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R.Wallace/BioSystems103 (2011) 18–2625

Appendix A.Mathematical Appendix

A.1.Basic ideas about groupoids

Following Weinstein(1996)closely,a groupoid,G,is defined by a base set A upon which some mapping–a morphism–can be defined.Note that not all possible pairs of states(a j,a k)in the base set A can be connected by such a morphism.Those that can define the groupoid element,a morphism g=(a j,a k)having the natural inverse g−1=(a k,a j).Given such a pairing,it is pos-sible to define‘natural’end-point maps˛(g)=a j,ˇ(g)=a k from the set of morphisms G into A,and a formally associative prod-uct in the groupoid g1g2provided˛(g1g2)=˛(g1),ˇ(g1g2)=ˇ(g2), andˇ(g1)=˛(g2).Then the product is defined,and associative (g1g2)g3=g1(g2g3).

In addition,there are natural left and right identity elements g, g such that g g=g=g g(Weinstein,1996).

An orbit of the groupoid G over A is an equivalence class for the relation a j∼Ga k if and only if there is a groupoid element g with ˛(g)=a j andˇ(g)=a k.Following Cannas Da Silva and Weinstein (1999),we note that a groupoid is called transitive if it has just one orbit.The transitive groupoids are the building blocks of groupoids in that there is a natural decomposition of the base space of a general groupoid into orbits.Over each orbit there is a transitive groupoid,and the disjoint union of these transitive groupoids is the original groupoid.Conversely,the disjoint union of groupoids is itself a groupoid.

The isotropy group of a∈X consists of those g in G with ˛(g)=a=ˇ(g).These groups prove fundamental to classifying groupoids.

If G is any groupoid over A,the map(␣,␤):G→A×A is a mor-phism from G to the pair groupoid of A.The image of(␣,␤)is the orbit equivalence relation∼G,and the functional kernel is the union of the isotropy groups.If f:X→Y is a function,then the kernel of f, ker(f)=[(x1,x2)∈X×X:f(x1)=f(x2)]defines an equivalence relation.

Groupoids may have additional structure.As Weinstein(1996) explains,a groupoid G is a topological groupoid over a base space X if G and X are topological spaces and␣,␤and multiplication are continuous maps.A criticism sometimes applied to groupoid the-ory is that their classification up to isomorphism is nothing other than the classification of equivalence relations via the orbit equiv-alence relation and groups via the isotropy groups.The imposition of a compatible topological structure produces a nontrivial interac-tion between the two structures.Below we will introduce a metric structure on manifolds of related information sources,producing such interaction.

In essence,a groupoid is a category in which all morphisms have an inverse,here defined in terms of connection to a base point by a meaningful path of an information source dual to a cognitive process.

As Weinstein(1996)points out,the morphism(␣,␤)suggests another way of looking at groupoids.A groupoid over A identifies not only which elements of A are equivalent to one another(iso-morphic),but it also parametizes the different ways(isomorphisms) in which two elements can be equivalent,i.e.,all possible information sources dual to some cognitive process.Given the informa-tion theoretic characterization of cognition presented above,this produces a full modular cognitive network in a highly natural manner.

Brown(1987)describes the fundamental structure as follows:

A groupoid should be thought of as a group with many objects,or

with many identities...A groupoid with one object is essentially just a group.So the notion of groupoid is an extension of that of groups.It gives an additional convenience,flexibility and range of applications...

Example1.A disjoint union[of groups]G=∪ G , ∈ ,is a groupoid:the product ab is defined if and only if a,b belong to the same G ,and ab is then just the product in the group G .

There is an identity1 for each ∈ .The maps␣,␤coincide and map G to , ∈ .

Example2.An equivalence relation R on[a set]X becomes a groupoid with␣,␤:R→X the two projections,and product (x,y)(y,z)=(x,z)whenever(x,y),(y,z)∈R.There is an identity, namely(x,x),for each x∈X...

Weinstein(1996)makes the following fundamental point:

Almost every interesting equivalence relation on a space B arises in a natural way as the orbit equivalence relation of some groupoid G over B.Instead of dealing directly with the orbit space B/G as an object in the category S map of sets and map-pings,one should consider instead the groupoid G itself as an object in the category G htp of groupoids and homotopy classes of morphisms.

The groupoid approach has become quite popular in the study of networks of coupled dynamical systems which can be defined by differential equation models(e.g.,Golubitsky and Stewart,2006).

A.2.11.2Global and local symmetry groupoids

Here we follow Weinstein(1996)fairly closely,using his exam-ple of afinite tiling.

Consider a tiling of the euclidean plane R2by identical2by 1rectangles,specified by the set X(one-dimensional)where the grout between tiles is X=H∪V,having H=R×Z and V=2Z×R, where R is the set of real numbers and Z the integers.Call each connected component of R2/C,that is,the complement of the two-dimensional real plane intersecting X,a tile.

Let be the group of those rigid motions of R2which leave X invariant,i.e.,the normal subgroup of translations by elements of the lattice =H∩V=2Z×Z(corresponding to corner points of the tiles),together with reflections through each of the points 1/2 =Z×1/2Z,and across the horizontal and vertical lines through those points.As noted by Weinstein(1996),much is lost in this coarse-graining,in particular the same symmetry group would arise if we replaced X entirely by the lattice of corner points. retains no information about the local structure of the tiled plane. In the case of a real tiling,restricted to thefinite set B=[0,2m]×[0, n]the symmetry group shrinks drastically:the subgroup leaving X∩B invariant contains just four elements even though a repetitive pattern is clearly visible.A two-stage groupoid approach recovers the lost structure.

We define the transformation groupoid of the action of on R2 to be the set

G( ,R2)={(x, ,y|x∈R2,y∈R2, ∈ ,x= y},

with the partially defined binary operation

(x, ,y)(y, ,z)=(x, ,z).

Here˛(x, ,y)=x,andˇ(x, ,y)=y,and the inverses are natural.

We can form the restriction of G to B(or any other subset of R2) by defining

G( ,R2)|B={g∈G( ,R2)|˛(g),ˇ(g)∈B}

(1)An orbit of the groupoid G over B is an equivalence class for the

relation

x∼Gy if and only if there is a groupoid element g with˛(g)=x andˇ(g)=y.