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Abstract

The theory of n-fold groupoids is contextualized by applications to solid composites. The concept of defects of misalignment between the various constituents of a multi-composite is introduced and a definition of a representative n-fold material groupoid is proposed. The hierarchical structure of the various ranks of participant groupoids is shown to be equivalent to the graph of a hypercube. Examples of possible applications are presented.

Keywords

Groupoids categories composites misalignment metamaterials group-weighted graphs

1. Introduction

Of the many mathematical concepts that have found applications in continuum mechanics, groupoids have stood as an exception until fairly recently, and are still not part of the standard repertory. Their existence seems to have been hardly detected by specialists in solid mechanics, even though the theory of continuous distributions of dislocations offers notable similarities with the concept of groupoid. The early works of Noll [1] and Wang [2], among others, come very close to it with the notion of a materially uniform body, without explicitly designating it as a transitive Lie groupoid, or at the very least as a G-structure. The first application of groupoids in the theory of inhomogeneity appears to be Epstein and de León [3], followed by their use in many articles and monographs [4,5].

The mathematical literature on the theory of groupoids is relatively rich and is the subject of various specialized treatises [6 –9]. This is not the case for n-fold groupoids, which are often only included in highly specialized treatises on n-fold categories [10]. As is the case with truly original and creative ideas in science, although their setting is usually very technical and demanding of considerable specialized background, the underlying notions can be elucidated and applied effectively without requiring an intimate knowledge of their rigorous apparatus, as long as it is treated with respect and recognition, and a fair amount of modesty. The close formal fit between double groupoids and binary composites [11] makes the exploration of further possibilities and generalizations pursued in this article appear worthwhile.

Section 2 is a rather elementary exposition of the hierarchical structure of a mixture in its relation with its constituent sub-mixtures, a discussion that results in the awareness of the close relation that exists between this structure and the graph of an n-dimensional hypercube. This essentially combinatoric result prepares the ground for the otherwise somewhat unexpected role that hypercubes play in n-fold groupoids. In Section 3, the general concept of groupoid is briefly reviewed and applied to single-constituent material bodies to define the material groupoid, whose arrows represent material isomorphisms derived from a given constitutive law. Section 4 introduces n-fold groupoids by means of a recursive definition. Accordingly, an n-fold groupoid is a set $Q$ endowed with n groupoid structures defined over n other $(n - 1)$ -fold groupoids. The elements of $Q$ are defined as objective skeletons of sorts that can be composed in n different ways that satisfy a compatibility condition. Finally, the longer Section 5 is devoted to the contextualization of n-groupoids within the application to composites and metamaterials.

2. Mixtures and hypercubes

2.1. The graph of a mixture

It is a truism that a mixture of n constituents is also a mixture of mixtures of lower orders $m = 1, \dots, n - 1$ . Thus, a quaternary mixture $(n = 4)$ can be conceived as a mixture containing single substances $(m = 1)$ , binary mixtures $(m = 2)$ , and ternary mixtures $(m = 3)$ . This simple observation can be readily quantified using elementary combinatorics. The number of m-ary mixtures that can be made from n different constituents is given by

C_{m}^{n} = \frac{n}{m} = \frac{n!}{m! (n - m)!} .

(1)

If, for the sake of elegance, we also count the mixture consisting of no components $(m = 0)$ , the total number of mixtures involved is obtained as

V_{n} = \sum_{m = 0}^{n} C_{m}^{n} = 2^{n} .

(2)

Setting $m = n - 1$ in equation (1), we observe that an n-ary mixture consists, if one wills, of n constituents, each of which is an $(n - 1)$ -ary mixture. By the same token, each of these mixed constituents is a mixture of $n - 1$ constituents, each of which is an $(n - 2)$ -ary mixture. Proceeding downwards until we reach $m = 0$ we obtain a whole hierarchy of mixtures and their interrelations as possible components of those in the immediately higher rank of the hierarchy.

To better visualize what is going on, Figure 1 shows the various hierarchies for the cases $n = 0, 1, 2, 3$ . An ingenious notation has been used to describe the various sub-mixtures. For a mixture with n basic simple constituents, the label of each sub-mixture consists of a sequence of n spaces, one for each basic constituent. Each space is then filled with either a 0 or a 1, according to whether the corresponding basic constituent is absent or present in the sub-mixture. The total mixture, therefore, consists of a sequence of n 1’s. We remark incidentally that this notation, when interpreted in the binary system, constitutes a systematic numbering of the various sub-mixtures. The arrows between sub-mixtures can be read as “is a constituent of,” as they point from one sub-mixture towards another one in the immediately higher rank. As expected from equation (1), for each n the number of mixtures in each rank abides by the entries of the corresponding row of the Pascal triangle. Thus, for $n = 3$ we have the pattern 1-3-3-1 as the rank in the hierarchy increases from 0 to 3.

Figure 1.

Hierarchy of constituent mixtures for $n = 0, 1, 2, 3$ , from left to right.

Each of the representations in Figure 1 is, technically speaking, a directed graph [12]. What happens for $n > 3$ ? The graph is expected to become correspondingly more encumbered, but one would be remiss not to notice that the representation looks like a point, a segment, a square, and a cube (in perspective), respectively, for $n = 0, 1, 2, 3$ .

2.2. The graph of a hypercube

The solid n-dimensional unit hypercube $Q_{n}$ (or n-cube for short) is defined as

Q_{n} = \underset{n times}{\underset{⏟}{J \times \dots \times J}} = J^{n} \subset ℝ^{n} .

(3)

where $J = [0, 1]$ is the oriented closed unit segment in ℝ. For each integer $0 \leq h < n$ , an h-face $F_{h}$ is a proper subset of $Q_{n}$ obtained by fixing $n - h$ coordinates of $ℝ^{n}$ to either 0 or 1. For a fixed h, the number of h-faces is obtained as

N_{h} = 2^{n - h} \frac{n}{h} = 2^{n - h} \frac{n!}{h! (n - h)!} .

(4)

A 0-face of $Q_{n}$ is called a vertex. The number of vertices is, therefore, $V = 2^{n}$ . The coordinates of every vertex consist of a sequence of zeros and ones. This obvious observation proves that there is a one-to-one correspondence between the vertices of $Q_{n}$ and the sub-mixture constituents of an n-ary mixture. We remark that the rank of a vertex can be defined as the number of ones in its coordinate representation.

A 1-face is called an edge. The number of edges of $Q_{n}$ is $E = 2^{n - 1} n$ . By their very definition, the edges of $Q_{n}$ are vectors (in the affine space of $ℝ^{n}$ ) joining two vertices that differ only in one of the coordinates. This coordinate runs from a value of 0 at the tail of the arrow to a value of 1 at its tip. In other words, each edge connects a vertex of rank k with a vertex of rank $k + 1$ , for $k = 0, \dots, n - 1$ .

The skeleton $S_{n}$ of $Q_{n}$ is defined as the collection of vertices and edges of $Q_{n}$ . The skeleton $S_{n}$ is clearly a directed graph. We have, therefore , shown that:

Proposition 2.1. The hierarchy of mixture constituents of an n-ary mixture is represented, for any given n, by the directed graph of the standard n-dimensional hypercube.

3. Groupoids

3.1. Groupoids in general

Broadly speaking, in their applications to physics, groups are the mathematical expression of symmetries, that is, transformations of a physical system into itself (automorphisms) that leave it essentially unchanged. Borrowing an example from Weinstein’s inspiring article [13], a tiling of the plane $ℝ^{2}$ with, say, identical square tiles enjoys all the symmetries afforded by all those rigid reflections, rotations and translations that are intuitively compatible with the preservation of the tiling. As soon, however, as the system is restricted by a finite boundary (such as the walls of a bathroom restrict the tiled floor) all (or, with luck, most) of these symmetries are lost. Nevertheless, our mind is not at ease with this loss. The individual tiles are still symmetric, and two separate tiles can be exchanged with each other without anyone being able to tell the difference. How can these local and distant symmetries be still captured by a well-defined mathematical entity?

The notion of groupoid made its first appearance almost a hundred years ago in an article aptly entitled “On a generalization of the concept of group” by Brandt [14]. This notion, later subsumed under category theory by Ehresmann, does the job of capturing the aforementioned local and distant symmetries of a system. In the case of continuum mechanics, the former entail the material symmetries of a given constitutive equation of a material point, while the latter describe what Noll [1] defines as material isomorphisms between two body points.

Definition 3.1. A groupoid $Z ⇉ B$ consists of a set $Z$ (the set of arrows) and a set $B$ called the base of the groupoid (the set of objects, or points). These sets are related by means of two surjective projection maps $α : Z \to B$ and $β : Z \to B$ called, respectively, the source map and the target map. A binary internal operation (product or composition) between two arrows $u, v \in Z$ is defined whenever the “tip-to-tail” condition

β (u) = α (v)

(5)

is satisfied. The result, denoted by $vu$ , is an arrow such that

α (vu) = α (u) β (vu) = β (v) .

(6)

This operation, whenever defined, is associative. For each object $X \in B$ a (unique) unit arrow $ϵ (X) \in Z$ exists such that

u ϵ (X) = u \forall u \in Z with α (u) = X

(7)

and

ϵ (X) u = u \forall u \in Z with β (u) = X .

(8)

Finally, for each $u \in Z$ there is an inverse arrow $u^{- 1} \in Z$ such that $α (u^{- 1}) = β (u)$ , $β (u^{- 1}) = α (u)$ , and

u^{- 1} u = ϵ (α (u)) u u^{- 1} = ϵ (β (u)) .

(9)

Remark 3.1. Pictorially, the projection maps α and β assign to each $u \in Z$ a tail (or source) $α (u) = X \in B$ and a tip (or target) $β (u) = Y \in B$ , respectively. For this reason, it is often convenient to think of u as an arrow joining the points X and $Y$ , although $Z$ and $B$ are mathematically different sets. This imagery is also useful to visualize equations (5)–(9).

Consider an object (point) $X \in B$ . It is not hard to show that the collection of all arrows $u \in Z$ such that $α (u) = β (u) = X$ is a group $G_{X}$ , known as the vertex group at X. This collection of “loop-shaped” arrows is never empty, since the unit $ϵ (X)$ is always well-defined. The vertex group at X is also known as the (local) symmetry group at X.

Given two different points $X, Y \in B$ , the collection $P_{XY}$ of all arrows u such that $α (u) = X$ and $β (u) = Y$ may or may not be empty. If it is not, let $u_{0}$ be such an arrow with source X and target $Y$ . Then, the whole collection is given by

P_{XY} = u_{0} G_{X} = G_{Y} u_{0} .

(10)

Moreover, the vertex groups $G_{X}$ and $G_{Y}$ are mutually conjugate, namely,

G_{Y} = u_{0} G_{X} u_{0}^{- 1} .

(11)

Definition 3.2. A groupoid $Z ⇉ B$ is said to be transitive if there is at least one arrow between every pair of objects.

In a transitive groupoid, all the vertex groups are mutually conjugate.

3.2. Material groupoids

A material body $B$ is endowed with a material response expressible in a given reference configuration by means of one or more constitutive equations. In the case of a simple elastic response, to which we confine our analysis for the sake of simplicity, the material response can be expressed in terms of the Cauchy stress tensor $t$ as a function of the deformation gradient $F$ and the point $X \in B$ , namely, $t = t (F, X)$ . If we consider two points $X_{1}, X_{2} \in B$ we may ask: are they made of the same material? The answer to this question can be obtained using the notion of material isomorphism.

Definition 3.3. Two points $X_{1}, X_{2} \in B$ are materially isomorphic (or made of the same material) if there is a linear map between their tangent spaces

P_{12} : T_{X_{1}} B \to T_{X_{2}} B

(12)

such that the equation

t (F, X_{2}) = t ({FP}_{12}, X_{1})

(13)

is satisfied identically for all $F \in GL (3, ℝ)$ .

Such a map is also called a material transplant by analogy with a surgical transplant, whereby a damaged piece of skin is replaced with a healthy counterpart harvested from another part of the body. This kind of transplant may involve not only a translation and a rotation but also a stretch required to produce a perfect graft that cannot be distinguished from the replaced original.

The material groupoid of a body $B$ with constitutive equation $t = t (F, X)$ is the groupoid $Z ⇉ B$ whose arrows are the material isomorphisms. We remark that since material isomorphism is an equivalence relation, the composition rule of tip-to-tail arrows is automatically enforced.

At each point $X \in B$ , the loop-shaped arrows are those material automorphisms $G : T_{X} B \to T_{X} B$ which, according to equation (13), satisfy

t (F, X) = t (FG, X) \forall F \in GL (3, ℝ) .

(14)

As expected, such arrows represent precisely the material symmetries at X. In other words, the vertex groups of the material groupoid are nothing but the local symmetry groups $G_{X}$ of the body at each point $X \in B$ .

Remark 3.2. Given a body $B$ , considered just as a continuum regardless of its material response, one can always define a groupoid $Π^{1} (B, B) ⇉ B$ whose arrows are all the possible transplants between all pairs of points. Thus, the collection of arrows from any point of $B$ to any other is the general linear group $GL (3, ℝ)$ or, equivalently, the collection of all non-singular $3 \times 3$ matrices. This groupoid is known as the 1-jet groupoid over $B$ . Clearly, every material groupoid is a subgroupoid of $Π^{1} (B, B) ⇉ B$ .

A body $B$ is uniform if, and only if, its material groupoid is transitive. Indeed, transitivity means that all its points are materially isomorphic (made of the same material). In particular, in a uniform body all the vertex groups are mutually conjugate.¹ In this case, it is convenient to think of an archetypal material element placed outside the body, as shown in Figure 2. Since all the vertex groups are mutually conjugate, it seems appropriate to say that the symmetry group $\bar{G}$ of the archetype can be considered as the archetypal symmetry group of the uniform body.² Upon implanting the archetype into each point $X \in B$ via a field of implant maps $P (X)$ , the transplant map $P_{12}$ between $X_{1}$ and $X_{2}$ is recovered as the composition

P_{12} = P (X_{2}) {(P (X_{1}))}^{- 1},

(15)

The symmetry group $G_{X}$ at $X \in B$ is obtained by conjugation as

G_{X} = P (X) \bar{G} {(P (X))}^{- 1} .

(16)

If the archetype’s constitutive equation is $\bar{W} (H)$ , the (point-dependent) constitutive equation of the uniform body is

W (F, X) = \bar{W} (FP (X)) .

(17)

Figure 2.

An archetype and its implants in a uniform body.

4. n-fold groupoids

4.1. General considerations

There is in principle no restriction on the nature of the set of objects (or base set) of a groupoid. We may consider, for example, a groupoid $Q$ whose base set is itself a groupoid $Z ⇉ B$ . In this case we may write $Q ⇉ (Z ⇉ B)$ . Also, the objects as far as $Q$ is concerned are arrows of $Z$ . Therefore, each arrow of $Q$ is an arrow (or hyper-arrow) between two arrows of $Z$ , one acting as a source and the other as a target. For $Q$ to be a groupoid, of course, one would have to define a composition law for these hyper-arrows, and also define the appropriate unit elements. This is an interesting construct, but not interesting enough for a potential application to mixtures.

We consider, instead, just like in a mixture, that we have as a point of departure n constituent groupoids $Z_{I} ⇉ B$ , with $I = 1, \dots, n$ , sharing the same base set $B$ . The objective is to endeavour to unify these n basic groupoid structures under the umbrella of what can be called an n-fold groupoid. This objective was first achieved by Ehresmann [15,16] within the general theory of n-fold categories. Adopting a rather tentative viewpoint, inasmuch as possible, one may define an n-fold groupoid recursively as follows:

Definition 4.1. An n-fold groupoid generated by n simple groupoids with common base $B$ is a set $Q$ equipped with n groupoid structures whose bases are $(n - 1)$ -fold groupoids, each of which is generated by one of the n combinations of $n - 1$ of the n simple groupoids of departure. Denoting the compositions of these groupoid structures of $Q$ by $⊙_{I} (I = 1, \dots n)$ , the following compatibility condition must be satisfied for every pair of indices $I \neq J$ :

(A ⊙_{I} B) ⊙_{J} (ℂ ⊙_{I} D) = (A ⊙_{J} ℂ) ⊙_{I} (B ⊙_{J} D) \forall A, B, ℂ, D \in Q,

(18)

whenever the operations are possible.

The compatibility condition (18) is crucial to the definition. We remark that, as required by the recursive process, the point of departure to arrive at an n-fold groupoid must of necessity involve n simple groupoids. Consequently, each of the possible $(n - 1)$ -fold groupoids mentioned in the definition leaves one of these basic simple groupoids out of the mix. Therefore, each of the n groupoid structures in the definition (say, the Ith structure) consists of “arrows” whose source and target belong to the $(n - 1)$ -fold groupoid that does not include the Ith basic groupoid. The compatibility condition ensures that the “arrows” of this Ith structure are somehow made up of arrows of the missing basic groupoid.

4.2. Groupoid hierarchy

Even without developing the theory any further, Definition 4.1 implies that an n-fold groupoid involves n other $(n - 1)$ -fold groupoids, each of which in turn involves $n - 1$ other $(n - 2)$ -fold groupoids, and so on, until one reaches n original simple groupoids with a common base $B$ . If we adopt a notation similar to the notation used in Section 2.1 for a mixture, and if then we invoke Proposition 2.1, we conclude that:

Proposition 4.1. The hierarchy of all the m-fold groupoids (with $m = 0, \dots, n$ ) involved in an n-fold groupoid is represented, for any given n, by the directed graph of the standard n-dimensional hypercube.

Example 4.1. The case n =4: Figure 3 represents the various hierarchies directly emerging from Definition 4.1. The notation indicates (with a subscript 1) which of the 4 basic groupoids participate in the formation of the various m-fold groupoids $(m = 0, \dots, n)$ . According to Proposition 4.1, this diagram should be the same as the graph of a 4-cube (or tesseract). Although this fact is not apparent (due to the fourth dimension involved), we will later have a chance to verify that the two graphs are identical.

Figure 3.

Hierarchical structure of groupoids in a 4-fold groupoid.

In Figure 3, we have placed a bar over the various groupoids to distinguish between the binary and the decimal numbering. For instance ${\bar{Z}}_{1000} = Z_{1}$ .

4.3. Back to the skeletons

The skeletons $S_{n}$ of the hypercubes $Q_{n}$ are naturally endowed with projections on n different pairs of $(n - 1)$ -skeletons. For this reason it is convenient to describe the arrows of each of the groupoid structures of an n-fold groupoid as objective n-skeletons, which will be introduced below.

In Section 2.2, having defined an h-face of $Q_{n}$ as a subset $F_{h} \subset Q_{n}$ obtained by fixing $n - h$ coordinates to either 0 or 1, we considered the cases $h = 0$ and $h = 1$ . They correspond, respectively to the vertices and the edges of $Q_{n}$ . The $2^{n}$ vertices enjoy a natural ordering derived from reading their coordinate representations in $ℝ^{n}$ , made up obviously by a sequence of zeros and ones, as a number in the binary system.

The $2^{n - 1} n$ edges of the skeleton $S_{n}$ can be regarded as vectors in the affine space of $ℝ^{n}$ . Having just one running coordinate, edges fall naturally into n classes according to the axis to which they are parallel. Each class contains exactly $2^{n - 1}$ parallel edges. Each edge e in the Ith class joins two vertices, say $V_{0}^{e}$ and $V_{1}^{e}$ , with all coordinates respectively equal except for the Ith coordinate which is 0 for $V_{0}^{e}$ and 1 for $V_{1}^{e}$ . Accordingly, we can define for each $I = 1, \dots, n$ a pair of projection maps ${\bar{α}}_{I}, {\bar{β}}_{I} (I = 1, \dots n)$ , which we may call the Ith source and target projections. They are defined as

{\bar{α}}_{I} (e) = V_{0}^{e} {\bar{β}}_{I} (e) = V_{1}^{e} .

(19)

At the other end of the spectrum, for $h = n - 1$ , we obtain the facets of $S_{n}$ . The number of facets, according to equation (4), is $2 n$ . Moreover, each facet is trivially congruent to $S_{n - 1}$ , since only one of the coordinates, $X_{I}$ say, is fixed (to either 0 or 1). Accordingly, there are n classes of facets, each consisting of a pair which we may call the back facet $F_{I}^{0}$ (with $X_{I} = 0$ ) and the front facet $F_{I}^{1}$ (with $X_{I} = 1$ ) of their class. Clearly, the facets in the Ith class are perpendicular to the edges in the Ith class.

We can, therefore, extend the definition of the projection maps (19) to the skeleton $S_{n}$ and obtain

{\bar{α}}_{I} (S_{n}) = F_{I}^{0} {\bar{β}}_{I} (S_{n}) = F_{I}^{1},

(20)

whereby each edge of class I is projected by ${\bar{α}}_{I}$ onto a vertex of $F_{I}^{0}$ , and by ${\bar{β}}_{I}$ onto a vertex of $F_{I}^{1}$ . We depict these projections for the cube $(n = 3)$ and the tesseract $(n = 4)$ in Figure 4.

Figure 4.

Projections of the cube (left) and the tesseract (right) along the second coordinate axis to produce two disjoint facets of each.

Remark 4.1. Comparing the graph of Figure 3 with that of the tesseract³ in Figure 4, we conclude that both graphs are equivalent: their edges connect the same corresponding nodes.

4.4. Objective skeletons

Recall that we have at our disposal n simple groupoids $Z_{I} ⇉ B (I = 1, \dots, n)$ , with source and target projections $α_{I} : Z_{I} \to B$ and $β_{I} : Z_{I} \to B$ , respectively. In each of these groupoids, the collection of arrows between two points $X, Y \in B$ is denoted by $P_{XY}^{I}$ .

To define a typical element (hyper-arrow) of an n-fold groupoid $Q$ , we start by choosing a tuple $W = {W_{0}, W_{1}, \dots, W_{2^{n} - 1}}$ of $2^{n}$ (not necessarily different) points of $B$ . Recalling that the set of vertices of the standard skeleton $S_{n}$ is endowed with a special natural ordering, it can also be regarded as a tuple $V = {V_{0}, V_{1}, \dots, V_{2^{n} - 1}}$ of the same length. We thus obtain a bijection $f : V \to W$ . We will refer to each $W_{i}$ as an objective vertex.

Consider now an edge of class I of $S_{n}$ , and let $V_{i}$ and $V_{j}$ be its tail and its tip, respectively. Their images by the bijection f are, respectively, $W_{i}$ and $W_{j}$ . If the set $P_{W_{i} W_{j}}^{I}$ is not empty, we choose one of its elements, that is, an arrow in the Ith groupoid with source at $W_{i}$ and target at $W_{j}$ . We will call this element of $P_{W_{i} W_{j}}^{I}$ an objective arrow. Making such a choice for every edge of $S_{n}$ (if possible) we have obtained an objective n-skeleton $T_{n}$ . It consists of $2^{n}$ objective vertices and corresponding $2^{n - 1} n$ objective arrows, namely, $2^{n - 1}$ from each of the constituent simple groupoids of departure.

By its very definition, every objective n-skeleton $T_{n}$ has objectiveh-faces for each $h = 0,,, n - 1$ . Each of these h-faces is an objective h-skeleton. For $h = n - 1$ , we obtain exactly $2 n$ objective facets, which can be classified into n classes, each containing a pair of facets complementary to the Ith basic groupoid. Just as in the case of $S_{n}$ , an objective n-skeleton is endowed with $2 n$ natural projections, ${\tilde{α}}_{I}$ and ${\tilde{β}}_{I}$ , which map the objective skeletons into their respective objective facets via the projections of the objective edges onto the respective objective vertices using the groupoid projections of the corresponding basic simple groupoid, as suggested in Figure 5.

Figure 5.

Schematic conception of $T_{n}$ and two objective facets.

Let $Q$ be the (never empty) set of all possible objective n-skeletons $T_{n}$ arising from the given n simple groupoids $Z_{I} ⇉ B (I = 1, \dots, n)$ . By the description just given, we can say that this set of objective n-skeletons is endowed with n source and n target projection maps onto the n sets of all objective $(n - 1)$ -skeletons that can be formed by excluding one of the basic simple groupoids of departure. We have, therefore, achieved a recursive construction of the first stipulation of Definition 4.1. To complete the construction, we still need to define the composition laws and verify that they satisfy the compatibility condition.

4.5. Composition of objective skeletons

Let $T_{n}$ and $T_{n}^{'}$ be two n-skeletons in $Q$ satisfying the condition

{\tilde{β}}_{I} (T_{n}^{'}) = {\tilde{α}}_{I} (T_{n}) .

(21)

Because of the way in which these projections were defined (in terms of the projections $α_{I}$ and $β_{I}$ ), the corresponding $X_{I}$ edges of $T_{n}$ and $T_{n}^{'}$ automatically satisfy the tip-to-tail condition as arrows of the Ith simple groupoid $Z_{I} ⇉ B$ and they can be composed accordingly as such. We define the composition $T_{n} ⊙_{I} T_{n}^{'}$ as the objective n-skeleton such that

\begin{array}{l} {\tilde{α}}_{I} (T_{n} ⊙_{I} T_{n}^{'}) & = {\tilde{α}}_{I} (T_{n}^{'}), \end{array}

(22)

\begin{array}{l} {\tilde{β}}_{I} (T_{n} ⊙_{I} T_{n}^{'}) & = {\tilde{β}}_{I} (T_{n}), \end{array}

(23)

and whose $X_{I}$ -edges are obtained as the composition in the simple groupoid $Z_{I}$ of the corresponding $X_{I}$ edges of the factors.

We need to show that the various compositions satisfy the compatibility conditions

(T_{n} ⊙_{I} T_{n}^{'}) ⊙_{J} (T_{n}^{″} ⊙_{I} T_{n}^{‴}) = (T_{n} ⊙_{J} T_{n}^{″}) ⊙_{I} (T_{n}^{'} ⊙_{J} T_{n}^{‴})

(24)

for each distinct pair $I, J$ whenever the operations are possible. The proof is straightforward, since each composition affects only the edges parallel to one direction. The units and inverses for each of these operations are defined by rendering the corresponding I edges as units and inverses, respectively, in the ordinary groupoid $Z_{I} ⇉ B$ .

The set $Q$ including its n groupoid structures is called the coarse n-fold groupoid formed from the n original simple groupoids $Z_{I} ⇉ B$ . The name derives from the fact that we have included in $Q$ all the possible objective n-skeletons. All other n-fold groupoids obtainable out of n simple groupoids will be subsets of the coarse n-fold groupoid.

Example 4.2. The case $n = 3$ : Two objective 3-skeletons, as depicted in Figure 6, are to be composed according to the composition operator $⊙_{2}$ . Condition (21) prescribes that for this composition to be possible we must have

â_{1} = a_{2} â_{3} = a_{4} ĉ_{1} = c_{2} ĉ_{3} = c_{4} .

(25)

The composition affects only the edges of class $I = 2$ . Conditions (22) and (23) are fulfilled.

Figure 6.

Composition $⊙_{2}$ of two objective 3-skeletons.

The core groupoid of an n-fold groupoid is obtained by setting all the edges, except those issuing from the vertex $W_{0}$ , of each objective n-skeleton to the unit of the corresponding groupoid. Effectively, all the vertices but $W_{0}$ collapse into a single vertex, $W_{7}$ say, and we obtain a 1-fold groupoid whose arrows are bundles of n arrows, one from each groupoid. This is schematically represented for the case $n = 3$ in Figure 7.

Figure 7.

An element in the core, where identity arrows are dashed.

5. Application to composites

5.1. Binary composites and double groupoids

The literature on double groupoids is confined to high-level mathematical articles [15,17 –19]. Physical applications do not abound, with the possible exception of quantum mechanics. In the more mundane context of classical continuum mechanics, however, it is not difficult to discern the natural appearance of double groupoids in such applications as solid binary mixtures [11]. Most materials, both natural and artificial, when examined at a microscopic or mesoscopic scale, reveal the coexistence of different constituent structures, whose combination results in the observable macroscopic behaviour. According to context or scope of application one can speak about composites, mixtures, additive manufacturing, or metamaterials. Each constituent of any of these complex media can be considered as a material in its own right. Nevertheless the technique of homogenization, widely used in the analysis of composites, provides a sophisticated approximation to a putative single material equivalent in its mechanical response to the original composite.

To illustrate what may be lost with a technique of homogenization of a composite, consider a two-dimensional universe in which two sheets of material have been glued together throughout their surface, just as it is done to create plywood from two (theoretically infinitesimally thin) sheets of wood.⁴ The two layers are made of the same material, but one of them has been deformed by some forces or couples applied in its plane before the gluing operation takes place. The assembly is then released, whereupon the misalignment between the two structures results in the appearance of residual stresses in each layer. If instead of two inert structures we had combined two biological membranes in a similar way, the appearance of a difference of chemical potential between the layers could result in a process of time evolution whereby some state of equilibrium is negotiated in terms of structural changes in both layers to compensate for the initial misalignment. It is precisely this possible misalignment that would be interesting to quantify.

Unsurprisingly, 2-fold groupoids, known more commonly as double groupoids, have been studied in great detail by mathematicians [17 –19], and have found applications in topology and differential geometry. A relatively recent application to the theory of defects in binary composites [11,20] promises to pave the way for the use of n-fold groupoids in further applications to complex metamaterials.

In the case $n = 2$ , it is traditional to designate the two basic simple groupoids as horizontal and vertical and to denote them as $H$ and $V$ , respectively. The respective groupoid structures of the double groupoid $Q$ are the vertical structure (over $H$ ) and the horizontal structure (over $V$ ). The hierarchical structure of a double groupoid, with the corresponding source and target projections, is depicted in the diagram (26) below. It has the graph of $S_{2}$ , namely, a square.

The elements of $Q$ are, as expected from the general theory of n-fold groupoids, objective squares. The role of each pair of opposite sides is fulfilled by arrows of one of the basic groupoids. Let $A$ be a square in $Q$ , as represented in equation (27). Its two horizontal sides s and t are obtained as source and target using the projections of the vertical structure, that is, with the maps ${\tilde{α}}_{V}$ and ${\tilde{β}}_{V}$ , respectively. Similarly, the vertical sides ŝ and $\hat{t}$ are the source and target projections of $A$ according to the horizontal-structure projections ${\tilde{α}}_{H}$ and ${\tilde{β}}_{H}$ .

Since $Q$ is endowed with two different groupoid structures, each structure must have its own well-defined product, which will be denoted by $⊙_{H}$ and $⊙_{V}$ , respectively, for the horizontal and vertical structures. In the horizontal structure, the product of two squares, $A ⊙_{H} A^{'}$ , can only be carried out if $ŝ = {\hat{t}}^{'}$ , in agreement with the usual concatenation criterion in a groupoid.

The composition law of the horizontal structure of $Q$ is obtained by applying the composition law of the horizontal side groupoid $H$ to each pair of adjacent horizontal arrows of the two squares. This is easily represented graphically as

The unit square of the horizontal structure at the vertical arrow û is the square

Similarly, in the vertical structure, the product $A ⊙_{V} A^{'}$ can only be carried out if $s = t^{'}$ , or graphically

The unit square of the vertical structure at the horizontal arrow u is the square

The definition of a double groupoid must be completed by the verification of a compatibility condition between the two structures, namely,

(A ⊙_{H} B) ⊙_{V} (ℂ ⊙_{H} D) = (A ⊙_{V} ℂ) ⊙_{H} (B ⊙_{V} D),

(32)

whenever the operations are possible. Graphically, this condition means that, in a square made of four squares whose edges in contact are matched, the result is the same whether one first composes horizontally and then vertically, or vice versa. In other words, equation (32) is verified in equation (33):

Finally, the double source map assigning to each square $A$ its sources in the horizontal and vertical structures, ${\tilde{α}}_{H} (A), {\tilde{α}}_{V} (A)$ , is often assumed to be surjective. The graphical representation of this assumption is that a right angle made of a horizontal and a vertical arrow issuing from a common corner in $B$ , can always be completed (not necessarily uniquely) to a square in $Q$ . This is known as the filling condition.

A useful example of a double groupoid is the coarse double groupoid, $□ (H, V)$ , generated by two groupoids, $H$ and $V$ , with a common base set. It consists of all the possible consistent squares that can be formed using the two groupoids as sides. Any double groupoid can be naturally mapped as an inclusion into the coarse double groupoid generated by its side groupoids.

Definition 5.1. The core of a double groupoid consists of the collection of all squares $A$ for which the two source arrows s and ŝ are, respectively, the units of $H$ and $V$ at the point $X = α_{H} ({\hat{α}}_{V} (A)) = α_{V} ({\hat{α}}_{H} (A))$ .

A square in the core groupoid, therefore, will have 3 coincident vertices, $W_{1} = W_{2} = W_{3}$ , as represented in Figure 8. For this reason, the core can be given a canonical groupoid structure by regarding its arrows as pairs of arrows $(t, \hat{t})$ with common source $X = W_{0}$ and target point $Z = β_{H} ({\hat{β}}_{V} (A)) = β_{V} ({\hat{β}}_{H} (A)) = W_{3}$ . In many respects the core groupoid characterizes the double groupoid [17], although the physical identity of the original side groupoids is not preserved when passing to the core.

Figure 8.

An element in the core, where identity arrows are dashed.

5.2. Measures of misalignment

A schematic representation of a binary composite whose constituents happen to be uniform, is shown in Figure 9. We ask ourselves: is this composite uniform? The answer to this question depends on the archetypal symmetry groups of the constituents. If the composite fails to be uniform, we recognize a new kind of defectivity not present in ordinary simple materials. Indeed, the presence of defects in a simple (single-constituent) material assumes that the body is already uniform while the defects manifest themselves as the inability of the body to find a homogeneous configuration whereby the transplants are mere translations. In the case of a binary composite, however, even when both constituents are individually uniform (and even homogeneous), if the mixture turns out not to be uniform we have a situation of a defect of misalignment between the constituents.

Figure 9.

A binary composite and its two archetypes.

Example 5.1. Two triclinic constituents: Recall that a triclinic material has no non-trivial symmetries, that is, no material symmetries except the identity. This being the case, the implant maps $P (X)$ and $\hat{P} (X)$ are uniquely defined. We conclude that the composite is materially uniform if, and only if, the transplant maps $P_{12}$ and ${\hat{P}}_{12}$ coincide for every pair of points $X_{1}$ and $X_{2}$ . Since

P_{12} = P (X_{2}) {(P (X_{1}))}^{- 1}

(34)

and

{\hat{P}}_{12} = \hat{P} (X_{2}) {(\hat{P} (X_{1}))}^{- 1},

(35)

we may state the following

Theorem 5.1. A binary composite made of uniform elastic constituents with no non-trivial symmetries is uniform if, and only if, the matrix field

A (X) = {(\hat{P} (X))}^{- 1} P (X) .

(36)

is constant.

Remark 5.1. For more general cases of two uniform constituents with various symmetry groups, similar criteria can be obtained for the composite to be uniform. These criteria have been studied at length elsewhere [11].

5.3. Construction and meaning of the material double groupoid

In the general case of materially uniform constituents, the groupoids $Z_{1} ⇉ B$ and $Z_{2} ⇉ B$ of the constituents are transitive subgroupoids of the 1-jet groupoid $Π^{1} (B, B)$ introduced in Remark 3.2. We can, accordingly, form the coarse double groupoid $□ (Z_{1}, Z_{2})$ . Out of this large generic double groupoid, whose squares include all possible pairs of arrows, one pair from each groupoid, joining every ordered set of 4 points of $B$ , we would like to select only those squares that convey a physical property of a binary composite. The following definition achieves this objective.

Definition 5.2. The material double groupoid of a binary composite consists of all commutative squares of $□ (Z_{1}, Z_{2})$ .

Remark 5.2. A square

is said to be commutative if

\hat{t} s = t ŝ .

(38)

It is important to emphasize that this equality is only possible when, as is the case in a material double groupoid, both side groupoids are subgroupoids of a common groupoid. Otherwise, there is no meaning that can be attributed to the composition of arrows in equation (38). In our case, the common underlying groupoid is the 1-jet groupoid $Π^{1} (B, B) ⇉ B$ , introduced in Remark 3.2.

To unravel the physical interpretation of the commutativity of a square, let us consider the horizontal arrows s and t. Their sources are indicated in equation (39) as $X_{0}, X_{1}$ , and their targets as $X_{2}, X_{3}$ .

Since the constituents have been assumed to be uniform, we may choose an arrow â belonging to the vertical side groupoid $Z_{2}$ , as shown in the diagram (39). In the same diagram, moreover, we have added the arrow $\hat{b} = \hat{t} â ŝ^{- 1}$ , also belonging to $Z_{2}$ . The arrows â and $\hat{b}$ are not part, in general, of a square of the double groupoid, but they obviously exist (not necessarily uniquely) since $Z_{2}$ is transitive. We define a loop-shaped arrow in $Π^{1} (B, B)$ at $X_{0}$ as

m_{0} = â^{- 1} s .

(40)

We refer to $m_{0}$ as the horizontal alignment of â with s. Similarly, at $X_{1}$ we define the corresponding alignment of $\hat{b}$ with t as

m_{1} = {\hat{b}}^{- 1} t .

(41)

We remark that a unit alignment, namely, $m_{0} = ϵ_{Z_{1}} (X_{0})$ , corresponds to the existence of a common arrow between $X_{0}$ and $X_{2}$ for the two material groupoids. In other words, $X_{0}$ and $X_{2}$ , considered as points of the composite, are materially isomorphic.

The following theorem is a direct consequence of the definitions just stated.

Theorem 5.2. A square is commutative (in accordance with equation (38)) if, and only if, the alignment $m_{1}$ is the conjugate of $m_{0}$ via ŝ, that is

m_{1} = ŝ m_{0} ŝ^{- 1} .

(42)

A similar theorem holds true for the vertical alignments ${\hat{m}}_{0} = a^{- 1} ŝ$ and ${\hat{m}}_{2} = b^{- 1} \hat{t}$ , where a is an arrow in the horizontal side groupoid and b is given by $b = t a s^{- 1}$ , as suggested in diagram (43).

The conjugacy condition is now

{\hat{m}}_{2} = s {\hat{m}}_{0} s^{- 1} .

(44)

Remark 5.3. The relation of conjugacy within a group is usually an indication of equivalence of meaning. In the case of the general linear group, whose elements are non-singular square matrices, conjugacy is known as similarity. Two similar matrices share the same characteristic polynomial. Thus, they can be regarded as the same linear operator expressed in two different bases. In solid mechanics, moreover, conjugacy of two quantities arises naturally upon changes of reference configuration of the body.

We may say that the relation of commutativity of a square introduces a new kind of symmetry. Indeed, local symmetries at a point are represented by groups. Distant symmetries between two points manifest themselves as arrows of a groupoid. The squares of a double groupoid can be regarded, therefore, as shared symmetries between two pairs of points. The physical meaning of these shared symmetries in the case of a binary composite consist of having “the same” degree of alignment between the arrows of both underlying materials.

5.4. The core condition

According to Definition 5.1, the core of a double groupoid consists of the objective squares of the form

By the very meaning of the identities, this kind of square can also be represented as

Since our double groupoid is made of commutative squares, we obtain the following result.

Theorem 5.3. A composite made of two materially uniform constituents is uniform if, and only if, the core groupoid is transitive.

Indeed, the commutativity of a square in the core implies that $\hat{t} = t$ . Recall that the core groupoid is an ordinary groupoid each of whose arrows consists of a pair $(t, \hat{t})$ . The transitivity of this groupoid means that for every $X, Y \in B$ there is a common material isomorphism for both constituent materials.

5.5. Designing distributions of defects of misalignment in metamaterials

Assuming that the fields $A (X)$ (or a similar field for other symmetry classes of the constituents) is differentiable, its gradient is a legitimate measure of the local lack of transitivity of the double groupoid. This is a somewhat unexpected result (possibly meaningful in other applications) providing a local measure of intransitivity of the core of certain double groupoids having two transitive side groupoids. Recent research [21,22] reveals that intentionally designed distributions of topological defects in metamaterials can have a beneficial effect in steering stresses and deformations. In terms of the continuous limit, this kind of design can be represented by, for example, a doubly periodic planar field $A (X)$ in some reference configuration. This deliberately introduced defectivity of misalignment between several constituents of a metamaterial will have repercussions in the solution of the associated boundary value problem in elasticity.

Example 5.2. Periodic distribution of defects of misalignment: A planar body consists of two uniform constituents made of the same orthotropic material, the only difference being the respective implant fields. A stress-free archetype of a solid can be regarded as the orthonormal natural basis $E_{α} (α = 1, 2)$ of $ℝ^{2}$ . A field of implants $P (X)$ for a uniform body is represented as a field of bases $P_{α} (α = 1, 2)$ given in Cartesian components by

P_{α} = P_{α}^{I} E_{I},

(47)

where we are adhering to the summation convention for diagonally repeated indices. For the first constituent, it will be assumed that the field of matrices $[P] (X)$ is everywhere equal to the unit matrix $[I]$ . The corresponding field of bases is represented in Figure 10(a). For the second constituent, made of the same orthotropic material, the implant field has been deliberately modulated according to the rotation matrix

[\hat{P}] = [\begin{array}{l} \cos θ (x, y) & - \sin θ (x, y) \\ \sin θ (x, y) & \cos θ (x, y) \end{array}]

(48)

with

θ (x, y) = 0.5 \sin^{2} (π x ∕ 2) + 0.3 \sin^{2} (π y) .

(49)

This field of bases is represented in Figure 10(b). The superposition of both fields is shown in Figure 10(c). Although both constituents are individually uniform, the resulting composite is not. The material has been tuned to perform in a desired way suited to the application at hand, or the misalignment between the constituents may be due to a fabrication defect. We remark the essential difference between defects of inhomogeneity in a simple uniform material and defects of misalignment in a composite. Even if the two constituents were not just uniform but also homogeneous, a misalignment could happen as the result of initial thermal strains (or other causes) in one of the constituents at the time of assembly.

Figure 10.

Lack of uniformity due to misalignment of uniform constituents: (a) Constituent 1; (b) Constituent 2; (c) Composite 1+2.

5.6. Material n-fold groupoids and multi-composites

Consider the special case of an n-fold groupoid $Q$ whose base set is a body manifold $B$ and each of the n basic groupoids $Z_{i} ⇉ B (i = 1, \dots, n)$ is a subgroupoid of the 1-jet groupoid $Π^{1} (B, B)$ . Each objective n-skeleton $T_{n}$ in $Q$ is a directed graph modelled after $S_{n}$ . But each objective edge of $T_{n}$ is an element of $GL (3, ℝ)$ , that is, a non-singular $3 \times 3$ real matrix. This fact suggests the possibility of considering these objective n-skeletons as directed weighted graphs, where the weights are such matrices.

We recall that a walk in a graph is a sequence of adjacent vertices $v_{i}$ . A finite walk p with m steps, therefore, can be explicitly given as an $(m + 1)$ -tuple

p = {v_{0}, v_{1}, \dots, v_{m}}

(50)

with the condition of adjacency between $v_{i - 1}$ and $v_{i}$ for each $i = 1, \dots, m$ . By adjacency, we mean that there is an edge joining these vertices, regardless of the orientation of the edge joining them. A walk is said to be closed if the first and last entries of the sequence p coincide.

Remark 5.4. In graph theory there is a need to distinguish between walks, trails, and paths, a distinction that will not be necessary at this point. In a walk, vertices and edges may be visited more than once along the way.

Definition 5.3. Let $G$ be a group and let S be a graph whose collection of edges we denote by $E$ . We say that S is a $G$ -weighted graph if a function

w : E \to G,

(51)

called a weight function has been defined. This function need not be injective.

In a significant departure from conventional graph theory, in the case of directed graphs, we will permit the consideration of all walks in the underlying non-directed graph. In the directed graph, however, the weight of an edge traversed against its orientation will be replaced with the inverse of the weight assigned by the weight function w.

Definition 5.4. The weight of a (finite) walkp in a group-weighted and directed graph S is the group composition of the weights of the successive edges of the walk, provided inverse weights are assigned to edges traversed against the current.

For lack of a better term we will refer to this policy as the direct-inverse weighting convention. According to this definition, if a walk consists of a return to the vertex of departure via the reversal of the steps of a previous walk, the combination of both walks will have a total weight equal to the unit of the group.

Theorem 5.4. The total weight of every closed walk in $T_{n} \in Q$ is the group identity if, and only if, every 2-face is commutative.

We will refer to such a conservativen-fold groupoid as a material n-fold groupoid. The “only if” part of the theorem is trivial. The proof of the “if” part can be obtained by induction as follows. The theorem is trivially true for $n = 2$ . Assume it to be true for some value $n - 1$ . Given a closed walk in $T_{n}$ ,

If the walk is completely contained in an ( $n - 1$ )-face of $T_{n}$ the theorem is obviously true.

If not, consider the partition of $T_{n}$ into 2 $X_{I}$ -facets. Since in a closed walk the initial and terminal vertices coincide, any closed walk must comprise an even number of $X_{I}$ edges, just like the number of bridge traversals between two shores connected by various bridges. Consider, therefore, a pair of successive traversals. If the connecting edge happens to be the same for both traversals, we conclude that their combined contribution to the total weight of the closed walk is the group identity, since the same edge is traversed in opposite directions.

Otherwise, if the connecting edges are different, let A and B denote the corresponding vertices in the two $X_{I}$ -facets joined by these two edges. We can join A and B by an identical finite path in each of the facets. Consider a single edge $A A^{'}$ of these paths, as shown in Figure 11. Together with the two edges, say $E_{1}$ and $E_{2}$ joining the corresponding vertices, they constitute a 2-face of $T_{n}$ . By commutativity, therefore, the weight contributed by $E_{1}$ is identical to that contributed by $E_{2}$ . Continuing along the path joining A with B, we can invoke the reasoning adduced in the previous item (where the two edges coincided).

Figure 11.

Independence of weight for parallel traversing edges.

Corollary 5.1. A mixture of nuniform constituents is materially uniform if, and only if, the core of its materialn-fold groupoid is transitive.

If the mixture is not uniform, we say that there are sources of misalignment, namely, a defectivity arising from an incompatibility between the constituents. We remark, however, that even if all the constituents are compatible with a fixed one, the mixture is not necessarily uniform. In other words, in the case of continuous symmetry groups, the core of the material n-fold groupoid may fail to be transitive, due to the extra degrees of freedom afforded by the symmetries. If this observation is correct, we can assert that a different kind of defectivity has been identified beyond that provided in the literature [11] for strictly binary mixtures.

In a mixture of n uniform constituents, each of which has the trivial symmetry group, if one of these constituents is aligned with the remaining $n - 1$ constituents the mixture is necessarily uniform. If, however, at least one of the constituents has a continuous symmetry group (such as full or transverse isotropy) uniformity requires further conditions. As an example, which may be applicable to metamaterials, we can consider a combination of three materials, one of which is an isotropic solid in a homogeneous configuration. The other two constituents have discrete symmetry groups and are in different states of contorted aeolotropy. This last term is used to describe a stress-free state (whereby all the material isomorphisms are pure rotations). The isotropic constituent is, therefore, well aligned with each of the other two constituents, since isotropy affords a degree of freedom of rotation. If the composite is achieved by building a regular scaffold of cubic boxes whose walls are made of the isotropic constituent, as shown in Figure 12, and if the boxes are alternately filled with one of the other materials (so that the latter are not in contact with each other), we obtain an inert mixture. If, on the other hand, the scaffold is made of one of the anisotropic materials, those boxes filled with the other anisotropic constituent may react and evolve in time (perhaps due to the existence of a chemical potential associated with the misalignment). This kind of situation may arise in applications to cell and tissue engineering [23].

Figure 12.

Alternating filling in metamaterial.

Footnotes

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Marcelo Epstein

Notes

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Mixtures and the hierarchical groupoid structure of n -fold groupoids

Abstract

Keywords

1. Introduction

2. Mixtures and hypercubes

2.1. The graph of a mixture

2.2. The graph of a hypercube

3. Groupoids

3.1. Groupoids in general

3.2. Material groupoids

4. n-fold groupoids

4.1. General considerations

4.2. Groupoid hierarchy

4.3. Back to the skeletons

4.4. Objective skeletons

4.5. Composition of objective skeletons

5. Application to composites

5.1. Binary composites and double groupoids

5.2. Measures of misalignment

5.3. Construction and meaning of the material double groupoid

5.4. The core condition

5.5. Designing distributions of defects of misalignment in metamaterials

5.6. Material n-fold groupoids and multi-composites

Footnotes

Funding

ORCID iD

Notes

References