m × ∆ q ) are compatible. If we denote by t the variables on the simplices ∆ p , by x the variables on the manifolds X p and by s the variables on the simplex ∆ q , then we can write with an obvious notation the differential of the complex x is the differential of the complex Ω * dR (X • ). SinceΘ q (θ 0 , . . . , θ q ) is closed, we have d t,xΘq (θ 0 , . . . , θ q ) = −d sΘq (θ 0 , . . . , θ q ).
In particular, for q = 1 we have that given any two connections on P • , θ 0 and θ 1 , and an invariant polynomial Φ, we can write in a canonical way In the sequel it will be convenient to consider formal series of invariant polynomials (like the total Chern class for example), which will then give under the Chern-Weil construction formal sums of differential forms. We now describe a notation (the same as in Karoubi [15,16]) to write formulae in this setting in a compact way. We can write a formal series of invariant polynomials Φ as a sum r Φ r with Φ r a homogeneous polynomial of degree r. Let F = {F r Ω * dR (X • )} be a filtration of the de Rham complex of X • and ω = r ω r , η = r η r be formal sums of forms in Ω * dR (X • ) (note that we do not require that ω r is of degree r, actually most of the times this will not be the case). We will write ω = η mod F if and only if for each r we have (1) ω r − η r ∈ F r Ω * dR (X • ). We will also write ω = η modF when for each r the above equation is satisfied modulo exact forms.
With this notation, all the constructions and proofs in the sequel will be formally the same both for the case of an invariant polynomial, where we will be dealing with forms, and for a formal series of invariant polynomials, in which case we will work with formal sums of forms homogeneous degree by homogeneous degree. Hence we will not distinguish between the two cases in what follows, writing just Φ and ω also for formal sums. Definition 3.3 Let Φ be an invariant polynomial (or a formal series) and F = {F r Ω * dR (X • )} a filtration of the de Rham complex of X • . An (F, Φ)-multiplicative bundle (or just a multiplicative bundle when F and Φ are understood) over X • is a triple (P • , θ, ω) where P • is a principal G-bundle over X • , θ is a connection on P • and ω is a (formal series of) form(s) in Ω * dR (X • ) such that Φ(θ) = dω mod F. An isomorphism f : (P • , θ, ω) → (P • , θ , ω ) between two multiplicative bundles is an isomorphism f of the underlying bundles P • , P • such that As in Karoubi [15], using Proposition 3.2 to prove transitivity, it follows that isomorphism is an equivalence relation on multiplicative bundles, so we can make the following definition.
Definition 3. 4 We denote by MK Φ (X • ; F) the set of isomorphism classes of multiplicative bundles, the multiplicative K-theory of X • with respect to (F, Φ).
As usual we will omit Φ and F from the notation when there is no risk of ambiguity.

Characteristic classes for secondary theories
Let G be a Lie group. Given a principal G-bundle on a simplicial smooth manifold X • with a connection θ , and an invariant polynomial Φ of homogeneous degree k (for the case of a formal series of invariant polynomials one has just to work degree by degree as in Section 3), we will associate characteristic classes with values in multiplicative cohomology groups and in groups of differential characters of X • associated to any filtration of the simplicial de Rham complex Ω * dR (X • ). This will generalize the secondary characteristic classes introduced by Karoubi in the case of smooth manifolds [15].
The connection θ on the principal G-bundle π • is given as 1-form θ ∈ Ω 1 dR (P • ; g) as in Section 1. The characteristic form of Theorem 1.1 dR (X • ) can also as usual be seen as a family of forms Φ(θ (n) ) ∈ Ω 2k dR (∆ n × X n ; g) satisfying the compatibility conditions.
Since Γ is a multiplicative bundle we have where the forms η and ω are also compatible sequences η = {η (n) } and ω = {ω (n) } of differential forms with ω ∈ F r Ω 2k dR (X • ) and η ∈ Ω 2k−1 dR (X • ). The connection θ is the pullback of a connection θ U•• on U •• by a map Ψ as in Theorem 1.2. Let Λ be again a subring of the complex numbers C, and assume that Φ corresponds under the Chern-Weil map to a Λ-valued cohomology class.
For every n the inclusion ı n : B n• → B •• induces isomorphisms in cohomology since B n• is homotopy equivalent to the classifying space of G. For every n we also have that ı * n θ U•• = θ Un• . Since the form Φ(θ Un• ) represents the class of Φ by Theorem 1.1, and ı * n Φ(θ U•• ) = Φ(θ Un• ), we have that the form Φ(θ U•• ) ∈ Ω * dR (B •• ) represents the class of Φ. Then it follows that there exist a compatible cocycle c ∈ C 2k (B •• ; Λ) and a compatible cochain v ∈ C 2k−1 (B •• ; C) such that we have (where for simplicity we omit from the notation the maps from Λ-cochains to complex cochains and the quasi-isomorphism with the de Rham complex).

Remark 4.2
The above characteristic classes can be slightly generalized in the following way. Suppose Φ and Φ are formal sums of invariant polynomials such that every (F, Φ)-multiplicative bundle is also a (F, Φ )-multiplicative bundle (the main example we have in mind is the Chern character ch and the total Chern class c). Then by the same procedure we can construct the classes associated to Φ of elements of MK Φ (X • ; F) with values in the multiplicative cohomology groups and in the groups of differential characters associated to F .