- Groupoids
- Moufang Loops and Malcev Algebras
- Loopoids
- Octonion x-product and xy-product
- S7 and NonAssociativity

Groupoids

Unfortunately, in the mathematics literature there are two very different definitions of "groupoid". One, that I do NOT like to use, is definition number 1 in Eric Weisstein's Encyclopedia of Mathematics: "... an algebraic structure on a set with a binary operator. The only restriction is closure ...".

The definition that I like to use is given by Alain Connes in his book Noncommutative Geometry:

"... a groupoid G ... a small categrory with inverses ... consists of

- a set G,
- a distinguished subset G0 of G,
- two maps r, s : G -> G0 and
- a law of composition o : G2 = {(g1,g2) in G x G ; s(g1) = r(g2)} -> G

such that

- (1) s(g1 o g2) = s(g2) , r(g1 o g2) = r(g1) for all (g1,g2) in G2
- (2) s(x) = r(x) = x for all x in G0
- (3) g o s(g) = g , r(g) o g = g for all g in G
- (4) (g1 o g2) o g3 = g1 o (g2 o g3)
- (5) Each g has a two-sided inverse g-1 , with g g-1 = r(g) , g-1 g = s(g)

The maps r,s are called the range and source maps. ... Here are a few important examples of groupoids. ...

- Given a group P one takes G = P , G0 = {e}, and the law of composition is the group law. ...
- A Lie group G is, in a trivial way, a groupoid with G0 = {e}. ...
- The groupoid G = M x M where M is a compact manifold, r and s are the two projections G -> M = G0 = {(x,x) ; x in M} and the composition is (x,y) o (y,z) = (x,z) for all x,y,z in M. ...".

Alan Weinstein says in his paper Groupoids: Unifying Internal and External Symmetry:

"... a groupoid with base B is a set G with mappings a and b from G onto B with a partially defined binary operation (g,h) --> gh satisfying ...

- gh is defined only when b(g) = a(h) ...
- if either of the products (g h) k or g (h k) is defined, then so is the other, and they are equal [associativity] ...
- For each g in G, there are left and right identity elements Lg and Rg such that Lg g = g = g Rg ...
- Each g in G has an inverse g-1 for which g g-1 = Lg and g-1 g = Rg ...

... We may think of each element g of G as an arrow pointing from b(g) to a(g) in B; arrows are multiplied by placing them head to tail ... a(b) and Lg ( or b(g) and Rg) determine one another, thus producting a bijective mapping I from the base B to the subset G0 of G consisting of the identity elements. ...

... a groupoid is just a ... small ... category in which every morphism has an inverse ...

... An orbit of the groupoid G over B is an equivalence class for the relation x ~G y if and only if there is a groupoid element g with a(g) = x and b(g) = y ...

... The isotropy group of x [in] B consists of those g in G with a(g) = x = b(g) ...".

The groupoid picture is something like

G | | r| |s | | \/ \/ G0

John Baez, on his web page at week74 and related pages, describes groupoids and applies them to category theory.

Another way that a groupoid usefully generalizes a group is that by taking G0 to be a larger subset of G than the singleton identity {e}, you can use a multiplication law (such as the Octonion x-product and xy-product) that varies over the underlying base space G0.

However, as John Baez pointed out to me in September 2003 e-mail messages,

when you try to use a Groupoid to describe the 7-sphere S7 of unit Octonions, you are prevented by groupoid axiom (4), which requires associativity of a groupoid composition law, and therefore prevents use of S7's natural nonassociative Octonion multiplications.

Moufang Loops and Malcev Algebras

If you want to use nonassociativity, you can consider Moufang Loops.

Jaak Lohmus, Eugene Paal, and Leo Sorgsepp, in their book Nonassociative Algebras in Phyiscs (Hadronic Press 1994), say:

"...Moufang loops and Mal'tsev[another transliteration for "Malcev"]algebras ... are ... natural (minimal) generalizations of Lie groups and Lie algebras, respectively. Because of the uniqueness of octonions ... octonionic Moufang loop and the corresponding simple (non-Lie) Mat'tsev algebra are of exceptional importance ...A

Moufang loopis a set G with a binary operation (multiplication) ... so that ...

- 1) in the equation g h = k, the knowledge of any two of g,h,k [in] G specifies the third one uniquely;
- 2) there is a distinguished unit or identity element e of G with the property e g = g e = g, [for all] g [in] G;
- 3) the Moufang identity holds: (g h)(k g) = g((h k)g), [for all] g,h,k [in] G.
A set with such a binary operation that only axioms 1/ and 2) are satisfied is called a loop. ... roughly speaking,

loops are the "nonassociative groups". ... The most remarkable property of Moufang loops is their diassociativity: the subloop generated by any two elements in a Moufang loop is associative (group). Hence, for any g,h in a Moufang loop G ... (h g)g = h g^2 , g(g h) = g^2 h , (g h)g = g(h g) ... thanks to [which]... the Moufang identity can be written ... (g h)(k g) = g(h k) g.... one can define the notion of the inverse element of g [in] G. The unique solution of the equation g x = e (x g = e) is called the right (left) inverse element of g [in] G and denoted as g-1R (g-1L). From ... diassociativity ... g-1R = g-1L = g-1 , g-1(g h) = (h g)g-1 = h , (g-1)-1 = g , (g h)-1 = h-1 g-1 ; [for all] g,h [in] G.

... The Moufang loop G is said to be analytic if G is a real analytic manifold so that both the Moufang lop operation G x G --> G : (g,h) --> g h and the inversion map G --> G : g --> g-1 are analytic ... denote the dimension of G by r ... introduce the antisymmetric quantities

c^i_jk := a^i_jk - a^i_kj ... i,j,k = 1, ... , r ,

called the structure constants of G. ... The tangent algebra

of G ...[with]... productG[X,Y]^i := c^i_jk X^j Y^k = -[Y,X]^i ; i,j,k = 1, 2, ... , r .

...[with the]... Mal'tsev identity ...

[[X,Y],[Z,X]] + [[[X,Y],Z],X] + [[[Y,Z],X],X] + [[[Z,X],X],Y] = 0

... is ... the ... Mal'tsev algebra. ... the Jacobi identity ...[may fail] in

...G... every Lie algebra is a Mal'tsev algebra ... In a Mal'tsev algebra

the Yamaguti triple product ... may be defined as ...G[x,y,z] := [x,[y,z]] - [y,[x,z]] + [[x,y],z]

... K. Yamaguti proved ... the possibility of embedding a Mal'tsev algebra into a Lie algebra ... every Mal'tsev algebra can be realized as a subspace of some Lie algebra so that the Mal'tsev operation is a projection of the Lie algebra operation to this subspace. ...

.... Every Mal'tsev algebra is also a ... Lie triple system ... Lie triple systems ... serve as tangent algebras for symmetric spaces ...

According to the paper Analytic Loops and Gauge Fields by E. K. Loginov at hep-th/0109206:

"...simple nonassociative Moufang loops ...[are]... analytically isomorphic to one of the spacesS7, S3 x R4, or S7 x R7 ...... Suppose A is a complex (real) Cayley-Dickson algebra, M is its commutator Malcev algebra, and L(A) is the enveloping Lie algebra of regular representation of A.

It is obvious that the algebra L(A) is generated by the operators Rx and Lx, where x [is in] A. We select in L(A) the subspaces R(A), L(A), S(A), P(A) and D(A) generated by the operators

- Rx,
- Lx,
- Sx = Rx + 2Lx,
- Px = Lx +2Rx and
- Dx,y = [Tx, Ty]+T[x,y],
where Tx = Rx -Lx, accordingly. ...

[Rx, Sy] = R[x,y] ...[and]... [Lx, Py] = L[y,x].

... the algebra L(A) is decomposed into the direct sums

- L(A) = D(A) + S(A) + R(A),
- L(A) = D(A) + P(A) + L(A),
of the Lie subalgebras D(A) + S(A), D(A) + P(A) and the vector spaces R(A), L(A) ... In addition, the map x --> Sx from M into S(A) is a linear representation of the algebra M, which transforms the space R(A) into M-module that is isomorphic ... to the regular Malcev M-module. ...

... the direct summands ... are orthogonal with respect to the scalar product tr{XY} on L(A) ...

[ From S7 to Spin(8) ] ...

Let A be the complex Cayley-Dickson algebra [of Octonions]. Then A supposes the base 1, e1, ..., e7 such thatei ej = -delta_ij + c_ijk ek,

where the structural constants c_ijk are completely antisymmetric and different from 0 only if

c_123 = c_145 = c_167 = c_246 = c_257 = c_374 = c_365 = 1.

It is easy to see that in such base the operators

- Rei = e[i0] - (1/2) c_ijk e[jk] ,
- Lei = e[i0] + (1/2) c_ijk e[jk] ,
where e[uv] are skew-symmetric matrices 8 x 8 with the elements (euv)ab = delta_ma delta_nb - delta_mb delta_na . Using the identity

c_ijk c_mnk = delta_im delta_jn - delta_in delta_jm + c_ijmn ,

where the completely antisymmetric tensor c_ijkl is defined by the equality

(ei,ej,ek) = 2 c_ijkl el ,

we have D_ei,ej = 8 e[ij] + 2 c_ijmn e[mn] . ...

... Its enveloping Lie algebra L(A) (in fixed base) consists of real skew-symmetric 8 x 8 matrices. Therefore we can connect every element F = F_mn e[mn] of L(A) with the 2-form F = F_mn dx^m /\ dx^n . ... the factors of F are such that

epsilon F0i + (1/2) c_ijk F_ jk = 0 , if F [is in] S(A) + D(A) ...[or]... P(A) + D(A) epsilon F0i = c_ijk F_ jk , if F [is in] R(A) ...[or]... L(A)where there is no summing over j,k in [the second equations], c_ijk 6= 0, and

- epsilon = 0, if F [is in] D(A),
- epsilon = 1, if F [is in] S(A) + D(A), F [is not in] D(A) or F [is in] R(A),
- epsilon = -1, if F [is in] P(A) + D(A), F [is not in] D(A) or F [is in] L(A).
For epsilon = -1 these are precisely the (anty-self-dual) equations of Corrigan et al. ... In addition, R(A) and L(A) are not Lie algebras. Therefore the [second] equations, in contrast to [the first equationa], are not Yang-Mills equations. Nevertheless, there are a solution of the [second] equations, which generalizes the known (anty-)instanton solution of Belavin et al. ...".

Jaak Lohmus, Eugene Paal, and Leo Sorgsepp, say, further, in their book Nonassociative Algebras in Phyiscs (Hadronic Press 1994):

"... The generators for the 8-dimensional irreps of the algebra D4 [of Spin(8) and SO(8)] in terms of the regbirep matrices of octonions ... can be written ... [f]or vector representation 8v ... as follows:

- Gij = [ Li + Ri , Lj + Rj ]; i,j = 1,2, ... , 7;
- Gij ej = ei , Gij ei = -ej , Gij ek = 0 ; i=/= j =/= k ...
... The generators for halfspinor irreps may be represented as follows:

- {Lij} : Li0 x = Li x = ei x , Lij x = Lj Li x = ej (ei x) ,
- {Rij} : Ri0 x = Ri x = x ei , Rij x = Rj Ri x = (x ei) ej .
Low-dimensional IRs of SO(8) ... Branching rules ... IR ... SO(7) ... G2 ... SU(3)/Z3 1 ... 1 ... 1 ... 1 8v ... 1,7 ... 1,7 ... 8 8s1 ... 8 ... 1,7 ... 8 8s2 ... 8 ... 1,7 ... 8 28 ... 7,21 ... 7,7,14 ... 8,10,10bar 35 ... 1,7,27 ... 1,7,27 ... 8,27 35' ... 35 ... " ... " 35'' ... 35 ... " ... " 56 ... 21,35 ... 1,7,7,14,27 ... 1,8,10,10bar,27 56' ... 8,48 ... " ... " 56'' ... 8,48 ... " ... "The subgroup SO(7) does not alter the unit element e0 ( 8v --> 1 + 7 ), automorphisms form a still narrower subgroup G2 [of] SO(7). In the transition to the subgroup SO(7) from generators of SO(8) the generators L0i (R0i) separate, forming the Mal'tsev algebra M7 with the multiplication

[L0i * Lj0] = Li0 * Lj0 - Lj0 * Li0 , where the *-multiplication is defined by ... Lx Ly + [Lx,Ry] = Lx * Ly

... Ry Rx + [Rx,Ly] = Rx * Ry ... the Ri matrices of the octonion regbirep ... will be used for the expression of Dirac gamma-matrices. Ri-matrices ... are anticommuting antisymmetric matrices forming the 64-dimensional Clifford algebra C6 with 6 generic elements ... The Ri matrices corresponding to different multiplication tables are different. The transition between two tables may be carried out by Li-matrices. ... The group SO(8), the invariance group of the octonion norm, is the largest group which can be related to the octonionic space with one dimension. The exceptional groups F4, E6, E7, E8 are groups of various geometries of the octonionic plane. ...".

Loopoids

If you want to use a nonassociative
multiplication law that varies over an underlying manifold, thus
using the 480 octonion multiplication
products, you need to use a combination of Loop structure for
nonassociativity and Groupoid
structure for variation over an underlying space - something that
John Baez calls a **Loopoid** - which is defined as a groupoid
except that its composition law is alternative (not necessarily
associative).

The S7 loopoid structure that I like is an xy-product groupoid with G = S7 and G0 = G = S7 and with a composition law that takes into account the two points x and y in G0 = S7 of the two arrows for w and z respectively, being defined as w o z = w oxy z, thus producing a diagram like

S7 = G | | r| |s | | \/ \/ S7 = G0 / \ \G2/

where the G2 loop is the autormorphism group of the octonions implicit in the xy-product composition law

The S7 and S7 and G2 combine to form the Lie Group Spin(8). Then you can look at the irreducible representations of Spin(8) of dimensions

1 8 28 56 35

and their duals

35* 56* 28* 8* 1*

and combine them to form the 2^8 = 256-dimensional Clifford Algebra Cl(8) with graded structure

1 8 28 56 (35+35*) 56* 28* 8* 1*

Then you can construct ANY Clifford algebra of the form Cl(8N) , and therefore, if you include subalgebras, ANY Clifford algebra, by tensor product and periodicity factoring:

Cl(8N) = Cl(8) x ...(N-fold tensor product)... x Cl(8)

The net result of all this is, in my view, that

As Geoffrey Dixon says in hep-th/9411063:

"... let e_a, a = 0, 1, ... , 7, be a basis for O ... the Octonion algebra ... and ...

- W0 = {+/-e_a} [16 elements],
- W1 = ((+/-e_a +/-e_b)/sqrt(2) : a,b distinct [112 elements],
- W2 = ((+/-e_a +/-e_b +/-e_c +/-e_d)/2 : a,b,c,d distinct, e_a(e_b(e_c e_d)) = +/-1} [224 elements],
- W3 = {(SUM^7_a=0 +/_e_a)/sqrt(8) : odd number of +'s}, a,b,c,d [in] {0, ... , 7} [128 elements] ...
The 240 elements of W0 u W2 are the nearest neighbors (first shell) to the origin of an E8 lattice (so are the 240 elements of W1 u W3).

Define E8^h = Gh[W0 u W2], h = 1, ... , 7, where Gh is the O(8) reflection taking e_0 <--> e_h.

These 7 sets are nearest neighbor points for 7 different E8 lattices ...[and they]... close under multiplication ...

Actually, E8^0 = W0 u W2 is not closed under the multiplication we started with ... If, however we modify the product, using the X-product ... then ... E8^0 is closed ...

there are 56 X-product variants of E8^7 (and by virtue of index cycling, all the E8^h ... [thus producing 8x56 = 448 variants]) ...

we have an exact sequence

D4 in ZS3 --> E8^0 in S7 --> Z5 in S4... where ... D4 ... is the [24-element] inner shell of a ... D4 lattice ... [and] ... Z5 is ...the 5-dimensional cubic lattice [with 32-element inner shell] ...".

I think that it might be interesting to write in loopoid formalism the reflexive/recursive equivalence between

- the 7 octonion imaginaries e_1, ... , e_7 and
- the 7 E8 lattices E8^1, ... , E8^7

7 Imaginary Octonions i j k E I J K / \ | | \ / 7 (Associative Triangles+1) x 16 (4 signs) = 112 / \ / / \ \ 7 E8 Lattices | 224 + 16 = 240 Units \ | / x2 (RL) \ \ / / || \ / 480 Octonion Products 7 Coassociative Squares x 16 (4 signs) = 112

showing the expanding-nesting-Russian-doll structure of

point --> lattice -(pick a point)-> point --> lattice --> ...

the Gray Code / Hilbert Curve nesting-Russian-doll structure

... --> Cl(8x8x8) -(pick a 64)-> Cl(8x8) -(pick an 8)-> Cl(8)

of 3-dimensional Euclidean spacelike slices of 4-dimensional physical spacetime that comes from the Clifford Tensor Product Structure of the periodicity of real Clifford algebras that is used in my physics model. That structure is described on a web page by William Gilbert:

"... We exhibit a direct generalization of Hilbert's curve that fills a cube. The first three iterates of this curve are shown.In constructing one iterate from the previous one, note that the direction of the curve determines the orientation of the smaller cubes inside the larger one.

The initial stage of this three dimensional curve can be considered as coming from the 3-bit reflected Gray code which traverses the 3-digit binary strings in such a way that each string differs from its predecessor in a single position by the addition or subtraction of 1. The kth iterate could be considered a a generalized Gray code on the Cartesian product set {0,1,2,...,2^k-1}^3.

The n-bit reflected binary Gray code will describe a path on the edges of an n-dimensional cube that can be used as the initial stage of a Hilbert curve that will fill an n-dimensional cube. ...".

For a 4-dimensional physical Minkowski spacetime neighborhood, add a 1-dimensional time to generalize the 3-dimensional spatial structure described above to get a structure with 16 vertices per 4-dimensional cell

... --> Cl(16x16x16) -(pick a 256)-> Cl(16x16) -(pick a 16)-> Cl(16) -> -(pick an 8)-> Cl(8) with the fundmental Cl(16) at each 4-dim vertex corresponding by Cl(16) = Cl(8) x Cl(8) to 2 copies of Cl(8), aligned along the time axis so that one is in the future of the other.

For a 6-dimensional physical Conformal Spin(2,4) linear spacetime neighborhood, add 2-dimensional time and 1-dimensional space to generalize the 3-dimensional spatial structure described above to get a structure with 64 vertices per 6-dimensional cell

... --> Cl(64x64x64) -(pick a 4,096)-> Cl(64x64) -(pick a 64)-> Cl(64) -> -(pick an 8)-> Cl(8) with the fundmental Cl(64) at each 6-dim vertex corresponding by Cl(64) = Cl(8) x Cl(8) x Cl(8) x Cl(8) x Cl(8) x Cl(8) x Cl(8) x Cl(8) to a configuration of 8 copies of Cl(8).

As Danny Ross Lunsford has noted, you can use the Traceless 4x4x2 - 2 = 30-real-dimensional Lie Algebra of SL(4,C) to construct a Lie Algebra for a 6-dimensional linear spacetime. By imposing the additional condition A* J = - J A where

0 0 0 1 0 0 1 0 J = 0 -1 0 0 -1 0 0 0

you get the 30/2 = 15-real-dimensional Lie Algebra of symplectic (not-quaternionic-symplectic) Sp(4,C) = Spin(3,3) = Spin(4,2) that corresponds to the Clifford Algebra Cl(3,3) = Cl(4,2) = M(8,R) = Cl(0,6), which, as A. Sudbery says in his paper Division algebras, (pseudo-) orthogonal groups and spinors,J.Phys A, 17 (1984), 939-955 (see also math.RA/0203010), leads to the usual 4x1 Dirac spinors and gamma-matrices.

Because I agree with John Baez that fermions should be quaternionic, I prefer (instead of taking SL(4,C) as fundamental) to use the Conformal Lie Algebra of Spin(2,4) with Clifford algebra Cl(2,4) = M(4,Q) = Cl(1,5) = Cl(5,1) = Cl(6,0) which has 2x4 = 8-real-dimensional half-spinors.

Octonion x-product and xy-product

The **octonion**** x-product**
mentioned above is defined by Martin Cederwall and Christian
Preitschopf in hep-th/9309030
and described by Geoffrey Dixon in hep-th/9410202
where he says:

"... Let A,B,X [be in the octonions] O, with X a unit octonion (X X* = 1) ... DefineA ox B = (A X)(X* B) = (A (B X)) X* = X ((X* A) B) the X-product of A and B. ... for fixed X, the algebra OX (O endowed with the X-product) is isomorphic to O itself. Modulo sign change each X gives rise to a distinct copy of O, so the orbit of copies of O arising from any given starting copy is S7 / Z2 = RP7 , the manifold obtained from the 7-sphere by identifying opposite points.

Moreover, composition of X-products is yet another X-product. ... there are 480 different [octonion] multiplication tables ... these fall into two groups of 240 ... related by the octonionic X-product. ...".

The **octonion**** xy-product**
is defined by Geoffrey Dixon in hep-th/9503053
where he says:

"... it is possible to modify the octonion product in such a way that e_0 is not the identity of the result. In particular, defineA oxy B = (AX)(Y*B) = A ox ((XY*) ox B) = (A oy (XY*)) oy B , where as usual we assume that both X,Y [are in] S7 ((the X-product is obtained by setting X+Y). Let OXY be O with this modified product.

The question is, is this still isomorphic [to] O itself? ... [yes] ... OXY [is isomorphic to] O. ...

The octonion X-product changes the octonion multiplication table, but does not change the role of the identity. The XY-product is very similar, but shifts the identity as well. ...

... the X-product ... generate[s] all the 480 renumberings of the e_a, a = 1, ..., 7 [imaginary octonion basis elements], which leave e_0 = 1 fixed as the identity.

There are 7680 renumberings of the entire collection, e_a, a = 0, ... , 7, and the XY product plays exactly the same role in this context. ...

... in the X-product case the 480 renumberings arose from a pair of octonion E8 lattices.

The XY-product renumberings are related in a similar fashion to the pair of octonion /\16 lattices ...".

Note that

the /\16 Barnes-Wall lattice (unimodular but odd) has 2x240 + 16x240 = 480 + 3,840 = 4,320 units, related to the 16-dimensional Reed-Muller code. The 7,680 octonion XY-product orbits are two sets of 3,840 units.

Further,

the variation of the octonion identity is equivalent to the action on the identity of the automorphism group of the octonions, G2.

S7 and NonAssociativity

R. M. Kane, in his book The Homology of Hopf Spaces (North-Holland 1988), says:

"... Stasheff's approach leads to the idea of an obstruction theory for loop spaces. The A_n forms are a series of intermediate stages between H-spaces (= A_2 spaces) and loop spaces (= A_infinity spaces). ... in the homotopy category of connected CW complexes we have the following identities:loop spaces = topological groups = associative H-spaces = A_infinity spaces ...

among the spheres, only S0, S1, S3, S7 possess A_2 forms ...[and so]... are H-spaces ... The topological groups S0,S1, and S3 are...A_infinity spaces...= loop space ...

[nonassociative octonionic] S7 ....[is a]... H-space ... (= A_2 space) ...

S7 is not ... a A_3 space ...[which is] a homotopy associative H-space...

S7_(3) is not homotopy associative ...[and]...

S7_(2) is not homotopy associative ... On the other hand,

S7_(p) is homotopy associative for p>= 5.

So the homotopy associativity obstruction is really mod 2 and mod 3. ...".

Note that, even though S7 is a Moufang loop, it is not a loop space in the sense used by Kane.

Kasper K. S. Andersen, Tilman Bauer, Jespar Grodal, and Erik Kjaer Pedersen, in math.AT/0306234, say:

"... Since the discovery of the Hilton-Roitberg 'criminal' in 1968 it has been clear that not every finite loop space is homotopy equivalent to a compact Lie group. The conjecture emerged however, that this should hold rationally ... In this paper we resolve this conjecture in the negative by exhibiting a concrete finite loop space of rank 66 whose rational cohomology does not agree with that of any compact Lie group. ...".

Since (in Kane's sense) loop space = topological group, their rank 66 loop space of dimension 1254 is an example of something that is a topological group but not a Lie group.

Theorem 56 of Leon Pontrjagin's book Topological Groups (Princeton 1939) says:

"Let G be a compact topological group satisfying the second axiom of countability. If G is locally connected ... and is of finite dimension, it is a Lie group.".

The 1254-dimensional rank 66 loop space is a "connected finite loop space", but it is only p-compact.

In a post to the sci.physics.research thread Re: nonassociative category and D-branes, John Baez says:

"... a simple example ... of A_infinity categories ... Take any topological space X. Take as objects the points of X, and take as morphisms f: x -> y ... parametrized paths ... we can compose them .. This composition is not associative!!! ...".

In a 1993 paper, Gregor Masbaum and Colin Roarke say:

"... A topological quantum field theory "TQFT" is a functor from a "cobordism category" to a category of modules which satisfies a number of properties ... Strictly speaking, a cobordism category C is not a category in the usual sense. This is because composition is not strictly associative nor are there strict identities ... This is because (X u_S Y) u_T Z is not strictly the same set as X u_S (Y u_T Z) where S [in] X,Y and T [in] Y,Z are sets (though of course they are in canonical bijection). Thus composition ... is only associative up to canonical diffeomorphism. ... There is the question of whether disjoint union can be defined in a coherent way in a cobordism category ... The category of manifolds and diffeomorphisms is certainly coherent and this might naively lead to the belief that a cobordism category is also coherent ...[ What is coherence? Noson S. Yanofsky in math.CT/0007033 says "... The history of coherence theory has its roots in homotopy theory. Saunders Mac Lane's foundational paper on coherence theory was an abstraction of earlier work by James Stasheff on H-spaces ..." and Peter Schauenburg in New York J. Math. 7 (2001) 257-265 says "... It is well-known that the tensor product of vector spaces is not associative on the nose, but it is so up to isomorphism. ... MacLane's coherence theorem ... says (... sloppily) that all diagrams one can build up from associativity isomorphisms commute. ...". ]... However this fails to work in detail for similar reasons to the reason why a general cobordism category has non-associative composition. ... Indeed the question of coherence is not really meaningful for a category with non-associative composition so it only really makes sense to ask whether our model categories can be made coherent. But coherence must be by an isomorphism in the category ... This is all very suggestive of an A_infinity structure ... We conjecture that there is a genuine obstruction to a cobordism category being strictly coherent ... and that there may well be some interesting mathematical structure which can be found by a careful investigation of the situation ...".

In hep-th/0102183, C. I. Lazariou says:

"... there are good reasons (coming from the analysis of the A-twisted topological string theory) to believe that A-type D-branes in a type II compactification give a nonassociative category (likely an A_infinity category), due to worldsheet instanton effects. ...".

Tony Smith's Home Page ......