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International Journal of Theoretical Physics, Vol. 40, No. 7, 2001

Synthetic Braided Geometry II Hirokazu Nishimura1 Received October 3, 2000 Braided geometry is a natural generalization of supergeometry and is intimately connected with noncommutative geometry. Synthetic differential geometry is a peppy dissident in the stale regime of orthodox differential geometry, just as Grothendieck’s category-theoretic revolution in algebraic geometry was in the middle of the 20th century. Our previous paper [Nishimura (1998) International Journal of Theoretical Physics, 37, 2833–2849] was a gambit of our ambitious plan to approach braided geometry from a synthetic viewpoint and to concoct what is supposedly to be called synthetic braided geometry. As its sequel this paper is intended to give a synthetic treatment of braided connections, in which the second Bianchi identity is established. Considerations are confined to the case that the braided monoidal category at issue is a category of vector spaces graded by a finite Abelian group with its nonsymmetric braiding being given by phase factors. Thus the present paper encompasses physical systems amenable to anyonic statistics.

0. INTRODUCTION Synthetic differential geometry is the vanguard of modern differential geometry, in which nilpotent infinitesimals are not only abundantly available as in the age of Riemann, Lie and Cartan, but also coherently organized with mathematical rigor. Synthetic differential geometry was pioneered by Lawvere, a famous category-theorist, in the middle of the 1960’s, while Grothendieck revolutionized algebraic geometry by exploiting ideas of category theory (e.g., representable functors). Although Grothendieck’s category-theoretic revolution in algebraic geometry during the middle of the 20th century is well appreciated among contemporary algebraic geometers, Lawvere’s corresponding one in differential geometry has not received more than studied indifference from orthodox differential geometers. The so-called tensor analysis on infinitesimal entities (e.g., vector fields) in orthodox differential geometry is often stodgy and factitious, concealing the truly infinitesimal nature of infinitesimal considerations under a topsy-turvy of lengthy calculations in a dull drone. Synthetic differential geometry enables us to endow differential geometry with an infinitesimal horizon relatively independent of local 1 Institute

of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan. 1363 C 2001 Plenum Publishing Corporation 0020-7748/01/0700-1363$19.50/0 °

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and global ones. The standard notion of smooth manifold in orthodox differential geometry is not susceptible of any reasonable or fruitful generalization to noncommutative or braided geometry, while the central notion of microlinear space in synthetic differential geometry is. For good textbooks on synthetic differential geometry, the reader is referred to Lavendhomme (1996) (devoted mainly to a consistently axiomatic presentation of synthetic differential geometry) and Moerdijk and Reyes (1991) (devoted to model theory of synthetic differential geometry) as well as to Kock’s bible of the field (Kock, 1981). We assume the reader to be familiar with Lavendhomme (1996) up to Chapter 5. Supergeometry enables us to deal with bosons and fermions on an equal footing by intermingling them. It is expected to play a central role in any possible unification of relativity and quantum theory. Supergeometry lies at the entrance to noncommutative geometry in the sense that the ring of real supernumbers is not commutative but graded-commutative. For good textbooks on supergeometry the reader is referred to Bartoci et al. (1991), Leites (1980), and Manin (1988). We have approached supergeometry from a synthetic viewpoint in Nishimura (1998a, 1999, 2000a,c). In particular, a synthetic treatment of superconnections was given in Nishimura (2000a). Braided geometry is an elegant and far-reaching generalization of supergeometry, in which the category of vector spaces is replaced by a braided monoidal category. It is pioneered and championed by Majid (1995a,b), Marcinek (1994), and others. The standard gadget for transmogrifying braided geometry into noncommutative geometry is bosonization, while the standard device for translating noncommutative geometry into braided geometry is transmutation. If the braiding is symmetric, braided geometry lies in the very periphery of supergeometry so that it is not truely braided. We will consider a (nonsymmetric) braided monoidal category of vector spaces graded by a finite Abelian group whose braiding is given by a phase factor. Our considerations do not only encompass color geometry but also anyonic geometry. For anyonic geometry the reader is referred to Majid (1993, 1994, 1997). As a sequel to Nishimura (1998b) this paper gives a synthetic treatment of braided connections by generalizing our synthetic treatment of superconnections in Nishimura (2000a). Basic definitions and basic properties will be presented in Section 3. Section 5 is devoted to a combinatorial treatment of the so-called second Bianchi identity. Sections 4 and 6 are devoted to induced braided connections. We have gathered some preliminaries in Section 1. We deal in haste with braided exterior differential calculus in Section 2. As is usual in synthetic differential geometry, the reader should presume throughout the paper that we are working in a (not necessarily Boolean) topos, so that the excluded middle and Zorn’s lemma have to be avoided. Objects of the topos go under such aliases as a “space,” a “set,” etc.

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1. PRELIMINARIES 1.1. Basic Braided Algebra We choose, once and for all, a braided monoidal category C = (C , ⊗, 1, 8, l , r , 9) abiding by the following conditions: (1.1) C is a subcategory of the category of all k -linear spaces with a field k . (1.2) ⊗ is the standard tensor product of k -linear spaces. (1.3) The unit object 1 is k , regarded as a k -linear space in the standard manner. (1.4) The associativity constraint 8, the left unit constraint l and the right unit constraint r are the standard ones of k -linear spaces. (1.5) There exists a finite set Π of mutually nonisomorphic objects of C including the unit object 1, say, Π = {1, 2, 3, . . . ,k}, such that (1.5.1) Every object p in Π is a one-dimensional k -linear space. (1.5.2) The set Π is closed under ⊗, i.e., for any objects p, q in Π, there exists an object r in Π such that p ⊗ q is isomorphic to r in the category C (we will constantly use p, q, r, . . . with or without subscripts as variables over Π). (1.5.3) Every direct sum of (possibly infinitely many) copies of objects in Π as well as all its associated canonical injections and projections belongs to C , and any object in C is a direct sum of copies of objects in Π. (1.6) For any morphism α : U → V in C , if α happens to be an isomorphism of k -linear spaces, then α −1 : V → U belongs to C so that α is an isomorphism in C . Note that we have not assumed 9 to be symmetric. As is the custom in dealing with monoidal categories, we will often proceed as if the monoidal category (C , ⊗, 1, 8, l , r ) were strict, which is justifiable by Theorem XI.5.3 of Kassel (1995). We will often write p + q for r isomorphic to p ⊗ q in (1.5.2). Then it is easy to see that Proposition 1.1. above.

Π is an abelian monoid with respect to the operation + defined

Proof: The associativity constraint 8p,q,r : (p ⊗ q) ⊗ r → p ⊗ (q ⊗ r) guarantees that Π is a semigroup. The left unit constraint l p : 1 ⊗ p → p and the right unit constraint rp : p ⊗ 1 → p warrants that Π is not only a semigroup but also a monoid. The commutativity of the monoid Π follows from the braiding 9p,q : p ⊗ q → q ⊗ p. ¤

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We choose an arbitrary nonzero element xp of each one-dimensional k -linear space p in Π once and for all. For p, q in Π there exists a unique δ p, q ∈ k such that (1.7) 9p,q (xp ⊗ xq ) = δ p,q (xq ⊗ xp ) It is easy to see that the numbers δ p,q do not depend on our particular choice of {xp }p∈Π . Proposition 1.2.

The numbers δ p,q satisfy the following identities:

(1.8) δ p,q+r = δ p,q δ p,r (1.9) δ p+q,r = δ p,r δ q,r (1.10) δ p,1 = δ 1,p = 1 Proof: (1.8) and (1.9) follow from the so-called hexagon axiom, which claims that 9p,q⊗r = (idq ⊗ 9p,r ) ◦ (9p,q ⊗ idr ) and 9p⊗q,r = (9p,r ⊗ idq ) ◦ (idp ⊗ 9q,r ) up to associativity and unit constraints. Since δ p,1 = δ p,1+1 = δ p,1 δ p,1 by (1.8), it follows that δ p,1 = 1. Similary it follows from (1.9) that δ 1,p = 1. ¤ Since we do not assume 9 to be symmetric, it does not follow that δ p,q δ q,p = 1. We require that (1.11) Π is not only an abelian monoid but even an abelian group, so that, if the braiding 9 is symmetric, the pair (Π, δ) is a signed group in terms of Marcinek (1991). Given an object U in C , the direct sum decomposition of U into objects in Π in (1.5.3) is not unique, but the p-component of U defined as the direct sum of the images of all the canonical injections from p into U with respect to a particular decomposition of U , will soon turn out to be independent of our choice of a particular decomposition of U . Therefore we can safely write U p for the p-component of U . Proposition 1.3. Let 0 and 0 0 be two direct sum decompositions of U in (1.5.3). p p Then for any p in Π, the p-components U0 and U00 of U with respect to 0 and 0 0 coincide. Proof: The proof uses a gimmick that is familiar in the proof of the well-known fact of algebra that although a direct sum decomposition of a semisimple module into simple ones is not unique, its homogeneous component affiliated to a particular simple module is well defined, for which the reader is referred, e.g., to Wisbauer (1991, Chapter 4). For any canonical injection ι of p into U in the decomposition 0 and any canonical projection π of U onto q in the decomposition 0 0 with

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p 6= q, π ◦ ι = 0, for otherwise p and q would be isomorphic in C by (1.7). This p p means that U0 ⊂ U00 for any p in Π. By interchanging the roles of 0 and 0 0 in the p p above discussion, we have that U00 ⊂ U0 for any p in Π. Therefore the desired conclusion follows. ¤ Corollary 1.4. U = U 1 ⊕ · · · ⊕ U k , so that each u ∈ U can be decomposed uniquely as u = u 1 + · · · + u k with u p ∈ U p for any p in Π. An element u of U which happens to consist in U p for some p in Π is called pure (of grade p), in which we will denote p by |u|. The same gadget used in the proof of Proposition 1.3 establishes the following: Proposition 1.5. Any morphism α : U → V in C preserves grading (i.e., α(U p ) ⊂ V p for each p in Π. We now enjoin that the class of morphisms in C be saturated with respect to this property in the following sense: (1.12) For any objects U, V in C , if a homomorphism α : U → V of k -linear spaces preserves grading (i.e., α(U p ) ⊂ V p for any p in Π), then α lies in C . The notion of an algebra in the braided monoidal category C, usually called a C-algebra, can be defined diagrammatically as in Kassel (1995, Section III.1). A C-algebra A with its product µA : A ⊗A → A is said to be C-commutative if µA ◦ 9A , A = µA . Given a C-algebra A , the notions of a left A -module and a right A -module in C, usually called a left A -C-module and a right A -C-module, respectively, can be defined diagrammatically as in Majid (1995a, Section 1.6). If A happens to be C-commutative, a left A -C-module M with its left action η : A ⊗ M → M can naturally be converted into a right A -C-module with its right action η ◦ 9M , A : M ⊗ A → M , and vice versa, so that the distinction between “left” and “right” is not essential in the C-commutative case. In this case any left A -C-module and, equivalently, any right A -C-module are naturally A -bimodules. A left (right, respectively) A -C-module M is said to be C-finite-dimensional if there exists a finite-dimensional k -linear space V in C such that A ⊗ V , (V ⊗ A , respectively) is isomorphic to M as left (right, respectively) A -C-modules. The notions of a left A -module algebra and a right A -module algebra in C, usually called a left A -C-algebra and a right A -C-algebra, respectively, can also be defined diagrammatically as in Majid (1995a, Section 1.6). An ideal of a C-algebra A is said to be a C-ideal if it belongs to C . Other standard notions such as that of a homomorphism of C-algebras that can easily be formulated diagrammatically will be used freely. Given a C-commutative C-algebra A and an A -C-algebra B , SpecA B denotes the totality of homomorphisms of A -C-algebra from B into A .

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Now we choose, once and for all, a C-commutative C-algebra R intended to play a role of real numbers in our braided mathematics. So we must enjoin the following axiom on R: (1.13) R is a C-commutative C-algebra. Another important axiom on R will be presented in the next subsection. Given a set Z , the totality of functions from Z to R is an R-C-algebra with componentwise operations whose p-component can naturally be identified with the totality of functions from Z to Rp . Given right R-C-modules M and N , the totality HomR (M , N ) of R-homomorphisms from M to N is a left R-module in the sense that for any a ∈ R, any u ∈ M and any f ∈ HomR (M , N ), (1.14) (af ) (u) = af (u). p

It is not difficult to see that the p-component HomR (M , N ) of HomR (M , N ) is the totality of f ∈ HomR (M , N ) such that | f (u)| = |u| + p for any u ∈ M . We define HomR (M , N ) to be Hom1R (M , N ) ⊕ · · · ⊕ HomK R (M , N ), which is an R-C-module. Given a finite sequence p1 , . . . , pn in Π, we can form the tensor C-algebra T (p1 ⊕ · · · ⊕ pn ) of the k -linear space p1 ⊕ · · · ⊕ pn . The quotient C-algebra of T (p1 ⊕ · · · ⊕ pn ) with respect to the C-ideal generated by {xp j xpi − δ p j , pi xpi xp j |1 ≤ i, j ≤ n} is a C-algebra called the polynomial C-algebra of variables xp1 , . . . , xpn and is denoted by k [xp1 , . . . , xpn ]. The R-C-algebra R ⊗2 k [xp1 , . . . , xpn ] is called the polynomial C-algebra of variables xp1 , . . . , xpn over R or the polynomial R-C-algebra of variables xp1 , . . . , xpn and is denoted by R [xp1 , . . . , xpn ]. The R-C-algebra R [xp1 , . . . xpn ] is characterized by the following universality property: Proposition 1.6. The R-C-algebra R [x p1 , . . . , xpn ] is C-commutative. For any C-commutative R-C-algebra A and any morphisms αi : pi → A in C (1 ≤ i ≤ n), there exists a unique homomorphism α of R-C-algebras from R[xp1 , . . . , xpn ] to A whose restriction to pi is αi (1 ≤ i ≤ n). 1.2. Weil C-Algebras and C-Microlinearity A Weil C-algebra is a C-commutative R-C-algebra M which, regarded as an R-module, is to be written as M = R ⊕ m with the first component being the R-C-algebra structure and the second being a finite-dimensional nilpotent C-ideal (called the C-ideal of augmentation). By way of example, the quotient C-algebra of the polynomial C-algebra R[X 1 , . . . , X n ] with respect to the C-ideal generated by {X i X j |1 ≤ i ≤ n} is a Weil C-algebra and is denoted by M(p1 , . . . , pn ) with 2⊗

denotes the braided tensor product.

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pi = |X i | (1 ≤ i ≤ n). Given Weil C-algebras M1 and M2 with their C-ideals of augmentation m1 and m2 respectively, a homomorphism of R-C-algebras ϕ : M1 → M2 is said to be a homomorphism of Weil C-algebras if it preserves their C-ideals of augmentation, i.e., if ϕ(m1 ) ⊂ m2 . A finite limit diagram of R-Calgebras is said to be a good finite limit diagram of Weil C-algebras if every object occurring in the diagram is a Weil C-algebra and every morphism occurring in the diagram is a homomorphism of Weil C-algebras. The diagram obtained from a good finite limit diagram of Weil C-algebras by taking SpecR is called a quasicolimit diagram of C-small objects. The Braided version of the general Kock axiom, called the general C-Kock axiom, is as follows: (1.15) For any Weil C-algebra M, the canonical homomorphism M → RSpecR (M) of R-C-algebras is an isomorphism. Spaces of the form SpecR (M) for some Weil C-algebra M are called Cinfinitesimal spaces or C-small objects. The C-infinitesimal space corresponding to Weil C-algebra M(p1 , . . . , pn ) is denoted by D(p1 , . . . , pn ). By Proposition 1.6, D(p1 , . . . , pn ) is to be identified with {(d1 , . . . , dn ) | di ∈ Rpi (1 ≤ i ≤ n), di d j = 0 (1 ≤ i, j ≤ n)}. We will often denote D(p) by D p . The C-infinitesimal space D(1, . . . , k) plays a very important role in our discussion of tangency. First we note that D(1, . . . , k) can be identified with the subset of R consisting of all d ∈ R such that dp dq = 0 for any p, q ∈ Π. Under this identification (d1 , . . . , dk ) ∈ D (1, . . . k) corresponds to d1 + · · · + dk ∈ R. What concerns us most about D(1, . . . , k) is that the space D(1, . . . , k), regarded as a subset of R, is closed under the left and right actions of R on itself. Just as the general Kock axiom paved to the introduction of microlinear spaces, its braided version invokes the notion of a C-microlinear space, which is by definition a space M abiding by the following condition: (1.16) For any good finite limit diagram of Weil C-algebras with its limit W , the diagram obtained by taking SpecR and then exponentiating over M is a limit diagram with its limit M SpecR W . The following proposition guarantees that we have a plenty of C-microlinear spaces. Proposition 1.7. (1) Rp is a C-microlinear space for any p ∈ Π. (2) The class of C-microlinear spaces is closed under limits and exponentiation by an arbitrary space. The above braided version of the general Kock axiom surely subsumes the following braided version of the Kock–Lawvere axiom to be called the

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C-Kock–Lawvere axiom: (1.17) For any function f : D p → R, there exists unique b ∈ R such that f (d) = f (0) + bd for any d ∈ D p . The axiom (1.17) is equivqlent to the following axiom: (1.18) For any function f : D p → R, there exists unique b0 ∈ R such that f (d) = f (0) + db0 for any d ∈ D p . We conclude this subsection by a definition. An R-C-module M is said to be C-Euclidean if it abides by the following equivalent of (1.17): (1.19) For any function f : D p → M , there exists unique x ∈ M such that f (d) = f (0) + xd for any d ∈ D p . 1.3. C-Microcubes A C-microlinear space M shall be chosen arbitrarily ones and for all. Given (p1 , . . . , pn ) ∈ Πn , a pure n-C-microcube of type (p1 , . . . , pn ) on M is a function from D p1 × · · · × D pn to M. We denote by Tp1 ,...,pn M the totality of pure n-Cmicrocubes of type (p1 , . . . , pn ) on M. We denote by Tn M the set-theoretic union of Tp1 ,...,pn M for all (p1 , . . . , pn ) ∈ Πn . In particular, T1 M is usually denoted by TM, and their elements are called pure C-vectors tangent to M. Given γ ∈ Tp1 ,...,pn M and e ∈ D pi , γei denotes the mapping (di , . . . , dn−1 ) ∈ D p1 × · · · × D pi−1 × D pi+1 × · · · × D pn | → γ (d1 , . . . , di−1 , e, di+1 , . . . , dn−1 ), which is surely a pure (n − 1)-C-microcube of type (p1 , . . . , pi−1 , pi+1 , . . . , pn ). An n-C-microcube on M is a mapping from D(1, . . . , k)n to M. We denote by n T M the totality of n-C-microcubes on M. In particular, T1 M is usually denoted by TM and their elements are called C-vectors tangent to M. Given x ∈ M, we p denote the sets {t ∈ Tp M | t(0) = x} and {t ∈ T M | t (0) = x} by Tx M and Tx M, respectively. We have shown (Nishimura, 1998b, Section 4) that Tx M is an p R-C-module and that its p-component can naturally be identified with Tx M. We have noted there also that the R-C-module Tx M is C-Euclidean. Given (p1 , . . . , pn ) ∈ Πn , the canonical injection of D p1 × · · · × D pn into D(1, . . . , k)n and the canonical projection of D(1, . . . , k)n onto D p1 × · · · × D pn are denoted by ιp1 ,...,pn and πp1 ,...,pn , respectively. The totality of γ¯ ∈ Tn M with γ¯ ◦ ιp1 ,...,pn ◦ πp1 ,...,pn = γ¯ can and shall hereafter be identified with Tp1 ,...,pn M. 1.4. C-Vector Bundles A mapping ζ : E → M of C-microlinear spaces is called a C-vector bundle providing that E x = ζ −1 (x) is a C-Euclidean R-C-module for any x ∈ M. We call M the base space of ζ and E x the fiber over x. The totality of mappings λ : M → E

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with ζ ◦ λ = id M (id M denotes the identity transformation of M) is denoted by Sec ζ . The totality of ¯t ∈ TE with ζ ◦ ¯t = 0 (the zero C-vector to M at ζ ◦ ¯t (0)) is to be put down as a C-vector bundle over E and is to be denoted by V (E). The tangent bundle τ M : M D(1,...,k) → M is a C-vector bundle, where τ M assigns, to each t ∈ M D(1,...,k) , t (0) ∈ M. If N is a C-Euclidean R-C-module, which is a C-microlinear space, then the trivial bundle M × N → M is a Cvector bundle. Various algebraic constructions in linear C-algebra can be carried over to C-vector bundles. If ζ : E → M and η : F → M are C-vector bundles over the same base space M, then their Whitney sum ζ ⊕ η and the natural projection πL (ζ,η) : L (ζ, η) → M are C-vector bundles, where L (ζ, η) denotes the settheoretic union of Hom(ζx , ηx ) for all x ∈ M. 2. BRAIDED EXTERIOR DIFFERENTIAL CALCULUS Given γ ∈ Tp1 ,...,pn M and a ∈ Rq , pure n-C-microcubes γi˙a and ai˙γ of type (p1 , . . . , pi − q, . . . , pn ) on M (1 ≤ i ≤ n) are defined to be (2.1) (γi˙a) (d1 , . . . , dn ) = γ (d1 , . . . , ad i , . . . , dn ) (2.2) (ai˙γ ) (d1 , . . . , dn ) = γ (d1 , . . . , di a, . . . , dn ) for any (d1 , . . . , dn ) ∈ D p1 × · · · × D pi −q × · · · × D pn . Let Symn be the symmetric group of the set {1, . . . , n}. Given γ ∈ Tp1 ,...,pn M and σ ∈ Symn , a pure n-C-microcube 6σ (γ ) of type (pσ −1 (1) , . . . , pσ −1 (n) ) on M is defined as follows: (2.3) 6σ (γ )(d1 , . . . , dn ) = γ (dσ (1) , . . . , dσ (n) ) for any (d1 , . . . , dn ) ∈ D pσ −1 (1) × · · · × D pσ −1 (n) . A (differential) n-C-preform on M is a mapping θ from Tn M to R abiding by the following conditions: 1 (2.4) θ(γi˙a) = θ(ai+1 ˙ γ ) (1 ≤ i ≤ n − 1) while θ (γn˙ a) = θ (γ ) a for any a ∈ R n and any γ ∈ T M; (2.5) If γ is a pure n-C-microsquare of type (p1 , . . . , pn ) on M, then θ (γ ) = −δ pi ,pi+1 θ(6(i,i+1) (γ )) (1 ≤ i ≤ n − 1), where (i, i + 1) is the transposition of i and i + 1.

A differential n-C-preform θ on M is called C-braided if it abides by the following condition: (2.6) θ(γi˙a) = θ(ai+1 ˙ γ ) (1 ≤ i ≤ n − 1) while θ (γn˙ a) = θ (γ ) a for any p ∈ Π, a ∈ Rp and any γ ∈ Tn M. (M) the totality of differential n-C-preforms on We denote by4n (M) and 4 ∼n M and that of C-braided differential n-C-preforms on M, respectively.

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Given γ¯ ∈ Tn M and a ∈ R, n-C-microcubes γi˙a and ai˙γ¯ on M (1 ≤ i ≤ n) are defined as in (2.1) and (2.2), respectively. Given γ¯ ∈ Tn M and σ ∈ Symn , an n-C-microcube 6σ (γ¯ ) on M is defined as in (2.3). A (differential) n-C-form on M is a mapping θ¯ from Tn M to R subject to the following conditions: ¯ i+1 ¯ γ¯i˙a) = θ(a (2.7) θ( ˙ γ¯ ) (1 ≤ i ≤ n − 1) while θ¯ (γ¯n˙ a) = θ¯ (γ¯ ) a for any a ∈ R and any γ¯ ∈ Tn M; (2.8) If γ¯ is a pure n-C-micosquare of type ( p1 , . . . , pn ) on M, then θ¯ (γ¯ ) = ¯ (i,i+1) (γ¯ )) (1 ≤ i ≤ n − 1). −δ pi ,pi+1 θ(6 ˜ n (M) the totality of differential n-C-forms on M. We denote by 4 Proposition 2.1. ˜ n (M). and 4

There is a natural bijective correspondence between 4 (M) ∼n

Proof: By the same token as in Nishimura (1999, Proposition 1.2).

¤

Therefore we can loosely and will often identify C-braided differential n-C(M) and preforms and differential n-C-forms, so that we will loosely denote 4 ∼n ˜ n (M) by the same symbol 4n (M). 4 A marked pure n-C-microcube of type (p1 , . . . , Pn ) on M is a pair (γ , e) of a pure n-C-microcube γ of type (p1 , . . . , pn ) on M and e = (e1 , . . . , en ) ∈ D p1 × ˜ p1 ,...,pn M the totality of marked pure n-C-microcubes of · · · × D pn . We denote by T ˜ n M the set-theoretic union of T ˜ p1 ,...,pn M type (p1 , . . . , pn ) on M. We denote by T n for all (p1 , . . . , pn ) ∈ Π . Given (p1 , . . . , pn ) ∈ Πn , (γ , e) ∈ T˜ p1 ,...,pn M and θ ∈ 4n (M) with e = (e1 , . . . , en ), ϕθ (γ , e) ∈ R is defined as follows: (2.9) ϕθ (γ , e) = θ(γ ) e1 , . . . , en Given e = (e1 , . . . , en ) ∈ D p1 × · · · × D pn and a ∈ Rq , elements ai˙e and and ei˙a of D p1 × · · · × D pi +q × · · · × D pn (1 ≤ i ≤ n) are defined to be (2.10) ai˙e = (e1 , . . . , aei , . . . , en ) (2.11) ei˙a = (e1 , . . . , ei a, . . . , en ) Given e = (e1 , . . . , en ) ∈ D p1 × · · · × D pn and σ ∈ Symn , 6σ (e) ∈ D pσ −1 (1) × · · · × D pσ −1 (n) is defined to be P (2.12) σ (e) = (eσ −1 (1) , . . . , eσ −1 (n) ) Given θ ∈ 4n (M), it is easy to see that the function ϕθ · T˜ n (M) → R satisfies ˜ p1 ,...,pn M and a ∈ R1 . the following properties with (γ , e ) ∈ T (2.13) ϕθ (γi˙a, e ) = ϕθ (ai˙γ , e ) = aϕθ (γ , e )

(1 ≤ i ≤ n)

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(2.14) ϕθ (γ , e i˙a) = ϕθ (γ , ai˙e ) = aϕθ (γ , e ) (1 ≤ i ≤ n) (2.15) ϕθ (γ , e ) = −δ pi ,pi+1 δ qi ,qi+1 ϕθ (6(i,i+1) (γ ), 6(i,i+1) ( e ))

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Now we have the following converse. ˜ n M to R abides by conditions (2.13)– Proposition 2.2. If a function ϕ from T (2.15), then there exists unique θ ∈ 4n (M ) such that ϕ = ϕθ . Proof: By the same token as in Lavendhomme (1996, Section 4.2, Proposition 2). ¤

Given θ ∈ 4n (M ), we define ψθ to be the function from T˜ n+1 (M ) to R such ˜ p1 ,...,pn+1 M with e = (e1 , . . . , en+1 ), that for any (γ , e ) ∈ T n+1 (2.16) ψθ (γ , e ) = 6i=1 (−1)i αi (θ (γ0i ) − θ (γeii ))e1 . . . eˆ i . . . en+1 pi ,p j pk ,pi where αi = (Πn+1 )(Πi−1 ). j=i+1 δ k=1 δ

Proposition 2.3. (2.13)–(2.15).

The above function ψθ : T˜ n+1 M → R satisfies conditions

Proof: By the same token as in Lavendhomme (1996, Section 4.2, Proposition 3). ¤

We denote by dθ the element of 4n+1 (M ) such that ϕdθ = ψθ . Its existence and uniqueness is guaranteed by Propositions 2.2 and 2.3. It is called the exterior C-derivative of θ. Now we have a family {d : 4n (M ) → 4n+1 (M )}n∈N of mappings, for which we have the following: Proposition 2.4.

d ◦ d = 0.

Proof: By the same token as in Lavendhomme (1996, Section 4.2, Proposition 1). ¤

Proposition 2.5. If a differential n-C-preform θ on M is C-braided, then so is dθ . Proof: By the same token as in Nishimura (1999, Proposition 2.5 and Lemmas 2.6 and 2.7; 2000a, Lemmas 2.3 and 2.4). ¤ Therefore the family {d : 4n (M ) → 4n+1 (M )}n∈N of mappings naturally gives rise to a family {d : 4n (M ) → 4n+1 (M )}n∈N of mappings.

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If ϕ : M → N a map of C-microlinear space and η : F → N is a C-vector bundle, then the notion of a (C-braided) differential n-C-preform on M and that of differential n-C-form on M discussed earlier can be generalized easily to that of a (C-braided) differential n-C-preform on M with values in η relative to ϕ and that of a differential n-form on M with values in η relative to ϕ, as in Lavendhomme (1996, ϕ ϕ ϕ ∼n n Section 5.3.1). We denote by 4n (M → N ; η),4 ∼ (M → N ; η), and 4 (M → N ; η) the totality of differential n-C-prefroms on M with values in η relative to ϕ, that of C-braided differential n-C-preforms on M with values in η relative to ϕ, and that of differential n-C-forms on M with values in η relative to ϕ, respectively. ϕ n (M → N ; η) A direct generalization of Proposition 2.1 enables us to identity 4 ∼ ϕ ϕ ∼n and 4 (M → N ; η), which we will often denote by 4n (M → N ; η). If N = M and ϕ ϕ is the identity map id M of M, then 4n (M → N ; η) is denoted also by 4n (M; η). If η is furthermore a trivial bundle M × R → M, then 4n (M; η) degenerates into 4n (M).

3. BRAIDED CONNECTIONS Let ζ : E → M be a C-vector bundle. A C-connection on ζ is a mapping ∇ : M D(1,...,k) × M E → E D(1,...,k) such that for any (t,v) ∈ M D(1,...,k) × M E, any a ∈ R and any d ∈ D(1, . . . , k) we have that (3.1) (3.2) (3.3) (3.4)

∇(t, v)(0) = v ∇(t a, v)(d) = ∇(t, v)(ad) ∇(t, va)(d) = (∇(t, v)(d))a The mapping u ∈ E t(0) | → ∇(t, u)(d) ∈ E t(d) , denoted by p∇(t,d) or p(t,d) , is p p bijective and preservs grades (i.e., p(t,d) (E t(0) ) = E t(d) for any p ∈ Π). Its ∇ inverse is denoted by q(t,d) = q(t,d) : E t,(d) → E t(0) . We call p(t,d) the parallel transport from t(0) to t(d) along t, while q(t,d) is called the parallel transport from t(d) to t(0) along t.

If the C-vector bundle ζ : E → M is a trivial bundle M × N → M, and if ∇(t, (t(0), x))(d) = (t(d), x) for any t ∈ M D(1,...,k) any x ∈ N and any d ∈ D(1, . . . , k), then the C-connection ∇ is called trivial. ¯ ¯t ∈ E D(1,...,k) to be Given ¯t ∈ E D(1,...,k) , we define ω( (3.5) ω( ¯ ¯t = ¯t − ∇(ζ ◦ ¯t, ¯t(0)) Since ω( ¯ ¯t) ∈ V (E), there exist unique ωp (¯t) ∈ E ζ ◦¯t(0) (p ∈ Π) such that (3.6) ω( ¯ ¯t)(d) = ¯t(0) + 6p∈Π ωp (¯t)dp for any d ∈ D(1, . . . , k). We define ω(¯t) to be 6p∈Π ωp (¯t).

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Proposition 3.1. Given v ∈ E and x ∈ M with x = ζ (V ), the mapping ¯t ∈ (E D(1,...,k) )V | → ω(¯t) ∈ E satisfies (2.6), so that ω is a differential 1-C-form on E with values in ζ relative to ζ . Proof: By (3.2) ω¯ satisfies (2.6), so that for any a ∈ R and any d ∈ D(1, . . . , k), (3.7) ω( ¯ ¯ta)(d) = ω( ¯ ¯t)(ad) = ω( ¯ ¯t)(6p∈Π 6q+r=p aq dr ) = ¯t(0) + 6p∈Π ωp (¯t)(6q+r=p aq dr ) = ¯t(0) + 6r∈Π (6p∈Π ωp (¯t)ap−r )dr Therefore (3.8) ω(¯ta) = 6r∈Π (6p∈Π ωp (¯t)ap−r ) = 6p,q∈Π ωp (¯t)aq = ω(¯t)a, as was claimed.

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We say that ω is the C-connection C-form of ∇. Proposition 3.2.

For any d ∈ D(1, . . . , k) and any ¯t ∈ E D(1,...,k) we have

(3.9) q(ζ ◦¯t,d) (¯t(d)) = ¯t(0) + 6p∈Π ωp (¯t)dp Proof: Consider the mapping (d, d0 ) ∈ D(1, . . . , k, 1, . . . , k) | → p(ζ ◦t,d) (¯t(0) + 6p∈Π ωp (¯t)d0p ) ∈ E, which coincides with ∇(ζ ◦ ¯t, ¯t (0)) on the first D(1, . . . , k) and with ω( ¯ ¯t) on the second D(1, . . . , k). Therefore the mapping d ∈ D(1, . . . , k) | → p(ξ ◦t,d) (¯t(0) + 6p∈Π ωp (¯t)dp ) ∈ E coincides with ¯t, which implies the desired proposition.

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Now suppose that we are given a mapping ϕ : M → N of C-microlinear spaces and a C-vector bundle η : F → N endowed with a C-connection ∇¯ , which shall be fixed throughout the rest of this section. Braided exterior differential calculus discussed in the previous section (particularly Propositions 2.2 and 2.3) can be generalized easily to braided covariant exterior differential calculus.

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Proposition 3.3. Given a differential n-C-form θ on M with values in η relative to ϕ, there exists a unique differential (n + 1)-C- form on M with values in η relative to ϕ, to be denoted by d∇¯ θ , such that for any (p1 , . . . , pn+1 ) ∈ Πn+1 and any (γ , e) ∈ T˜ p1 ,...,pn+1 M with e = (e1 , . . . , en+1 ), we have (3.10) d∇¯ θ(γ )e1 . . . en+1

¡ ¡ ¢ ¡ ¡ ¢¢¢ ¯ n+1 = 6i=1 (−1)i αi θ γ0i − q∇(ϕ◦γi ,ei ) θ γeii e1 . . . ei . . . en+1 ,

where γi is the tangent C-vector to M assigning γ (0, . . . , 0, d, 0, . . . , 0) (d is pi ,p j pk ,pi )(n i−1 ). positioned at the ith slot) to each d ∈ D pi and αi = (n n+1 j=i+1 δ k=1 δ Proof: By a direct generalization of Proposition 2.3.

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We call d∇¯ θ the covariant exterior C-derivative of θ . 4. INDUCED BRAIDED CONNECTIONS I Let us define some induced C-connections. Let ζ : E → M and η : F → M be C-vector bundles over the same base space M with C-connection ∇ and ∇ 0 bestowed upon them. First we define an induced C-connection ∇ ⊕ ∇ 0 on the Whitney sum ζ ⊕ η as follows: (4.1) (∇ ⊕ ∇ 0 )(t, v ζ ⊕ v η )(d) = ∇(t, v ζ )(d) ⊕ ∇ 0 (t, v η )(d) for any t ∈ M D(1,...,k) , any v ζ ∈ E t(0) , any v η ∈ Ft(0) an any d ∈ D(1, . . . , k). Proposition 4.1. For any ¯tζ ∈ E D(1,...,k) and any ¯tη ∈ F D(1,...,k) with ζ D(1,...,k) (¯tζ ) = η D(1,...,k) (¯tη ), we have (4.2) ωζ ⊕η (¯tζ ⊕ ¯tη ) = ωζ (¯tζ ) ⊕ ωη (¯tη ), where ωζ ⊕η , ωζ , and ωη denote the C-connection C-forms of ∇ ⊕ ∇ 0 , ∇ and, ∇ 0 , respectively. Proof: Let t = ζ D(1,...,k) (¯tζ ) = η D(1,...,k) (¯tη ). For any d ∈ D(1, . . . , k), we have, by Proposition 3.2, that 0 ¯ ¯ (4.3) q∇⊕∇ (t,d) (tζ (d) ⊕ tη (d))

= (¯tζ (0) + 6p∈Π ωζ,p (¯tζ )d p ) ⊕ (¯tη (0) +

P p∈Π

ωη,p (¯tζ )dp )

= (¯tζ (0) ⊕ ¯tη (0)) + 6p∈Π (ωζ,p (¯tζ ) ⊕ ωη,p (¯tη ))dp Therefore the desired proposition obtains by Proposition 3.2 again. Corollary 4.2. For any µ ∈ Sec ζ and any ν ∈ Sec η, we have

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(4.4) d∇⊕∇ 0 (µ ⊕ ν) = d∇ µ ⊕ d∇ 0 ν b on πL (ζ,η) as follows: We now difine an induced C-connection ∇ b vˆ )(d)(v) = p ∇ (ˆv (q ∇ (v))) (4.5) ∇(t, (t,d) (t,d) for any t ∈ M D(1,...,k) , any d ∈ D(1, . . . , k), any vˆ ∈ L (ζ, η)t(0) and any v ∈ E t(0) . Proposition 4.3. For any ˆt ∈ L (ζ, η) D(1,...,k) and any ¯t ∈ E D(1,...,k) with (πL (ζ,η) ) D(1,...,k) (ˆt) = ζ D(1,...,k) (¯t), we have (4.6) ωη (ˆt(¯t)) = 6p,q,r∈Π δ p,r δ q,p δ p,q (ωˆ p (ˆt))q (¯t(0)r ) + ˆt(0)(ωζ (¯t)), b and ˆt(¯t) denotes the mapping d ∈ where ωˆ denote the C-connection form of ∇, D(1, . . . , k) | → ˆt(d)(¯t(d)). In particular, if the braiding 9 happens to be symmetric, then we have (4.7) ωη (ˆt(¯t)) = 6p,r∈Π δ p,r ωˆ p (ˆt)(¯t(0)r ) + ˆt(0)(ωζ (¯t)). Proof: Let t = (πL (ζ,η) ) D(1,...,k) (ˆt) = ζ D(1,...,k) (¯t). For any d ∈ D(1, . . . , k), we have, by Proposition 3.2, that 0 (4.8) q∇(t,d) (ˆt(d)(¯t(d)))

b = q∇(t,d) (ˆt(d))(q∇(t,d) (¯t(d)))

= (ˆt(0) + 6p∈Π ωˆ p (ˆt)dp )(¯t(0) + 6p∈Π ωζ,p (¯t)dp ) = (ˆt(0) + 6p,q∈Π δ q,p dp (ωˆ p (ˆt))q )(¯t(0) + 6p∈Π ωζ,p (¯t)dp ) = ˆt(0)(¯t(0)) + 6p,q∈Π δ q,p dp (ωˆ p (ˆt))q (¯t(0)) + 6p∈Πˆt(0)(ωζ,p (¯t))dp = ˆt(0)(¯t(0)) + 6p,q∈Π δ q,p dp (ωˆ p (ˆt))q (6r∈Π¯t(0)r ) + 6p∈Πˆt(0)(ωζ,p (¯t))dp = ˆt(0)(¯t(0)) + 6p,q,r∈Π δ q,p δ p,q+r (ωˆ p (ˆt))q (¯t(0)r )dp + 6p∈Πˆt(0)(ωζ,p (¯t))dp = ˆt(0)(¯t(0)) + 6p∈Π {6q,r∈Π δ p,r δ q,p δ p,q (ωˆ p (ˆt))q (¯t(0)r ) + ˆt(0)(ωζ,p (¯t))}dp Therefore the desired proposition obtains by Proposition 3.2 again. Corollary 4.4. For any µ ∈ Sec ζ and any ι ∈ Sec πL (ζ,η) , we have p

(4.9) d∇ 0 (ι(µ)) = 6p,q,r,∈Π δ p,r δ q,p δ p,q (d∇b ι)q (µr ) + ι(d∇ µ) In particular, if the braiding 9 is symmetric, then we have p

(4.10) d∇ 0 (ι(µ)) = 6p,r,∈Π δ p,r (d∇b ι)(µr ) + ι(d∇ µ)

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If η is the trivial bundle M × R → M and the C-connection ∇ 0 is trivial, b is usually denoted by ∇ ∗ . If ζ = η and ∇ = ∇ 0 , then the then the C-connection ∇ b ˜ C-connection ∇ is usually denoted by ∇. 5. CURVATURE Let ζ : E → M be a C-vector bundle endowed with a C-connection ∇, which shall be fixed throughout this section. The principal objective of this section is to introduce a sort of curvature abiding by the so-called second Bianchi identity. First let us introduce a preliminary version of curavature somewhat disobedient to the second Bianchi identity, from which our desired curvature naturally follows. The ζ C-connection C-form ω is surely an element of 41 (E → M; ζ ), and its covariant ζ exterior C-derivative d∇ ω ∈ 42 (E → M; ζ ) is called the curvature C-form of the first kind and denoted by Ä, for which we have Proposition 5.1. For any γ¯ ∈ E D(p)×D(q) and any (d1 , d2 ) ∈ D(p) × D(q) with γ = ζ ◦ γ¯ , t1 = γ (·, 0), t2 = γ (d1 , ·), t3 = γ (0, ·) and t4 = γ (·, d2 ), we have (5.1) (δ p,q )−1 Ä(γ¯ )d1 d2 = q(t1 ,d1 ) ◦ q(t2 ,d2 ) (γ¯ (d1 , d2 )) − q(t3 ,d2 ) ◦ q(t4 ,d1 ) (γ¯ (d1 , d2 )) Proof: By the very definition of covariant exterior C-differentiation, we have (5.2) (δ p,q )−1 Ä(γ¯ )d1 d2 = ω(γ¯ (·, 0))d1 + q(t1 ,d1 ) (ω(γ¯ (d1 , ·)))d2 − q(t3 ,d2 ) (ω(γ¯ (·, d2 )))d1 − ω(γ¯ (0, ·))d2 By Proposition 3.2 we have (5.3) ω(γ¯ (·, 0))d1 = q(t1 ,d1 ) (γ¯ (d1 , 0)) − γ¯ (0, 0) (5.4) q(t1 ,d1 ) (ω(γ¯ (d1 , ·)))d2 = q(t1 ,d1 ) {q(t2 ,d2 ) (γ¯ (d1 , d2 )) − γ¯ (d1 , 0)} = q(t1 ,d1 ) ◦ q(t2 ,d2 ) (γ¯ (d1 , d2 )) − q(t1 ,d1 ) (γ¯ (d1 , 0)) (5.5) q(t3 ,d2 ) (ω(γ¯ (·, d2 )))d1 = q(t3 ,d2 ) {q(t4 ,d1 ) (γ¯ (d1 , d2 )) − γ¯ (0, d2 )} = q(t3 ,d2 ) ◦ q(t4 ,d1 ) (γ¯ (d1 , d2 )) − q(t3 ,d2 ) (γ¯ (0, d2 )) (5.6) ω(γ¯ (0, ·))d2 = q(t3 ,d2 ) (γ¯ (0, d2 )) − γ¯ (0, 0) Therefore the desired conclusion follows.

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Now we introduce another curvature C-form, to be called the curvature C˜ as follows: form of the second kind and to be denoted by Ä, ˜ γ¯ ) = Ä(h(γ¯ )) for any C-microsquare γ¯ on E, where h(γ¯ ) denotes the (5.7) Ä( horizontal component of γ¯ (cf. Moerdijk and Reyes, 1991, Chap.V. Section 6) in the sense that for any (d1 , d2 ) ∈ D(1, . . . , k)2 , (5.8) h(γ¯ )(d1 , d2 ) = p(γ (d1 ,·),d2 ) ◦ p(γ (·,0),d1 ) (γ¯ (0, 0)) with γ = ζ ◦ γ¯ . For the curvature C-form of the second kind, we have Proposition 5.2. (5.9) (δ

Using the same notation as in Proposition 5.1, we have

p,q −1

˜ γ¯ )d1 d2 = γ¯ (0, 0) − q(t3 ,d2 ) ◦ q(t4 ,d1 ) ◦ p(t2 ,d2 ) ◦ p(t1 ,d1 ) (γ¯ (0, 0)), ) Ä(

˜ γ¯ ) depends only on γ = ζ ◦ γ¯ and v = γ¯ (0, 0), which enables us to so that Ä( ˜ )(v) = Ä( ˜ γ¯ ). ˜ regard Ä as a function from T2 M to L (ζ ) in the sense that Ä(γ Proof: Simply put h(γ¯ ) in place of γ¯ in Proposition 5.1.

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˜ as a function from M D(1,...,k)2 to L (ζ ) in the canonical We now reckon Ä way, for which we have ˜ : M D(1,...,k)2 → L (ζ ) is a differential 2-C-form Proposition 5.3. The function Ä ˜ ∈ 42 (M; πL (ζ ) ). on M with values in πL (ζ ) . I.e., Ä 2

2

Proof: We define a function h : M D(1,...,k) × E → E D(1,...,k) as follows: M

2

(5.10) h(γ , v)(d1 , d2 ) = p(γ (d1 ,·),d2 ) ◦ p(γ (·,0),d1 ) (v) for any (γ , v) ∈ M D(1,...,k) × E and any (d1 , d2 ) ∈ D(1, . . . , k)2 .

M

Then it is easy to see that (5.11) h(γi˙ a, v) = h(γ , v)i˙ a for any a ∈ R(i = 1, 2). ˜ )(v) = Ä(b(γ , v)) and Ä satisfies (2.6), Ä ˜ also satisfies (2.6). Now we Since Ä(γ ˜ satisfies (2.5), use the same notation as in Proposition 5.1 and 5.2. To show that Ä we Let v 0 = v and define v 1 and v 2 in order as follows: (5.12) v 1 = q(t3 ,d2 ) ◦ q(t4 ,d1 ) ◦ p(t2 ,d2 ) ◦ p(t1 ,d1 ) (v 0 ) (5.13) v 2 = q(t1 ,d1 ) ◦ q(t2 ,d2 ) ◦ p(t4 ,d1 ) ◦ p(t3 ,d2 ) (v 1 ) On the one hand it follows directly from (5.12) and (5.13) that (5.14) v 2 = v 0 On the other hand we can calculate v 1 and v 2 in order by making use of Proposition 5.2:

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˜ )(v 0 )d1 d2 (5.15) v 1 = v 0 − (δ p,q )−1 Ä(γ ˜ (5.16) v 2 = v 1 − (δ q,p )−1 Ä(6(γ ))(v 1 )d2 d1 p,q −1 ˜ = v 0 − (δ ) Ä(γ )(v 0 )d1 d2 ˜ ˜ )(v 0 )d1 d2 )d2 d1 [(5.15)] − (δ q,p )−1 Ä(6(γ ))(v 0 − (δ p,q )−1 Ä(γ ˜ )(v 0 )d1 d2 − (δ q,p )−1 Ä(6(γ ˜ ))(v 0 )d2 d1 = v 0 − (δ p,q )−1 Ä(γ q,p −1 ˜ ˜ )(v 0 )d2 d1 − (δ ) Ä(6(γ ))(v 0 )d2 d1 = v 0 − Ä(γ [since d1 d2 = δ p,q d2 d1 ] It follows from (5.14) and (5.16) that ˜ ˜ )(v 0 ) + Ä(6(γ ))(v 0 ) = 0, (5.17) δ q,p Ä(γ ˜ satisfies (2.5). which means that Ä

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The above proof is a prototype of the proof of Theorem 5.5 to come. Now we give a braided, cubical version of Kock’s simplicial and combinatorial Bianchi identity (Kock, 1996, Theorem 2). Theorem 5.4. Let γ ∈ M D(p)×D(q)×D(r) . Let (d1 , d2 , d3 ) ∈ D(p) × D(q) × D(r). We denote points γ (0, 0, 0), γ (d1 , 0, 0), γ (0, d2 , 0), γ (0, 0, d3 ), γ (d1 , d2 , 0), γ (d1 , 0, d3 ), γ (0, d2 , d3 ), and γ (d1 , d2 , d3 ) by O, A, B, C, D, E, F, and G, respectively. These eight points are depicted figuratively as the eight vertices of a cube: C

E

F

G B

O A

D

Then we have (5.18)

PAO ◦ PDA ◦ PGD ◦ RGFBD ◦ RGECF ◦ RGDAE ◦ PDG ◦ PAD ◦ POA ◦ ROCEA ◦ ROBFC ◦ ROADB = idO ,

where (5.19) for any adjacent vertices X, Y of the cube, PXY denotes the parallel transport from X to Y along the line connecting X and Y (e.g., POA and PAO denote p(γ ( · ,0),d1 ) and q(γ ( · ,0),d1 ) , respectively),

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(5.20)

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for any four vertices X, Y, Z, W of the cube rounding one of the six facial squares of the cube, RXYZW denotes PWX ◦ PZW ◦ PYZ ◦ PXY (e.g., ROADB denotes q(γ (0,·, 0),d2 ) ◦ q(γ ( ·, d2 ,O),d1 ) ◦ P(γ (d1 ,·,O),d2 ) ◦ P(γ ( · ,O,O),d1 )) and idO is the identity transfotmation of E O .

Proof: Write over (5.18) exclusively in terms of PXY ’s, and write off all consective PXY ◦ PYX ’s ¤ The above theorem gives rise to the following form of the second Bianchi identity in our braided context. Theorem 5.5. We have ˜ = 0, (5.22) d∇˜ Ä where d∇˜ is the covariant exterior C-differentiation with respect to the induced ˜ ∈ 42 (M; πL (ξ ) ), as was explained in C-connection ∇˜ on πL (ξ ) , and recall that Ä Proposition 5.3. Proof: The proof is carried out by the same method as in Proposition 5.3. Let γ , d1 , d2 , d3 , O, A, B, C, D, E, F, and G be as in Theorem 5.4. Given v 0 ∈ E γ (0, 0, 0) , we define v i ∈ E γ (0, 0, 0) (i = 1, 2, 3, 4, 5, 6) in order as follows: = = = = = (5.27) v 5 = = = = (5.23) (5.24) (5.25) (5.26)

v1 v2 v3 v4

= (5.28) v 6 = = =

ROADB (v 0 ) ROBFC (v 1 ) ROCEA (v 2 ) PAO ◦ PDA ◦ PGD ◦ RGDAE ◦ PDG ◦ PAD ◦ POA (v 3 ) PAO ◦ RAEGD ◦ POA (v 3 ) PAO ◦ PDA ◦ PGD ◦ RGECF ◦ PDG ◦ PAD ◦ POA (v 4 ) PAO ◦ RAEGD ◦ PEA ◦ RECFG ◦ PAE ◦ RADGE ◦ POA (v 4 ) PAO ◦ PEA ◦ REGDA ◦ RECFG ◦ READG ◦ PAE ◦ POA (v 4 ) ROCEA ◦ PCO ◦ PEC ◦ REGDA ◦ RECFG ◦ READG ◦ PCE ◦ POC ◦ ROAEC (v 4 ) ROCEA ◦ PCO ◦ PEC ◦ REGDA ◦ PCE ◦ RCFGE ◦ PEC ◦ READG ◦ PCE ◦ POC ◦ ROAEC (v 4 ) PAO ◦ PDA ◦ PGD ◦ RGFBD ◦ PDG ◦ PAD ◦ POA (v 5 ) PAO ◦ PDA ◦ RDGFB ◦ PAD ◦ POA (v 5 ) ROBDA ◦ PBO ◦ RBDGF ◦ POB ◦ ROADB (v 5 )

Now we calculate v i (i = 1, . . . , 6) in order. It follows directly from Proposition 5.2 that ˜ (·, ·, 0))(v 0 )d1 d2 (5.29) v 1 = v 0 − (δ p,q )−1 Ä(γ

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The calculations of v 2 and v 3 are similar, so we present details of the former calculation but simply register the result of the latter calculation, safely leaving its details to the reader. ˜ (0, ·, ·)) (v 1 )d2 d3 [Proposition 5.2] (5.30) v 2 = v 1 − (δ q,r )−1 Ä(γ p,q −1 ˜ ˜ (0, ·, ·)) = v 0 − (δ ) Ä(γ (·, ·, 0)) (v 0 )d1 d2 − (δ q,r )−1 Ä(γ ˜ (·, ·, 0)) (v 0 )d1 d2 )d2 d3 [(5.29)] × (v 0 − δ p,q Ä(γ ˜ (·, ·, 0)) (v 0 )d1 d2 − (δ q,r )−1 Ä(γ ˜ (0, ·, ·)) (v 0 )d2 d3 = v 0 − (δ p,q )−1 Ä(γ ˜ (·, ·, 0)) (v 0 )d1 d2 − (δ q,r )−1 Ä(γ ˜ (0, ·, ·)) (v 0 )d2 d3 (5.31) v 3 = v 0 − (δ p,q )−1 Ä(γ p,r −1 ˜ + (δ ) Ä(γ (·, 0, ·)) (v 0 )d1 d3 The three calculations of v 4 , v 5 , and v 6 are similar, so we present their details only in case of the first, leaving details of the other two calculations to the reader. ˜ (·, ·, 0)) (v 0 )d1 d2 (5.32) v 4 = PAO ◦ RAEGD ◦ POA (v 0 − (δ p,q )−1 Ä(γ q,r −1 ˜ − (δ ) Ä(γ (0, ·, ·)) (v 0 )d2 d3 ˜ (·, 0, ·)) (v 0 )d1 d3 ) [(5.31)] + (δ p,r )−1 Ä(γ ˜ (·, ·, 0)) (v 0 ))d1 d2 = PAO ◦ RAEGD (POA (v 0 ) − (δ p,q )−1 POA (Ä(γ q,r −1 ˜ (0, ·, ·)) (v 0 ))d2 d3 − (δ ) POA (Ä(γ p,r −1 ˜ (·, 0, ·)) (v 0 ))d1 d3 ) + (δ ) POA (Ä(γ ˜ (·, ·, 0)) (v 0 ))d1 d2 = PAO (POA (v 0 ) − (δ p,q, )−1 POA (Ä(γ ˜ (0, ·, ·)) (v 0 ))d2 d3 − (δ q,r )−1 POA (Ä(γ p,r −1 ˜ (·, 0, ·)) (v 0 ))d1 d3 + (δ ) POA (Ä(γ q,r −1 ˜ + (δ ) Ä(γ (d1 , ·, ·)) (POA (v 0 ) ˜ (·, ·, 0)) (v 0 ))d1 d2 − (δ q,p )−1 POA (Ä(γ ˜ (0, ·, ·)) (v 0 ))d2 d3 − (δ r,q )−1 POA (Ä(γ r,p −1 ˜ (·, 0, ·)) (v 0 )))d1 d3 )d2 d3 + (δ ) POA (Ä(γ [Propositions 5.2 and 5.3] ˜ (·, ·, 0)) (v 0 ))d1 d2 = v 0 − (δ q,p )−1 Ä(γ ˜ (0, ·, ·)) (v 0 )d2 d3 + (δ r,p )−1 Ä(γ ˜ (·, 0, ·)) (v 0 )d1 d3 − (δ r,q )−1 Ä(γ q,r −1 ˜ (d1 , ·, ·)) (POA (v 0 )))d2 d3 + (δ ) PAO (Ä(γ ˜ (·, ·, 0)) (v 0 )d1 d2 − (δ q,r )−1 Ä(γ ˜ (0, ·, ·)) (v 0 )d2 d3 (5.33) v 5 = v 0 − (δ p,q )−1 Ä(γ ˜ (·, 0, ·)) (v 0 ))d1 d3 + (δ p,r )−1 Ä(γ q,r −1 ˜ (d1 , ·, ·)) (POA (v 0 )))d2 d3 + (δ ) PAO (Ä(γ p,q −1 ˜ (·, ·, d3 )) (POC (v 0 ))d1 d2 + (δ ) PCO (Ä(γ

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˜ (·, ·, 0)) (v 0 ))d1 d2 (5.34) v 6 = v 0 − (δ p,q )−1 Ä(γ ˜ (0, ·, ·)) (v 0 )d2 d3 − (δ q,r )−1 Ä(γ p,r −1 ˜ + (δ ) Ä(γ (·, 0, ·)) (v 0 )d1 d3 ˜ (d1 , ·, ·)) (POA (v 0 )))d2 d3 + (δ q,r )−1 PAO (Ä(γ p,q −1 ˜ (·, ·, d3 )) (POC (v 0 )))d1 d2 + (δ ) PCO (Ä(γ ˜ (·, d2 , ·)) (POB (v 0 )))d1 d3 − (δ p,r )−1 PBO (Ä(γ It should be the case by Theorem 5.4 that v 6 = v 0 . Therefore ˜ (·, ·, 0)) (v 0 )d1 d2 + (δ q,r )−1 Ä(γ ˜ (0, ·, ·)) (v 0 )d2 d3 (5.35) (δ p,q )−1 Ä(γ p,r −1 ˜ − (δ ) Ä(γ (·, 0, ·)) (v 0 )d1 d3 ˜ (d1 , ·, ·)) (POA (v 0 )))d2 d3 − (δ q,r )−1 PAO (Ä(γ p,q −1 ˜ (·, ·, d3 ,)) (POC (v 0 )))d1 d2 − (δ ) PCO (Ä(γ p,r −1 ˜ (·, d2 , ·)) (POB (v 0 ))))d1 d3 = 0 + (δ ) PBO (Ä(γ By multiplying δ p,q δ p,r δ q,r upon (5.35), we have ˜ (·, ·, 0)) (v 0 )d1 d2 + δ p,q δ p,r Ä(γ ˜ (0, ·, ·)) (v 0 )d2 d3 (5.36) δ p,r δ q,r Ä(γ p,q q,r ˜ − δ δ Ä(γ (·, 0, ·)) (v 0 )d1 d3 ˜ (d1 , ·, ·)) (POA (v 0 )))d2 d3 − δ p,q δ p,r PAO (Ä(γ p,r q,r ˜ − δ δ PCO (Ä(γ (·, ·, d3 )) (POC (v 0 )))d1 d2 ˜ (·, d2 , ·)) (POB (v 0 ))))d1 d3 = 0 + δ p,q δ q,r PBO (Ä(γ Since v 0 ∈ E γ (0,0,0) was chosen arbitrarily, the proof is complete.

¤

6. INDUCED BRAIDED CONNECTIONS II We calculate the curvature of the second kind of the induced C-connections dealt with in Section 4. Let ζ : E → M and η : F → M be C-vector bundles over the same base space M embellished with C-connections ∇ and ∇ 0 , as in that section. Proposition 6.1.

2

For any γ ∈ M D(1,...,k) we have

˜ ζ (γ ) ⊕ Ä ˜ η (γ ), ˜ ζ ⊕η (γ ) = Ä (6.1) Ä ˜ ζ , and Ä ˜ η denote the curvature forms of the second kind of C˜ ζ ⊕η , Ä where Ä 0 connections ∇ ⊕ ∇ , ∇, and ∇ 0 , respectively. Proof: Let v ⊕ v 0 ∈ (E ⊕ F)γ (0,0) . We assume that γ ∈ M D(p)×D(q) . Let d1 ∈ D(p) and d2 ∈ D(q). Let t1 = γ (·, 0), t2 = γ (d1 , ·), t3 = γ (0, ·) and t4 = γ (·, d2 ).

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By Proposition 5.2, we have ˜ ζ ⊕η (γ )(v ⊕ v 0 )d1 d2 (6.2) (δ q,p )−1 Ä 0

0

0

0

∇⊕∇ ∇⊕∇ ∇⊕∇ 0 = (v ⊕ v 0 ) − q∇⊕∇ (t3 ,d2 ) ◦ q(t4 ,d1 ) ◦ p(t2 ,d2 ) ◦ p(t1 ,d1 ) (v ⊕ v ) ª © = v − q∇(t3 ,d2 ) ◦ q∇(t4 ,d1 ) ◦ p∇(t2 ,d2 ) ◦ p∇(t1 ,d1 ) (v) © ª 0 0 0 0 ⊕ v 0 − q∇(t3 ,d2 ) ◦ q∇(t4 ,d1 ) ◦ p∇(t2 ,d2 ) ◦ p∇(t1 ,d1 ) (v 0 )

˜ ζ (γ )(v)d1 d2 ⊕ (δ q,p )−1 Ä ˜ η (γ 0 )(v 0 )d1 d2 = (δ q,p )−1 Ä ˜ ζ (γ )(v) ⊕ Ä ˜ η (γ 0 )(v 0 )}d1 d2 = (δ q,p )−1 {Ä ¤

Therefore the desired proposition obtains.

Let γ ∈ M D(p)×D(q) and vˆ ∈ L (ζ, η)γ (0,0) . Then we have

Proposition 6.2.

ˆ˜ )(ˆv ) = −ˆv ◦ Ä ˜ ζ (γ ) + 6r,s∈Π δ 0,p+q δ p+q,r+s (Ä ˜ η (γ ))r ◦ vˆ s (6.3) Ä(γ ˆ˜ Ä ˜ η denote the curvature forms of the second kind of C˜ ζ , and Ä where Ä, ˆ ∇, and ∇ 0 , respectively. In particular, if the braiding 9 happens connections ∇, to be symmetric, then we have ˆ˜ )(ˆv ) = −ˆv ◦ Ä ˜ η (γ ) ◦ vˆ s ˜ ζ (γ ) + 6s∈Π δ p+q,s Ä (6.4) Ä(γ Proof: Let d1 ∈ D(p) and d2 ∈ D(q). By Proposition 5.2 we have ˆ˜ )(ˆv )d d (6.5) (δ q,p )−1 Ä(γ 1 2 b

b

b

b

= vˆ − q∇(t3 ,d2 ) ◦ q∇(t4 ,d1 ) ◦ p∇(t2 ,d2 ) ◦ p∇(t1 ,d1 ) (ˆv ) = vˆ − q∇(t3 ,d2 ) ◦ q∇(t4 ,d1 ) ◦ p∇(t2 ,d2 ) ◦ p∇(t1 ,d1 ) ◦ vˆ ◦ q∇(t1 ,d1 ) ◦ q∇(t2 ,d2 ) ◦ p∇(t4 ,d1 ) ◦ p∇(t3 ,d2 ) © ª ª ˜ η (γ )d1 d2 ◦ vˆ ◦ id Eγ (0,0) + (δ q,p )−1 Ä ˜ ζ (γ)d1 d2 = vˆ − id Fγ (0,0) − (δ q,p )−1 Ä ª © ˜ η (γ ))r ◦ {6s∈Π vˆ s }◦ = vˆ − id Fγ (0,0) − (δ q,p )−1 60∈Π δ r,p+q d1 d2 (Ä ˜ ζ (γ )d1 d2 } {id Eγ (0,0) + (δ q,p )−1 Ä ˜ ζ (γ )d1 d2 = −(δ q,p )−1 vˆ ◦ Ä ˜ η (γ ))r ◦ vˆ s d1 d2 + (δ q,p )−1 6r,s∈Π δ 0,p+q δ p+q,r+s (Ä ˜ ζ (γ ) + 6r,s∈Π δ r,p+q δ p+q,r+s (Ä ˜ η (γ ))r ◦ vˆ s }d1 d2 = (δ q,p )−1 {−ˆv ◦ Ä Therefore the desired proposition obtains.

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