Theory and Applications of Categories, Vol. 14, No. 4, 2005, pp. 60–74.

THIN ELEMENTS AND COMMUTATIVE SHELLS IN CUBICAL ω-CATEGORIES

PHILIP J. HIGGINS

Abstract. The relationships between thin elements, commutative shells and connections in cubical ω-categories are explored by a method which does not involve the use of pasting theory or nerves of ω-categories (both of which were previously needed for this purpose; see [2], Section 9). It is shown that composites of commutative shells are commutative and that thin structures are equivalent to appropriate sets of connections; this work extends to all dimensions the results proved in dimensions 2 and 3 in [7, 6].

Introduction

Thin structures in simplicial sets were introduced by Dakin in [8] and were applied to cubical sets in [3, 4, 5]. In the cubical case a thin structure is equivalent to an ω-groupoid structure.

In this paper we use the term thin structure in a weaker sense which is appropriate for cubical ω-categories and does not imply the existence of inverses. This concept was introduced in the 2-dimensional case in [13], as arising from a pair of connections in dimension 2, and the equivalence of these notions in this dimension was shown in [7]. We extend this result to all dimensions (Theorem 3.1).

The definition of thin structure depends on the notion of commutative cube or commutative shell. This was studied in the 3-dimensional case in [13] for certain ‘special’ double categories, and in [6] as part of the proof of a van Kampen theorem for homotopy double groupoids. A key result there (see also [13, Proposition 3.11]) was that any composition of commutative 3-shells is commutative; we prove this result in all dimensions. We also prove in the general case that thin elements are precisely those which can be expressed as composites using only identity elements and connections (Theorem 2.8).

THIN ELEMENTS AND COMMUTATIVE SHELLS IN CUBICAL ω-CATEGORIES  61

These results, which were proved for ω-groupoids in [4], can be deduced for ω-categories from results in [2]. However, the methods used in [2] depend on the use of pasting theory and nerves of ω-categories which tend to obscure the intuitive nature of thinness and commutativity. The approach used below is simpler and more direct. The basic simplification is the use of a “partial folding operation” Ψ in place of the full folding operation Φ_{n} used in [2]. The operation Φ_{n} is needed to prove the equivalence between cubical ω-categories with connections and globular ω-categories, but the simpler Ψ suffices for a detailed study of thinness and commutativity. This direct approach should facilitate applications to homotopy theory (cf.[7, 6, 12, 13]) and to concurrency theory in computer science (cf. [10, 11]).

Received by the editors 2004-09-03 and, in revised form, 2005-01-25.

Transmitted by R. Paré. Published on 2005-02-28.

2000 Mathematics Subject Classification: 18D05.

Key words and phrases: cubical ω-category, connections, thin elements, thin structure, folding operations, commutative shells.

\begin{matrix}c \\ ◯ \end{matrix} Philip J. Higgins, 2005. Permission to copy for private use granted.

1. Composing the faces of a cube

Let C be a cubical ω-category as defined in [1] and [2]; for the moment we do not assume the existence of connections. If x ∈ C_{n} is an n-cube in C one may ask which of its (n − 1)- faces have common (n − 2)-faces and can be composed in C_{n−1}. The answer is that the following pairs of faces (and in general only these pairs) can be composed:

(∂_{i}^{−}x, ∂_{i+1}^{+}x),  (∂_{i}^{+}x, ∂_{i+1}^{−}x),  i = 1, 2, …, n − 1.

Thus the faces of x (by which we mean its (n−1)-faces) divide naturally into two sequences

(∂_{1}^{−}x, ∂_{2}^{+}x, ∂_{3}^{−}x, …, ∂_{n}^{±}x) and (∂_{1}^{+}x, ∂_{2}^{−}x, ∂_{3}^{+}x, …, ∂_{n}^{∓}x)

in which neighbouring pairs can be composed. We call these respectively the negative and the positive faces of x. This agrees with the terminology of [2].

We will frequently use 2-dimensional arrays of elements of C_{n}, and these will be shown in tabular form such as

a b c d f g [figure: i; j]

When using this notation we will always assume that pairs of adjacent elements in the table have a common face in the relevant direction, in this case i or j. Thus in the array above we assume that ∂_{j}^{+}a = ∂_{j}^{−}b, ∂_{i}^{+}a = ∂_{i}^{−}d, etc. The array can then be composed by rows: (a ∘_{j} b ∘_{j} c) ∘_{i} (d ∘_{j} f ∘_{j} g), or by columns: (a ∘_{i} d) ∘_{j} (b ∘_{i} f) ∘_{j} (c ∘_{i} g). It is a consequence of the interchange law that these two elements of C_{n} are equal, and we call their common value the composite of the array. To emphasise our implicit assumptions, we will use the term composable array.

More general composites in the form of rectangular partitions of a rectangle will also be used. The simplest is of the form

a b c  [figure: i; j]

Here the implicit assumptions are that ∂_{j}^{+}a = ∂_{j}^{−}b and ∂_{i}^{+}(a ∘_{j} b) = ∂_{i}^{−}c, and so the composite (a ∘_{j} b) ∘_{i} c can be formed. For a more general (finite) partition of a rectangle by rectangles, labelled by members of C_{n}, we will assume that two elements, or composites of elements, with a common edge in the partition, have a common face at this edge, so allowing the composition in the corresponding direction. For example, in the partition

d a b f c  [figure: i; j]

we assume, in addition to the relations above, that

∂_{i}^{+}d = ∂_{i}^{−}f and ∂_{j}^{+}[(a ∘_{j} b) ∘_{i} c] = ∂_{j}^{−}(d ∘_{i} f)

and so the composite [(a ∘_{j} b) ∘_{i} c] ∘_{j} (d ∘_{i} f) can be formed. We shall call such diagrams composable partitions provided that some sequence of compositions exist which combines all the elements to form a composite of the partition. We will specify the sequence if there is any ambiguity.

Now suppose that C is a cubical ω-category with connections, that is, it has for each n and for i = 1, 2, …, n, extra structure maps Γ_{i}^{+}, Γ_{i}^{−} : C_{n} → C_{n+1} (called connections) satisfying the identities set out, for example, in [1] and in Section 2 of [2]. We shall make free use of the defining identities in [2] without further comment. (In fact, the existence of connections in C implies that all composites of a given composable partition are equal. This is because, using connections, any composable partition can be refined to a composable array, and any composite of the partition must be equal to the unique composite of this array: see [7] and [9]. We do not need to use this general theorem.)

The ‘degenerate’ elements Γ_{k}^{+}x, Γ_{k}^{−}x and ε_{k}x will sometimes be represented in composable arrays by the symbols , and ( or ) respectively. The symbol will be used to denote an element that is an identity for both the horizontal and vertical compositions. These symbols will be used only where the elements of C_{n} they represent are uniquely determined by the composability of the array. (The lines in these symbols are designed to indicate that the corresponding faces are identities in the direction of these edges.)

For example, the array

[figure: x; i; j; denotes; _{x  ε_{j}∂_{j}^{+}x ε_{i}∂_{i}^{+}x ε_{i}∂_{i}^{+}ε_{j}∂_{j}^{+}x}; i; j]

whose composite is x itself; and the array

[figure: _{ab a b}; i; i+1; denotes; _{Γ_{i}^{+}a ε_{i+1}a ε_{i}a Γ_{i}^{+}b}; i; i+1]

whose composite, according to the transport law [2, (2.6)], is Γ_{i}^{+}(a ∘_{i} b). (Here the labels on the edges denote the corresponding faces of elements of the array.)

The following identities (see [2, (2.7)]) will be important in what follows:

\begin{aligned}& a \\ a \end{aligned} = \begin{matrix}a \\ a \end{matrix} [figure: i; i+1] and a a a = a a [figure: i; i+1]

that is, Γ_{i}^{+}a ∘_{i+1} Γ_{i}^{−}a = ε_{i}a and Γ_{i}^{+}a ∘_{i} Γ_{i}^{−}a = ε_{i+1}a.

We now define the elementary folding operations ψ_{i} : C_{n} → C_{n}, (i = 1, 2, …, n − 1) by

ψ_{i}x =  x  [figure: i; i+1]

These operations ψ_{i} were introduced in [2] as a categorical version of the folding operations Φ_{j} defined for ω-groupoids in [4]. The chief identities satisfied by the ψ_{i} are set out in Proposition 3.3(i) of [2]; we shall not need the “braid relations” proved in Theorem 5.2 of that paper. As the picture suggests, the effect of ψ_{i} is to “fold” the faces ∂_{i+1}^{−}x and ∂_{i+1}^{+}x to the i^{th} direction so that they abut the faces ∂_{i}^{+}x and ∂_{i}^{−}x respectively, and to compose the two pairs of faces. The composition of all the negative (and positive) faces of x, together with certain faces of the connections, can therefore be achieved by the folding operation Ψ : C_{n} → C_{n} defined by Ψ = ψ_{1}ψ_{2} … ψ_{n−1}.

We note that Ψ is not the full folding operation Φ_{n} of [2] (which maps C_{n} into its globular part (see Proposition 3.5 of [2]), but Ψ is sufficient for the study of thin elements and commutative shells. We will return to this point later. For now, the two most important faces of Ψx, namely

P x = ∂_{1}^{+}Ψx and Nx = ∂_{1}^{−}Ψx,

are to be viewed as convenient embodiments of the positive and negative boundaries of x.

1.1. Lemma.

(i) ψ_{1}ψ_{2} … ψ_{r−1}ε_{r} = ε_{1} : C_{n−1} → C_{n} for 1 ⩽ r ⩽ n − 1.

(ii) If y ∈ C_{n−1} then Ψε_{j}y ∈ ε_{1}C_{n−1} for 1 ⩽ j ⩽ n − 1.

Proof. (i) This follows immediately from the identities ψ_{i}ε_{i+1} = ε_{i}. (ii) similarly, using ψ_{i}ε_{j} = ε_{j}ψ_{i−1} for j < i and ψ_{j}ε_{j} = ε_{j}, we have

Ψε_{j} = ψ_{1}ψ_{2} … ψ_{n−1}ε_{j}
= ψ_{1}ψ_{2} … ψ_{j}ε_{j}ψ_{j}ψ_{j+1} … ψ_{n−2}
= ψ_{1}ψ_{2} … ψ_{j−1}ε_{j}ψ_{j}ψ_{j+1} … ψ_{n−2}
= ε_{j}ψ_{j}ψ_{j+1} … ψ_{n−2} by (i).  □

If x ∈ C_{n}, the shell ∂x of x is the family consisting of all its faces ∂_{i}^{α}x (i = 1, 2, …, n; α = +, −).

1.2. Proposition. Let C be a cubical ω-category with connections and x ∈ C_{n}. Then

(i) all faces ∂_{i}^{α}Ψx with i ⩾ 2 are of the form ε_{1}z_{i}^{α}, where z_{i}^{α} ∈ C_{n−2};

(ii) ∂Nx = ∂P x;

(iii) ∂Ψx is uniquely determined by Nx and P x.

Proof.

(i) We have ∂_{i}^{α}ψ_{j} = ψ_{j}∂_{i}^{α} for i > j + 1, so for i ⩾ 2,

∂_{i}^{α}Ψx = ∂_{i}^{α}ψ_{1} … ψ_{n−1}x = ψ_{1}ψ_{2} … ψ_{i−2}∂_{i}^{α}ψ_{i−1}, …, ψ_{n−1}x,

(if i = 2, the ψ_{1} …, ψ_{i−2} are missing). But ∂_{i}^{α}ψ_{i−1} = ε_{i−1}∂_{i−1}^{α}∂_{i}^{α}, so

∂_{i}^{α}Ψx = ψ_{1}ψ_{2} … ψ_{i−2}ε_{i−1}z_{i}^{α} where z_{i}^{α} ∈ C_{n−2}
= ε_{1}z_{i}^{α} by Lemma1.1 (i) .

(ii) This follows from (i) because

∂_{i}^{α}Nx = ∂_{i}^{α}∂_{1}^{−}Ψx = ∂_{1}^{−}∂_{i+1}^{α}Ψx = ∂_{1}^{−}ε_{1}z_{i+1}^{α} = z_{i+1}^{α}

and similarly ∂_{i}^{α}P_{x} = z_{i+1}^{α}.

(iii) The faces of Ψx are Nx, P x and the elements ε_{1}z where z is a face of Nx (or P x).

In the abstract, an n-shell in C is a family s = {s_{i}^{α}; s_{i}^{α} ∈ C_{n−1}; i = 1, 2, …, n; α = +, −} where the s_{i}^{α} satisfy the incidence relation

∂_{j}^{β}s_{i}^{α} = ∂_{i−1}^{α}s_{j}^{β} for 1 ⩽ j < i ⩽ n and α, β ∈ {+, −}.

We denote by □ C_{n−1} the set of such shells. The usual cubical incidence relations imply that ∂x ∈ □ C_{n−1} for all x ∈ C_{n}.

If {C_{0}, C_{1}, …, C_{n−1}} is a cubical (n − 1)-category with connections, then □C_{n−1} has naturally defined operations ∘_{i} (1 ⩽ i ⩽ n), and connections Γ_{j}^{α} : C_{n−1} → □C_{n−1} (1 ⩽ j ⩽ n − 1) which, together with the obvious structure maps ∂_{i}^{α} : □ C_{n−1} → C_{n−1} and ε_{j} : C_{n−1} → □ C_{n−1}, make {C_{0}, C_{1}, …, C_{n−1}, □ C_{n−1}} a cubical n-category with connections (cf. [4], section 5). Thus we can define folding maps ψ_{i}, Ψ : □C_{n−1} → □C_{n−1} which obviously satisfy:

1.3. Lemma. In a cubical n-category (C_{1}, C_{2}, …, C_{n}) with connections, the map

x ↦ ∂x : C_{n} → □C_{n−1},

together with identity maps in lower dimensions, gives a morphism of cubical n-categories with connections from (C_{1}, C_{2}, …, C_{n}) to (C_{1}, C_{2}, …, C_{n−1}, □C_{n−1}). In particular

Γ_{i}^{α}x = ∂Γ_{i}^{α}x, ψ_{j}∂x = ∂ψ_{j}x, Ψ∂x = ∂Ψx,
N∂x = ∂Nx, P ∂x = ∂P x and ∂ε_{j}x = ε_{j}x.

1.4. Theorem. Let C be a cubical ω-category (or a cubical m-category) with connections. Let a ∈ C_{n} and s ∈ □C_{n−1}. A necessary and sufficient condition for the existence of x ∈ C_{n} such that

∂x = s and Ψx = a

is that

Ψs = ∂a.

If x exists, it is unique.

Proof. The necessity of Ψs = ∂a follows from ∂Ψx = Ψ∂x (Lemma 1.2). The existence and uniqueness will be deduced from:

1.5. Lemma. Let a ∈ C_{n} and s ∈ □C_{n−1} satisfy ∂a = ψ_{j}s for some j ∈ {1, 2, …, n−1}. Then there is a unique x in C_{n} such that ∂x = s and ψ_{j}x = a.

Proof. First suppose that x exists, and consider the array

A :  x  [figure: j; j+1]

where the elements surrounding x are determined by the faces of x so that all rows and columns are composable. The composite of the middle row is ψ_{j}x = a. The elements of the first and third rows are determined by the faces of x, i.e. by s. Hence the composite of A is determined by a and s. But if we compose A by columns and use the law Γ_{j}^{+}t ∘_{j} Γ_{j}^{−}t = ε_{j+1}t, we see that the composite of A is x itself. Hence x is unique.

To prove existence, we note that the array A gives a formula for x in terms of a and s. So, given a and s we define x to be the composite of the composable partition

x =  ε_{j}s_{j}^{−}  Γ_{j}^{+}s_{j+1}^{+} a Γ_{j}^{−}s_{j+1}^{−}  ε_{j}s_{j}^{+}  [figure: j; j+1]

Here the first and third rows are the same as those in the array A, except that the 2fold identities have been omitted, being redundant. Because we are assuming that ∂a = ψ_{j}s, the faces ∂_{j}^{−}a and ∂_{j}^{+}a are the same as the upper and lower faces of the composite x in A. Hence the partition is composable and we can compute x from a and s, by composing rows first. It remains to verify that ψ_{j}x = a and ∂x = s.

The faces ∂_{j+1}^{±}x of x are ∘_{j}-composites as indicated in the diagram

x = ee s_{j+1}^{−}  s_{j+1}^{+} ee a  [figure: j; j+1]

where the faces labelled e are identities for ∘_{j}. (Note that ∂_{j+1}^{±}a are identities because ∂a = ψ_{j}s). Hence

ψ_{j}x =  ε_{j}s_{j}^{−}  Γ_{j}^{+}s_{j+1}^{+} Γ_{j}^{−}s_{j+1}^{+} a Γ_{j}^{+}s_{j+1}^{−} Γ_{j}^{−}s_{j+1}^{−}  ε_{j}s_{j}^{+}  [figure: j; j+1]

Here, the first and last columns are the relevant connections expanded by the transport law and then simplified. The diagram can be viewed as a 3×3 array in which two elements happen to be horizontal composites; therefore we may compute ψ_{j}x by composing the rows first instead of the columns. Using the law Γ_{j}^{+}t ∘_{j+1} Γ_{j}^{−}t = ε_{j}t, we find that ψ_{j}x = a, as required.

Finally ∂x is the composite (by rows) of

∂ε_{j}s_{j}^{−}  ∂Γ_{j}^{+}s_{j+1}^{+} ∂a ∂Γ_{j}^{−}s_{j+1}^{−}  ∂ε_{j}s_{j}^{+}  [figure: j; j+1]

Since ∂a = ψ_{j}s, this is the composite in □ C_{n−1} of the array

which, as before, is just s.

s  [figure: j; j+1]

We now use induction on r ⩽ n − 1 to show that if ∂a = ψ_{1}ψ_{2} … ψ_{r}s, then there is a unique x ∈ C_{n} such that ψ_{1}ψ_{2} … ψ_{r}x = a and ∂x = s. The case r = 1 is covered by Lemma 1.5. Suppose that the result is true when r = t − 1 < n − 1 and that ∂a = ψ_{1}ψ_{2} … ψ_{r}s. Then, by induction hypothesis, there is a unique y ∈ C_{n} such that ψ_{1}ψ_{2} … ψ_{r−1}y = a and ∂y = ψ_{t}s. But, again by Lemma 1.5, there is then a unique x ∈ C_{n} with ψ_{t}x = y and ∂x = s, completing the induction. The case r = n − 1, completes the proof of the theorem.

2. Thin elements and commutative shells

We say that an element x ∈ C_{n} is thin if Ψx ∈ ε_{1}C_{n−1}. Then Ψx = ε_{1}Nx = ε_{1}P x. We say that a shell s ∈ □ C_{n−1} is commutative if Ns = P s.

2.1. Proposition. Let C be a cubical ω-category (or cubical m-category) with connections.

(i) The shell of a thin element of C_{n} is a commutative n-shell.

(ii) A commutative n-shell is the same thing as a thin element of □ C_{n−1}.

(iii) Any commutative n-shell s has a unique thin filler (i.e. a thin element x ∈ C_{n} with ∂x = s).

Proof. (i) Let x ∈ C_{n} be thin. Then Ψx = ε_{1}z for some z ∈ C_{n−1}, so Nx = P x = z. Therefore N∂x = ∂Nx = ∂P x = P ∂x, by Lemma 1.3. (ii) If s ∈ □ C_{n−1} is commutative, then Ns = P s = u, say. The other faces of Ψs are all of the form ε_{1}v, where v is a face of u. These faces determine the shell Ψs and identify it as ε_{1}u, so s is thin. The converse is obvious. (iii) Let s be a commutative n-shell. Let u = Ns = P s and put a = ε_{1}u ∈ C_{n}. Then ∂a = ε_{1}u = Ψs, by (ii) and Lemma 1.3. By Theorem 1.4, there is a unique x ∈ C_{n} with ∂x = s and Ψx = a = ε_{1}u.

2.2. Proposition. Let C be a cubical ω-category (or a cubical m-category) with connections.

(i) Elements in C of the form ε_{i}c or Γ_{i}^{α}c are thin.

(ii) If a, b ∈ C_{n} are thin and c = a ∘_{i} b then c is thin.

Proof. (i) The thinness of ε_{i}c has been proved in Lemma 1.1. To prove thinness of Γ_{i}^{α}c, we first establish some formulae involving the ψ_{i} and Γ_{i}^{α}.

2.3. Lemma.

(i) ψ_{i}Γ_{i}^{α} = ε_{i}.

(ii) ψ_{j}Γ_{i}^{α} = Γ_{i}^{α}ψ_{j−1}, if j > i + 1.

(iii) ψ_{i}ψ_{i+1}Γ_{i}^{+}c = ε_{i}(Γ_{i}^{+}∂_{i+1}^{−}c ∘_{i+1} c). ψ_{i}ψ_{i+1}Γ_{i}^{−}c = ε_{i}(c ∘_{i+1} Γ_{i}^{−}∂_{i+1}^{+}c).

Proof. (i) ψ_{i}Γ_{i}^{+}c = Γ_{i}^{+}∂_{i+1}^{−}Γ_{i}^{+}c ∘_{i+1} Γ_{i}^{+}c ∘_{i+1} Γ_{i}^{−}∂_{i+1}^{+}Γ_{i}^{+}c.

= Γ_{i}^{+}ε_{i}∂_{α}^{−}c ∘_{i+1} Γ_{i}^{+}c ∘_{i+1} Γ_{i}^{−}c
= ε_{i+1}ε_{i}∂_{α}^{−}c ∘_{i+1} Γ_{i}^{+}c ∘_{i+1} Γ_{i}^{−}c
= Γ_{i}^{+}c ∘_{i+1} Γ_{i}^{−}c
= ε_{i}c.

The proof for Γ_{i}^{−} is similar. (ii) This is proved similarly by using standard laws from pp. 80-81 of [2]. (iii) When we try to compute ψ_{i+1}Γ_{i}^{+}, we are hindered by the lack of a simple law involving Γ_{i+1}^{−}Γ_{i}^{+}c. However, when we compute ψ_{i}ψ_{i+1}Γ_{i}^{+}c, this difficulty disappears. We note that if i + 1 < j then ψ_{i}(a ∘_{j} b) = ψ_{i}a ∘_{j} ψ_{i}b. So

ψ_{i}ψ_{i+1}Γ_{i}^{+}c = ψ_{i}(Γ_{i+1}^{+}∂_{i+2}^{−}Γ_{i}^{+}c ∘_{i+2} Γ_{i}^{+}c ∘_{i+2} Γ_{i+1}^{−}∂_{i+2}^{+}Γ_{i}^{+}c)
= ψ_{i}Γ_{i+1}^{+}∂_{i+2}^{−}Γ_{i}^{+}c ∘_{i+2} ψ_{i}Γ_{i}^{+}c ∘_{i+2} ψ_{i}Γ_{i+1}^{−}∂_{i+2}^{+}Γ_{i}^{+}c
= ψ_{i}Γ_{i+1}^{+}Γ_{i}^{+}∂_{i+1}^{−}c ∘_{i+2} ε_{i}c ∘_{i+2} ψ_{i}Γ_{i+1}^{−}Γ_{i}^{+}∂_{i+1}^{+}c.

We calculate the first and last terms separately:

ψ_{i}Γ_{i+1}^{+}Γ_{i}^{+} = ψ_{i}Γ_{i}^{+}Γ_{i}^{+} = ε_{i}Γ_{i}^{+} by (i)

and

ψ_{i}Γ_{i+1}^{−}Γ_{i}^{+}y = Γ_{i}^{+}∂_{i+1}^{−}Γ_{i+1}^{−}Γ_{i}^{+}y ∘_{i+1} Γ_{i+1}^{−}Γ_{i}^{+}y ∘_{i+1} Γ_{i}^{−}∂_{i+1}^{+}Γ_{i+1}^{−}Γ_{i}^{+}y
= (Γ_{i}^{+}Γ_{i}^{+}y ∘_{i+1} Γ_{i+1}^{−}Γ_{i}^{+}y) ∘_{i+1} Γ_{i}^{−}ε_{i+1}∂_{i+1}^{+}Γ_{i}^{+}y
= (Γ_{i+1}^{+}Γ_{i}^{+}y ∘_{i+1} Γ_{i+1}^{−}Γ_{i}^{+}y) ∘_{i+1} Γ_{i}^{−}ε_{i+1}y
= ε_{i+2}Γ_{i}^{+}y ∘_{i+1} ε_{i+2}Γ_{i}^{−}y
= ε_{i+2}(Γ_{i}^{+}y ∘_{i+1} Γ_{i}^{−}y)
= ε_{i+2}ε_{i}y.

Hence

ψ_{i}ψ_{i+1}Γ_{i}^{+}c = (ε_{i}Γ_{i}^{+}∂_{i+1}^{−}c ∘_{i+2} ε_{i}c) ∘_{i+2} ε_{i+2}ε_{i}∂_{i+1}^{+}c
= ε_{i}(Γ_{i}^{+}∂_{i+1}^{−}c ∘_{i+1} c).

The proof is similar for Γ_{i}^{−}. This completes the proof of Lemma 2.3. □

Returning to Proposition 2.2(i), the proof that Γ_{i}^{α}c (for c ∈ C_{n−1}) is thin is now straightforward:

ΨΓ_{i}^{α}c = ψ_{1}ψ_{2} … ψ_{n−1}Γ_{i}^{α}c

= ψ_{1}ψ_{2} … ψ_{i}ψ_{i+1}Γ_{i}^{α}cψ_{i+1} … ψ_{n−2}c, by 2.3(ii) = ψ_{1}ψ_{2} … ψ_{i−1}ε_{i}z for some z ∈ C_{n−1}, by 2.3(iii) = ε_{1}z by 1.1(i), as required.

(ii) To prove that composites of thin elements are thin we introduce a subsidiary definition: if x ∈ C_{n} and 1 ⩽ j ⩽ n − 1, we say that x is j-thin if ψ_{1}ψ_{2} … ψ_{j}x ∈ ε_{1}C_{n−1}. Thus, for elements of C_{n}, (n − 1)-thin means thin. We take “x ∈ C_{n} is 0-thin” to mean x ∈ ε_{1}C_{n−1}.

2.4. Lemma. If j ⩾ 1, then x is j-thin if and only if ψ_{j}x is (j − 1)-thin.

2.5. Lemma. If x = y ∘_{i} z in C_{n} and y, z ∈ ε_{j} C_{n−1} then x ∈ ε_{j} C_{n−1}.

(Note that in the case i = j, the hypotheses imply x = y = z.)

2.6. Lemma. If x ∈ ε_{k} C_{n−1}, then x is (k − 1)-thin.

Proof. ψ_{1}ψ_{2} … ψ_{k−1}x = ψ_{1}ψ_{2} … ψ_{k−1}ε_{k}y = ε_{1}y by (1.1)(i).

We now use induction on j to prove:

If a, b ∈ C_{n} are j-thin and c = a ∘_{i} b for some 1 ⩽ i ⩽ n then c is j-thin.  (*)

The case j = 0 is contained in Lemma 2.5.

Suppose that (*) is true for j = 0, 1, …, k − 1, where 1 ⩽ k ⩽ n − 1. We will deduce (*) for j = k.

Let c = a ∘_{i} b in C_{n} and assume that a and b are k-thin. We examine ψ_{k}c = ψ_{k}(a ∘_{i} b) in order to prove that it is (k − 1)-thin. Case 1: k < i−1 or k > i. In this case ψ_{k}c = ψ_{k}a ∘_{i} ψ_{k}b and ψ_{k}a and ψ_{k}b are (k−1)-thin. By induction hypothesis, ψ_{k}c is (k − 1)-thin and so c is k-thin by 2.4.

Case 2: k = i − 1. Then

ψ_{k}c = ψ_{i−1}(a ∘_{i} b)
=  a b  [figure: i−1; i]

=  \begin{aligned}& b \\ a \end{aligned}  [figure: i−1; i]

= ε_{i−1}u ψ_{i−1}b ψ_{i−1}a ε_{i−1}v [figure: i−1; i]

= ε_{k}u ψ_{k}b ψ_{k}a ε_{k}v [figure: i−1; i]

where u = ∂_{i−1}^{−}a, v = ∂_{i−1}^{+}b. Now ψ_{k}a and ψ_{k}b are (k − 1)-thin, by 2.4, and ε_{k}u, ε_{k}v are (k − 1)-thin, by 2.6. So ψ_{k}c, being a composite of these, is (k − 1)-thin by induction hypothesis. Hence c is k-thin. Case 3: k = i. This is similar using the formula

ψ_{k}c = ψ_{i}(a ∘_{i} b)

=  \begin{matrix}a \\ b \end{matrix}  [figure: i; i+1]

=  \begin{matrix}a \\ b \end{matrix}  [figure: i; i+1]

= ψ_{i}a ε_{i}s ε_{i}t  ψ_{i}b  [figure: i; i+1]

where s = ∂_{i+1}^{+}b, t = ∂_{i+1}^{−}a.

Thus, in all cases, c is k-thin, so the induction is complete. The case j = n − 1 of (*) completes the proof of Proposition 2.2.

2.7. Corollary. Let C be a cubical ω-category (or m-category) with connections.

(i) n-shells of the form ε_{i}c or Γ_{i}^{α}c for c ∈ C_{n−1} are commutative.

(ii) Composites of commutative shells are commutative.

From Proposition 2.2 we easily deduce

2.8. Theorem. Let C be a cubical ω-category (or m-category) with connections. An element c in C_{n} is thin if and only if it is a composite of elements of the form ε_{i}a or Γ_{i}^{α}a (a ∈ C_{n−1}).

Proof. Proposition 2.2 shows that any such composite is thin. For the converse, suppose that c ∈ C_{n} is thin. Then Ψc = ψ_{1}ψ_{2} … ψ_{n−1}c = ε_{1}z for some z ∈ C_{n−1}. Now we saw in the proof of Lemma 1.5 that any element x ∈ C_{n} can be written as a composite of ψ_{j}x and elements of type ε_{i}a, Γ_{i}^{α}a, namely

x =  x  [figure: j; j+1]

(here the dotted segments indicate that ψ_{j}x is first partitioned as shown and the 3 × 3 array is then completed so as to be composable.)

By iteration, x can be written as a composite of ε_{1}z and elements of type ε_{i}a, Γ_{i}^{α}a.

2.9. Corollary. An n-shell is commutative if and only if it can be written as a composite of shells of type ε_{i}a, Γ_{i}^{α}a, where a ∈ C_{n−1}.

Remark 1. It is not clear from the proof of 2.8 whether a thin element can always be written as a composite of an array of elements of type ε_{i}a, Γ_{i}^{α}a.

Remark 2. The particular folding map Ψ used to define thin elements depends on a number of choices and conventions. Theorem 2.8 shows that the notion of thinness is intrinsic and does not depend on these choices. Thus, for example, one might use Ψ′ = ψ_{n−1}ψ_{n−2} … ψ_{1} instead of Ψ but, by symmetry and Theorem 2.8, this would give the same concept. Similarly, the more complicated full folding operation Φ_{n} used in [2] gives the same concept of thinness (see Section 9 of that paper, especially Proposition 9.2 and Theorem 9.3). It is particularly reassuring that the concept of commutative shell is independent of the choice of foldings.

3. Thin structures and connections

We now consider cubical ω-categories (or m-categories) without the assumption of the extra structure of connections. Of course elements ε_{i}a and shells ε_{i}a exist for such ωcategories so, in view of Theorem 2.8, it is not surprising that there is a close relationship between existence of thin elements and the existence of connections. An equivalence between them in the 2-dimensional case was proved in [7]. We extend this result to all dimensions.

Let C be a cubical ω-category (or m-category). Suppose that C has connections Γ_{i}^{−}, Γ_{i}^{+} : C_{k−1} → C_{k}, defined for all k = 1, 2, …, n − 1, satisfying the usual laws, up to that dimension (see [2].) We aim to characterize possible definitions of thin elements in C_{n} without first introducing more connections there.

As mentioned in Section 1, the n-category (C_{0}, C_{1}, …, C_{n−1}, □ C_{n−1}) does have connections in dimension n as well as those in lower dimension, so we can define folding operations ψ_{1}, ψ_{2}, …, ψ_{n−1}, Ψ : □ C_{n−1} → □ C_{n−1}. As a result, the idea of a commutative n-shell is available, and we denote by □^{⧹} C_{n−1} the set of commutative n-shells in □ C_{n−1}. Clearly, by Corollary 2.7, (C_{0}, C_{1}, …, C_{n−1}, □^{⧹} C_{n−1}) is a sub-(cubical n-category with connections) of (C_{0}, C_{1}, …, C_{n−1}, □ C_{n−1}).

Definition A thin structure on C_{n} is a morphism

θ : (C_{0}, C_{1}, …, C_{n−1}, □^{⧹} C_{n−1}) → (C_{0}, C_{1}, …, C_{n−1}, C_{n})

of cubical n-categories which is the identity on C_{0}, C_{1}, …, C_{n−1}. Such a thin structure defines “thin” elements in C_{n}, namely elements of the form θ(s) for a commutative nshell s. Note that θ is necessarily injective on □^{⧹} C_{n−1} (because it preserves faces) and the image of θ must be a sub-n-category of C. Consequently, every commutative n-shell in C has a unique “thin” filler in C_{n}, and the composites of “thin” elements are “thin”. Furthermore, we may now define Γ_{i}^{α} : C_{n−1} → C_{n} by Γ_{i}^{α}a = θΓ_{i}^{α}a. Because θ preserves the lower dimensional Γ_{i}^{α}, ∘_{i}, ∂_{i}^{α} and ε_{i}, these newly defined Γ_{i}^{α} satisfy the required laws making (C_{0}, C_{1}, …, C_{n−1}, C_{n}) a cubical n-category with connections. The thin elements of C_{n} defined using these Γ_{i}^{α} are precisely the same as the “thin” elements defined by θ, because of Proposition 2.1.

Conversely, if we are given connections Γ_{i}^{α} : C_{n−1} → C_{n} making (C_{0}, C_{1}, …, C_{n}) a cubical n-category with connections, there is a unique thin structure θ with θΓ_{i}^{α}a = Γ_{i}^{α}a for all a ∈ C_{n−1}, α ∈ {+, −}, i ∈ {1, 2, …, n − 1}. This is because the morphism θ must map ε_{i}a to ε_{i}a and therefore, when it is defined on the Γ_{i}^{α}a, it is uniquely determined on all commutative shells, by Corollary 2.9. That such a θ exists is easily deduced from Proposition 2.1, and the thin elements defined by the two methods again coincide. Hence

3.1. Theorem. Let C = (C_{0}, C_{1}, …, C_{n}) be a cubical n-category and suppose that

(C_{0}, C_{1}, …, C_{n−1})

has the structure of cubical (n−1)-category with connections. Then there is a natural bijection between thin structures θ : □^{⧹} C_{n−1} → C_{n} and sets of connections Γ_{i}^{+}, Γ_{i}^{−} : C_{n−1} → C_{n} making (C_{0}, C_{1}, …, C_{n}) a cubical n-category with connections. The thin elements defined by the connections coincide in all cases with the thin elements defined by the corresponding θ.

Remark It would be useful to have a simple description of what is meant by a cubical T -complex, that is a weak thin structure (in all dimensions) on a cubical complex. The aim would be to impose axioms on the set of “thin” elements in each dimension which would be equivalent to the existence of a cubical ω-category structure with connections. A simple description does exist in the groupoid case (see [8, 5]), but seems to be more difficult in the category case.

Acknowledgement It is a pleasure to record my indebtedness to Ronnie Brown for many discussions on the content of this paper, for support of its typing through his Leverhulme Emeritus Fellowship, and for care to the production of the diagrams in xypic. Our long and fruitful collaboration in the area of higher dimensional algebra and homotopy has been a major source of inspiration to me over the years.

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