Let \(\mathsf{C}\) be a single-sorted \(\mathcal{I}\)-category and \(m \in \mathcal{I}\). Then, for every \(k \in \mathcal{I}\) such that \(k {\lt} m\), the composition operation \(\operatorname{\# }^k\) birestricts to the set of \(m\)-cells \(C_m\).

Let \(\mathsf{C}\) and \(\mathsf{D}\) be single-sorted \(\mathcal{I}\)-categories, \(F \colon \mathsf{C} \to \mathsf{D}\) be a single-sorted functor, and \(m \in \mathcal{I}\). Then \(F\) birestricts to a function from \(C_m\) to \(D_m\).

Let \(\mathsf{C}\) be a single-sorted \(\mathcal{I}\)-category and \(m \in \mathcal{I}\). Then, for every \(k \in \mathcal{I}\) such that \(k {\lt} m\), the source operation \(\operatorname{sc}^k\) birestricts to the set of \(m\)-cells \(C_m\).

Let \(\mathsf{C}\) be a single-sorted \(\mathcal{I}\)-category and \(m \in \mathcal{I}\). Then, for every \(k \in \mathcal{I}\) such that \(k {\lt} m\), the target operation \(\operatorname{tg}^k\) birestricts to the set of \(m\)-cells \(C_m\).

Let \(\mathsf{C}\) be a single-sorted \(\mathcal{I}\)-category and \(k \in \mathcal{I}\). We say that a morphism \(f \in C\) is a \(k\)-cell or identity morphism at dimension \(k\) if and only if

We will denote the set of \(k\)-cells of \(\mathsf{C}\) by \(C_k\).

Let \(\mathsf{C}\) be a single-sorted \(\mathcal{I}\)-category and \(k \in \mathcal{I}\). A morphism \(f \in C\) is a target-based \(k\)-cell if and only if

A many-sorted \(\mathcal{I}\)-category is a pre-many-sorted \(\mathcal{I}\)-category \(\mathsf{C} = ((C_k)_{k \in \mathcal{I}}, (\operatorname{sc}^{(k,j)}, \operatorname{tg}^{(k,j)}, \operatorname{i}^{(k,j)}, \circ ^{(k,j)})_{k, j \in \mathcal{I}, \, j {\lt} k})\) satisfying, in addition to the axioms of pre-many-sorted categories, the following cross-dimensional axioms, for all indices \(k, j, i \in \mathcal{I}\) with \(i {\lt} j {\lt} k\):

1. For all \(f \in C_k\),

   \begin{align*} \operatorname{sc}^{(j,i)} (\operatorname{sc}^{(k,j)} (f)) = \operatorname{sc}^{(k,i)} (f), \qquad & \operatorname{sc}^{(j,i)} (\operatorname{tg}^{(k,j)} (f)) = \operatorname{sc}^{(k,i)} (f), \\ \operatorname{tg}^{(j,i)} (\operatorname{tg}^{(k,j)} (f)) = \operatorname{tg}^{(k,i)} (f), \qquad & \operatorname{tg}^{(j,i)} (\operatorname{sc}^{(k,j)} (f)) = \operatorname{tg}^{(k,i)} (f). \end{align*}

2. For all \(f, g \in C_k\), if \(g\) and \(f\) are \((k,i)\)-composable, then \(\operatorname{sc}^{(k,j)} (g)\) and \(\operatorname{sc}^{(k,j)} (f)\) and \(\operatorname{tg}^{(k,j)} (g)\) and \(\operatorname{tg}^{(k,j)} (f)\) are also \((j,i)\)-composable, and

   \begin{align*} \operatorname{sc}^{(k,j)} (g \circ ^{(k,i)} f) = \operatorname{sc}^{(k,j)} (g) \circ ^{(j,i)} \operatorname{sc}^{(k,j)} (f), \\ \operatorname{tg}^{(k,j)} (g \circ ^{(k,i)} f) = \operatorname{tg}^{(k,j)} (g) \circ ^{(j,i)} \operatorname{tg}^{(k,j)} (f). \end{align*}

3. For all \(f \in C_i\),

   \[ \operatorname{i}^{(k,j)} (\operatorname{i}^{(j,i)} (f)) = \operatorname{i}^{(k,i)} (f). \]

4. For all \(f, g \in C_j\), if \(g\) and \(f\) are \((j,i)\)-composable, then \(\operatorname{i}^{(k,j)} (g)\) and \(\operatorname{i}^{(k,j)} (f)\) are \((k,i)\)-composable, and

   \[ \operatorname{i}^{(k,j)} (g \circ ^{(j,i)} f) = \operatorname{i}^{(k,j)} (g) \circ ^{(k,i)} \operatorname{i}^{(k,j)} (f). \]

5. For all \(f_1, f_2, g_1, g_2 \in C_k\), suppose that:

   • \(g_2\) and \(g_1\) are \((k,j)\)-composable,

   • \(f_2\) and \(f_1\) are \((k,j)\)-composable,

   • \(g_2\) and \(f_2\) are \((k,i)\)-composable, and

   • \(g_1\) and \(f_1\) are \((k,i)\)-composable.

   Then, the morphisms \((g_2 \circ ^{(k,i)} f_2)\) and \((g_1 \circ ^{(k,i)} f_1)\) are \((k,j)\)-composable, and the morphisms \((g_2 \circ ^{(k,j)} g_1)\) and \((f_2 \circ ^{(k,j)} f_1)\) are \((k,i)\)-composable, and

   \[ (g_2 \circ ^{(k,i)} f_2) \circ ^{(k,j)} (g_1 \circ ^{(k,i)} f_1) = (g_2 \circ ^{(k,j)} g_1) \circ ^{(k,i)} (f_2 \circ ^{(k,j)} f_1). \]

   That is, composing first at dimension \(j\) and then at dimension \(i\) yields the same result as composing first at dimension \(i\) and then at dimension \(j\).

The category of many-sorted \(\mathcal{I}\)-categories, denoted by \(\mathcal{I}\mathsf{Cat}_{\mathrm{ms}}\), is defined as follows:

1. Its objects are many-sorted \(\mathcal{I}\)-categories.

2. Its morphisms are many-sorted \(\mathcal{I}\)-functors between many-sorted \(\mathcal{I}\)-categories.

3. Composition of morphisms is given by composition of many-sorted functors.

4. The identity morphism for each object is given by the identity many-sorted functor.

Let \((C_k)_{k \in \mathcal{I}}\) be a family of sets whose elements will be called \(k\)-morphisms. The required data for defining a many-sorted \(\mathcal{I}\)-category structure on \((C_k)_{k \in \mathcal{I}}\) consists of, for each pair of indices \(k, j \in \mathcal{I}\) with \(j {\lt} k\):

1. A map \(\operatorname{sc}^{(k,j)} \colon C_k \to C_j\), called the \((k,j)\)-source.

2. A map \(\operatorname{tg}^{(k,j)} \colon C_k \to C_j\), called the \((k,j)\)-target.

3. A map \(\operatorname{i}^{(k,j)} \colon C_j \to C_k\), called the \((k,j)\)-identity.

4. A partial binary operation \(\circ ^{(k,j)} \colon C_k \times C_k \rightharpoonup C_k\), called the \((k,j)\)-composition.

The composition \(g \circ ^{(k,j)} f\) is defined if and only if \(\operatorname{sc}^{(k,j)} (g) = \operatorname{tg}^{(k,j)} (f)\), in which case we say that \(g\) and \(f\) are \((k,j)\)-composable.

We will represent the above data as a tuple \(\mathsf{C} = ((C_k)_{k \in \mathcal{I}}, (\operatorname{sc}^{(k,j)}, \operatorname{tg}^{(k,j)}, \operatorname{i}^{(k,j)}, \circ ^{(k,j)})_{k, j \in \mathcal{I}, \, j {\lt} k})\).

Let \(\mathsf{C}\) be a many-sorted \(n\)-category on a family \((C_k)_{k \leq n}\), and let \(m \in \mathbb {N}\) with \(n {\lt} m\). The discrete finite family \((C^{{\gt}m}_k)_{k \leq m}\) is defined by

Let \(\mathsf{C}\) be a many-sorted \(n\)-category on a family \((C_k)_{k \leq n}\). The discrete omega family \((C^{{\gt}\omega }_k)_{k \in \mathbb {N}}\) is defined by

Let \(n, m \in \overline{\mathbb {N}}\) with \(n \leq m\). The many-sorted discretization functor \(\operatorname{\mathcal{D}}_{\mathrm{ms}}^{(n, m)}\) from \(n\mathsf{Cat}_{\mathrm{ms}}\) to \(m\mathsf{Cat}_{\mathrm{ms}}\) is the functor that maps every many-sorted \(n\)-category \(\mathsf{C}\) to its discrete \(m\)-category \(\mathsf{C}^{{\gt}m}\) and every many-sorted \(n\)-functor \(F \colon \mathsf{C} \to \mathsf{D}\) to its discrete \(m\)-functor \(F^{{\gt}m} \colon \mathsf{C}^{{\gt}m} \to \mathsf{D}^{{\gt}m}\). If \(n = m\), then \(\operatorname{\mathcal{D}}_{\mathrm{ms}}^{(n, n)}\) stands for the identity functor on \(n\mathsf{Cat}_{\mathrm{ms}}\).

Let \(\mathsf{C} = ((C_k)_{k \in \mathcal{I}}, (\operatorname{sc}^{(k,j)}, \operatorname{tg}^{(k,j)}, \operatorname{i}^{(k,j)}, \circ ^{(k,j)})_{k, j \in \mathcal{I}, \, j {\lt} k})\) and \(\mathsf{D} = ((D_k)_{k \in \mathcal{I}}, (\operatorname{sc}^{(k,j)}, \operatorname{tg}^{(k,j)}, \operatorname{i}^{(k,j)}, \circ ^{(k,j)})_{k, j \in \mathcal{I}, \, j {\lt} k})\) be many-sorted \(\mathcal{I}\)-categories. A many-sorted \(\mathcal{I}\)-functor from \(\mathsf{C}\) to \(\mathsf{D}\) is a family of maps \(F = (F_k \colon C_k \to D_k)_{k \in \mathcal{I}}\), written \(F \colon \mathsf{C} \to \mathsf{D}\), satisfying, for each pair of indices \(k, j \in \mathcal{I}\) with \(j {\lt} k\), the following preservation properties:

1. For all \(f \in C_k\),

   \[ F_j (\operatorname{sc}^{(k,j)} (f)) = \operatorname{sc}^{(k,j)} (F_k (f)), \qquad F_j (\operatorname{tg}^{(k,j)} (f)) = \operatorname{tg}^{(k,j)} (F_k (f)). \]

2. For all \(f \in C_j\),

   \[ F_k (\operatorname{i}^{(k,j)} (f)) = \operatorname{i}^{(k,j)} (F_j (f)). \]

3. For all \(f, g \in C_k\), if \(g\) and \(f\) are \((k,j)\)-composable, then \(F_k (g)\) and \(F_k (f)\) are also \((k,j)\)-composable, and

   \[ F_k (g \circ ^{(k,j)} f) = F_k (g) \circ ^{(k,j)} F_k (f). \]

Let \(\mathsf{C}, \mathsf{D}, \mathsf{E}\) be many-sorted \(\mathcal{I}\)-categories, and let \(F = (F_k)_{k \in \mathcal{I}} \colon \mathsf{C} \to \mathsf{D}\) and \(G = (G_k)_{k \in \mathcal{I}} \colon \mathsf{D} \to \mathsf{E}\) be many-sorted \(\mathcal{I}\)-functors. Then, the family \(G \circ F = (G_k \circ F_k)_{k \in \mathcal{I}} \colon \mathsf{C} \to \mathsf{E}\) is also a many-sorted \(\mathcal{I}\)-functor.

Let \(\mathsf{C}\) be a many-sorted \(\mathcal{I}\)-category. Then, the family of identity functions \(\operatorname{id}_{\mathsf{C}} = (\operatorname{id}_{C_k})_{k \in \mathcal{I}}\), written \(\operatorname{id}_{\mathsf{C}} \colon \mathsf{C} \to \mathsf{C}\), is a many-sorted \(\mathcal{I}\)-functor from \(\mathsf{C}\) to itself.

Given a natural number \(n \in \mathbb {N}\), a many-sorted \(n\)-category is a many-sorted category indexed by the finite set \(n + 1 = \left\{ 0, 1, \ldots , n \right\} \).

Given a natural number \(n \in \mathbb {N}\), the category of many-sorted \(n\)-categories, denoted by \(n\mathsf{Cat}_{\mathrm{ms}}\), is the particular case of Definition 5.13 where \(\mathcal{I}\) is the finite set \(n + 1 = \left\{ 0, 1, \ldots , n \right\} \). Its objects are many-sorted \(n\)-categories and its morphisms are many-sorted \(n\)-functors.

Let \(\mathsf{C}\) be a many-sorted \(n\)-category on a family \((C_k)_{k \leq n}\), and let \(m \in \mathbb {N}\) with \(n {\lt} m\). The discrete \(m\)-category of \(\mathsf{C}\) is the many-sorted \(m\)-category \(\mathsf{C}^{{\gt}m}\) on the discrete family \((C^{{\gt}m}_k)_{k \leq m}\) of Definition 6.1, where the operations at each pair \((k, j)\) with \(j {\lt} k\) are defined as follows:

• If \(k \leq n\) (and hence \(j {\lt} k \leq n\)): the source \(\operatorname{sc}^{(k,j)}\), target \(\operatorname{tg}^{(k,j)}\), identity \(\operatorname{i}^{(k,j)}\), and composition \(\circ ^{(k,j)}\) are those of \(\mathsf{C}\).

• If \(k {\gt} n\) and \(j \leq n\): we have \(C^{{\gt}m}_k = C_n\) and \(C^{{\gt}m}_j = C_j\), so the source and target maps \(\operatorname{sc}^{(k,j)}, \operatorname{tg}^{(k,j)} \colon C_n \to C_j\) are defined as \(\operatorname{sc}^{(n,j)}\) and \(\operatorname{tg}^{(n,j)}\) respectively; the identity map \(\operatorname{i}^{(k,j)} \colon C_j \to C_n\) is defined as \(\operatorname{i}^{(n,j)}\); and composition \(\circ ^{(k,j)}\) is defined as \(\circ ^{(n,j)}\).

• If \(j {\gt} n\) (and hence \(k {\gt} j {\gt} n\)): we have \(C^{{\gt}m}_k = C^{{\gt}m}_j = C_n\). The source and target maps are the identity on \(C_n\), the identity map \(\operatorname{i}^{(k,j)}\) is also the identity, and composition is defined only between equal morphisms and returns the morphism itself.

Let \(\mathsf{C}\) be a many-sorted \(n\)-category on a family \((C_k)_{k \leq n}\). The discrete \(\omega \)-category of \(\mathsf{C}\) is the many-sorted \(\omega \)-category \(\mathsf{C}^{{\gt}\omega }\) on the discrete omega family \((C^{{\gt}\omega }_k)_{k \in \mathbb {N}}\) of Definition 6.3, defined by the same case distinction as in Definition 6.2, now with the indices \(k\) and \(j\) ranging over all of \(\mathbb {N}\).

Let \(\mathsf{C}\) be a many-sorted \(n\)-category on a family \((C_k)_{k \leq n}\), and let \(m \in \mathbb {N}\) with \(m {\lt} n\). The underlying \(m\)-category of \(\mathsf{C}\) is the many-sorted \(m\)-category \(\mathsf{C}^{{\lt}m}\) on the restricted family \((C_k)_{k \leq m}\), with the source, target, identity, and composition operations at each pair \((k, j)\) with \(j {\lt} k \leq m\) being those of \(\mathsf{C}\).

Given a natural number \(n \in \mathbb {N}\), a many-sorted \(n\)-functor is a many-sorted functor between many-sorted \(n\)-categories.

Let \(\mathsf{C}\) and \(\mathsf{D}\) be many-sorted \(n\)-categories, and let \(F = (F_k)_{k \leq n} \colon \mathsf{C} \to \mathsf{D}\) be a many-sorted \(n\)-functor. For any \(m \in \mathbb {N}\) with \(n {\lt} m\), the family \((F^{{\gt}m}_k)_{k \leq m}\) defined by

is a many-sorted \(m\)-functor \(F^{{\gt}m} \colon \mathsf{C}^{{\gt}m} \to \mathsf{D}^{{\gt}m}\).

Let \(\mathsf{C}\) and \(\mathsf{D}\) be many-sorted \(n\)-categories, and let \(F = (F_k)_{k \leq n} \colon \mathsf{C} \to \mathsf{D}\) be a many-sorted \(n\)-functor. The family \((F^{{\gt}\omega }_k)_{k \in \mathbb {N}}\) defined by

is a many-sorted \(\omega \)-functor \(F^{{\gt}\omega } \colon \mathsf{C}^{{\gt}\omega } \to \mathsf{D}^{{\gt}\omega }\).

Let \(\mathsf{C}\) and \(\mathsf{D}\) be many-sorted \(n\)-categories, and let \(F = (F_k)_{k \leq n} \colon \mathsf{C} \to \mathsf{D}\) be a many-sorted \(n\)-functor. For any \(m \in \mathbb {N}\) with \(m {\lt} n\), the restricted family \((F_k)_{k \leq m}\) is a many-sorted \(m\)-functor \(F^{{\lt}m} \colon \mathsf{C}^{{\lt}m} \to \mathsf{D}^{{\lt}m}\).

A many-sorted \(\omega \)-category is a many-sorted category indexed by the set of natural numbers \(\mathbb {N}\).

The category of many-sorted \(\omega \)-categories, denoted by \(\omega \mathsf{Cat}_{\mathrm{ms}}\), is the particular case of Definition 5.13 where \(\mathcal{I}\) is the set of natural numbers \(\mathbb {N}\). Its objects are many-sorted \(\omega \)-categories and its morphisms are many-sorted \(\omega \)-functors.

Let \(\mathsf{C}\) be a many-sorted \(\omega \)-category on a family \((C_k)_{k \in \mathbb {N}}\), and let \(m \in \mathbb {N}\). The underlying \(m\)-category of \(\mathsf{C}\) is the many-sorted \(m\)-category \(\mathsf{C}^{{\lt}m}\) on the restricted family \((C_k)_{k \leq m}\), defined by the same construction as in Definition 7.1.

A many-sorted \(\omega \)-functor is a many-sorted functor between many-sorted \(\omega \)-categories.

Let \(\mathsf{C}\) and \(\mathsf{D}\) be many-sorted \(\omega \)-categories, and let \(F = (F_k)_{k \in \mathbb {N}} \colon \mathsf{C} \to \mathsf{D}\) be a many-sorted \(\omega \)-functor. For any \(m \in \mathbb {N}\), the restricted family \((F_k)_{k \leq m}\) is a many-sorted \(m\)-functor \(F^{{\lt}m} \colon \mathsf{C}^{{\lt}m} \to \mathsf{D}^{{\lt}m}\).

Let \(n, m \in \overline{\mathbb {N}}\) with \(m \leq n\). The many-sorted underlying functor \(\operatorname{\mathcal{U}}_{\mathrm{ms}}^{(n, m)}\) from \(n\mathsf{Cat}_{\mathrm{ms}}\) to \(m\mathsf{Cat}_{\mathrm{ms}}\) is the functor that maps every many-sorted \(n\)-category \(\mathsf{C}\) to its underlying \(m\)-category \(\mathsf{C}^{{\lt}m}\) and every many-sorted \(n\)-functor \(F \colon \mathsf{C} \to \mathsf{D}\) to its underlying \(m\)-functor \(F^{{\lt}m} \colon \mathsf{C}^{{\lt}m} \to \mathsf{D}^{{\lt}m}\). If \(n = m\), then \(\operatorname{\mathcal{U}}_{\mathrm{ms}}^{(n, n)}\) stands for the identity functor on \(n\mathsf{Cat}_{\mathrm{ms}}\).

A pre-many-sorted \(\mathcal{I}\)-category is a tuple

satisfying, for each pair of indices \(k, j \in \mathcal{I}\) with \(j {\lt} k\), the following axioms:

1. For all \(f, g \in C_k\), if \(g\) and \(f\) are \((k,j)\)-composable, then

   \begin{align*} \operatorname{sc}^{(k,j)} (g \circ ^{(k,j)} f) = \operatorname{sc}^{(k,j)} (f), \qquad & \operatorname{tg}^{(k,j)} (g \circ ^{(k,j)} f) = \operatorname{tg}^{(k,j)} (g). \end{align*}

2. For all \(f \in C_j\),

   \[ \operatorname{sc}^{(k,j)} (\operatorname{i}^{(k,j)} (f)) = f, \qquad \operatorname{tg}^{(k,j)} (\operatorname{i}^{(k,j)} (f)) = f. \]

3. For all \(f \in C_k\), the morphisms \(f\) and \(\operatorname{i}^{(k,j)} (\operatorname{sc}^{(k,j)} (f))\) are \((k,j)\)-composable and

   \[ f \circ ^{(k,j)} \operatorname{i}^{(k,j)} (\operatorname{sc}^{(k,j)} (f)) = f. \]

   Similarly, \(\operatorname{i}^{(k,j)} (\operatorname{tg}^{(k,j)} (f))\) and \(f\) are \((k,j)\)-composable and

   \[ \operatorname{i}^{(k,j)} (\operatorname{tg}^{(k,j)} (f)) \circ ^{(k,j)} f = f. \]

4. For all \(f, g, h \in C_k\), if \(h\) and \(g\) are \((k,j)\)-composable, and \(g\) and \(f\) are \((k,j)\)-composable, then \(h \circ ^{(k,j)} g\) and \(f\) and \(h\) and \(g \circ ^{(k,j)} f\) are \((k,j)\)-composable, and

   \[ (h \circ ^{(k,j)} g) \circ ^{(k,j)} f = h \circ ^{(k,j)} (g \circ ^{(k,j)} f). \]

Any pre-many-sorted \(1\)-category is, in fact, a many-sorted \(1\)-category. Since the index set \(1 + 1 = \left\{ 0, 1 \right\} \) has exactly two elements, there are no triples of distinct indices \(i {\lt} j {\lt} k\), so all cross-dimensional axioms are vacuously satisfied.

A pre-single-sorted \(\mathcal{I}\)-category is a tuple

satisfying, for each index \(k \in \mathcal{I}\), the following axioms:

1. For all \(f \in C\),

   \begin{align*} \operatorname{sc}^k (\operatorname{sc}^k (f)) = \operatorname{sc}^k (f), \qquad & \operatorname{tg}^k (\operatorname{sc}^k (f)) = \operatorname{sc}^k (f), \\ \operatorname{sc}^k (\operatorname{tg}^k (f)) = \operatorname{tg}^k (f), \qquad & \operatorname{tg}^k (\operatorname{tg}^k (f)) = \operatorname{tg}^k (f). \end{align*}

2. For all \(f, g \in C\), if \(g\) and \(f\) are composable, then

   \begin{align*} \operatorname{sc}^k (g \operatorname{\# }^k f) = \operatorname{sc}^k (f), \qquad & \operatorname{tg}^k (g \operatorname{\# }^k f) = \operatorname{tg}^k (g). \end{align*}

3. For all \(f \in C\), the morphisms \(f\) and \(\operatorname{sc}^k (f)\) are composable and

   \[ f \operatorname{\# }^k \operatorname{sc}^k (f) = f. \]

4. For all \(f \in C\), the morphisms \(\operatorname{tg}^k (f)\) and \(f\) are composable and

   \[ \operatorname{tg}^k (f) \operatorname{\# }^k f = f. \]

5. For all \(f, g, h \in C\), if \(h\) and \(g\) are composable, and \(g\) and \(f\) are composable, then \(h \operatorname{\# }^k g\) and \(f\) and \(h\) and \(g \operatorname{\# }^k f\) are composable, and

   \[ (h \operatorname{\# }^k g) \operatorname{\# }^k f = h \operatorname{\# }^k (g \operatorname{\# }^k f). \]

Any pre-single-sorted \(1\)-category is, in fact, a single-sorted \(1\)-category. Since the index set \(1 = \left\{ 0 \right\} \) has exactly one element, there are no pairs of distinct indices \(j {\lt} k\), so all cross-dimensional axioms are vacuously satisfied.

A single-sorted \(\mathcal{I}\)-category is a pre-single-sorted \(\mathcal{I}\)-category \(\mathsf{C} = (C, (\operatorname{sc}^k, \operatorname{tg}^k, \operatorname{\# }^k)_{k \in \mathcal{I}})\) satisfying, in addition to the axioms of pre-single-sorted categories, the following cross-dimensional axioms, for all indices \(k, j \in \mathcal{I}\) with \(j {\lt} k\):

1. For all \(f \in C\),

   \begin{align*} \operatorname{sc}^k (\operatorname{sc}^j (f)) = \operatorname{sc}^j (f), \qquad & \operatorname{sc}^j (\operatorname{sc}^k (f)) = \operatorname{sc}^j (f), \\ \operatorname{sc}^j (\operatorname{tg}^k (f)) = \operatorname{sc}^j (f), \qquad & \operatorname{tg}^k (\operatorname{tg}^j (f)) = \operatorname{tg}^j (f), \\ \operatorname{tg}^j (\operatorname{tg}^k (f)) = \operatorname{tg}^j (f), \qquad & \operatorname{tg}^j (\operatorname{sc}^k (f)) = \operatorname{tg}^j (f). \end{align*}

2. For all \(f, g \in C\), if \(g\) and \(f\) are composable at dimension \(j\), then \(\operatorname{sc}^k (g)\) and \(\operatorname{sc}^k (f)\) and \(\operatorname{tg}^k (g)\) and \(\operatorname{tg}^k (f)\) are also composable at dimension \(j\), and

   \begin{align*} \operatorname{sc}^k (g \operatorname{\# }^j f) = \operatorname{sc}^k (g) \operatorname{\# }^j \operatorname{sc}^k (f), \\ \operatorname{tg}^k (g \operatorname{\# }^j f) = \operatorname{tg}^k (g) \operatorname{\# }^j \operatorname{tg}^k (f). \end{align*}

3. For all \(f_1, f_2, g_1, g_2 \in C\), suppose that:

   • \(g_2\) and \(f_2\) are composable at dimension \(j\),

   • \(g_1\) and \(f_1\) are composable at dimension \(j\),

   • \(g_2\) and \(g_1\) are composable at dimension \(k\), and

   • \(f_2\) and \(f_1\) are composable at dimension \(k\).

   Then, the morphisms \((g_2 \operatorname{\# }^j f_2)\) and \((g_1 \operatorname{\# }^j f_1)\) are composable at dimension \(k\), and the morphisms \((g_2 \operatorname{\# }^k g_1)\) and \((f_2 \operatorname{\# }^k f_1)\) are composable at dimension \(j\), and

   \[ (g_2 \operatorname{\# }^j f_2) \operatorname{\# }^k (g_1 \operatorname{\# }^j f_1) = (g_2 \operatorname{\# }^k g_1) \operatorname{\# }^j (f_2 \operatorname{\# }^k f_1). \]

   That is, composing first at dimension \(j\) and then at dimension \(k\) yields the same result as composing first at dimension \(k\) and then at dimension \(j\).

The category of single-sorted \(\mathcal{I}\)-categories, denoted by \(\mathcal{I}\mathsf{Cat}\), is defined as follows:

1. Its objects are single-sorted \(\mathcal{I}\)-categories.

2. Its morphisms are single-sorted \(\mathcal{I}\)-functors between single-sorted \(\mathcal{I}\)-categories.

3. Composition of morphisms is given by composition of single-sorted functors.

4. The identity morphism for each object is given by the identity single-sorted functor.

Let \(C\) be a set whose elements will be called morphisms. The required data for defining a single-sorted \(\mathcal{I}\)-category structure on \(C\) consists of:

1. A unary operation \(\operatorname{sc}^k \colon C \to C\) for each index \(k \in \mathcal{I}\), called the source at dimension \(k\).

2. A unary operation \(\operatorname{tg}^k \colon C \to C\) for each index \(k \in \mathcal{I}\), called the target at dimension \(k\).

3. A partial binary operation \(\operatorname{\# }^k \colon C \times C \rightharpoonup C\) for each index \(k \in \mathcal{I}\), called the composition at dimension \(k\).

The composition \(g \operatorname{\# }^k f\) is defined if and only if \(\operatorname{sc}^k (g) = \operatorname{tg}^k (f)\), in which case we say that \(g\) and \(f\) are composable at dimension \(k\).

We will represent the above data as a tuple \(\mathsf{C} = (C, (\operatorname{sc}^k, \operatorname{tg}^k, \operatorname{\# }^k)_{k \in \mathcal{I}})\).

Let \(n, m \in \overline{\mathbb {N}}\) with \(n \leq m\). The discretization functor \(\operatorname{\mathcal{D}}^{(n, m)}\) from \(n\mathsf{Cat}\) to \(m\mathsf{Cat}\) is the functor that maps every single-sorted \(n\)-category \(\mathsf{C}\) to its discrete \(m\)-category \(\mathsf{C}^{{\gt}m}\) and every single-sorted \(n\)-functor \(F \colon \mathsf{C} \to \mathsf{D}\) to its discrete \(m\)-functor \(F^{{\gt}m} \colon \mathsf{C}^{{\gt}m} \to \mathsf{D}^{{\gt}m}\). If \(n = m\), then \(\operatorname{\mathcal{D}}^{(n, n)}\) stands for the identity functor on \(n\mathsf{Cat}\).

Let \(\mathsf{C} = (C, (\operatorname{sc}^k, \operatorname{tg}^k, \operatorname{\# }^k)_{k \in \mathcal{I}})\) and \(\mathsf{D} = (D, (\operatorname{sc}^k, \operatorname{tg}^k, \operatorname{\# }^k)_{k \in \mathcal{I}})\) be single-sorted \(\mathcal{I}\)-categories. A single-sorted \(\mathcal{I}\)-functor from \(\mathsf{C}\) to \(\mathsf{D}\) is a map \(F \colon C \to D\), written \(F \colon \mathsf{C} \to \mathsf{D}\), satisfying, for each index \(k \in \mathcal{I}\), the following preservation properties:

1. For all \(f \in C\),

   \[ F (\operatorname{sc}^k (f)) = \operatorname{sc}^k (F (f)), \qquad F (\operatorname{tg}^k (f)) = \operatorname{tg}^k (F (f)). \]

2. For all \(f, g \in C\), if \(g\) and \(f\) are composable at dimension \(k\), then \(F (g)\) and \(F (f)\) are also composable at dimension \(k\), and

   \[ F (g \operatorname{\# }^k f) = F (g) \operatorname{\# }^k F (f). \]

Let \(\mathsf{C}, \mathsf{D}, \mathsf{E}\) be single-sorted \(\mathcal{I}\)-categories, and let \(F \colon \mathsf{C} \to \mathsf{D}\) and \(G \colon \mathsf{D} \to \mathsf{E}\) be single-sorted \(\mathcal{I}\)-functors. Then, the composition \(G \circ F \colon \mathsf{C} \to \mathsf{E}\) is also a single-sorted \(\mathcal{I}\)-functor.

Let \(\mathsf{C}\) be a single-sorted \(\mathcal{I}\)-category. Then, the identity function \(\operatorname{id}_C \colon C \to C\), written \(\operatorname{id}_{\mathsf{C}} \colon \mathsf{C} \to \mathsf{C}\), is a single-sorted \(\mathcal{I}\)-functor from \(\mathsf{C}\) to itself.

Given a natural number \(n \in \mathbb {N}\), a single-sorted \(n\)-category is a single-sorted category indexed by the finite set \(n = \left\{ 0, 1, \ldots , n-1 \right\} \).

Given a natural number \(n \in \mathbb {N}\), the category of single-sorted \(n\)-categories, denoted by \(n\mathsf{Cat}\), is the particular case of Definition 2.15 where \(\mathcal{I}\) is the finite set \(n\). Its objects are single-sorted \(n\)-categories and its morphisms are single-sorted \(n\)-functors.

Let \(\mathsf{C}\) be a single-sorted \(n\)-category, and let \(m \in \mathbb {N}\) with \(n {\lt} m\). The discrete \(m\)-category of \(\mathsf{C}\) is the single-sorted \(m\)-category \(\mathsf{C}^{{\gt}m}\) defined on the same underlying set \(C\), where ¹ :

• For \(k {\lt} n\), the source, target, and composition at dimension \(k\) are those of \(\mathsf{C}\).

• For \(n \leq k {\lt} m\), the source and target at dimension \(k\) are the identity, and composition is defined only between equal morphisms, returning the morphism itself.

Let \(\mathsf{C}\) be a single-sorted \(n\)-category. The discrete \(\omega \)-category of \(\mathsf{C}\) is the single-sorted \(\omega \)-category \(\mathsf{C}^{{\gt}\omega }\) defined by the same case distinction as in Definition 3.1, now with \(k\) ranging over all of \(\mathbb {N}\).

Let \(\mathsf{C}\) be a single-sorted \(n\)-category, and let \(m \in \mathbb {N}\) with \(m {\lt} n\). The underlying \(m\)-category of \(\mathsf{C}\) is the single-sorted \(m\)-category \(\mathsf{C}^{{\lt}m}\) defined on the set of \(m\)-cells \(C_m\), with the source, target, and composition operations given by the birestrictions of those of \(\mathsf{C}\) to \(C_m\).

Given a natural number \(n \in \mathbb {N}\), a single-sorted \(n\)-functor is a single-sorted functor between single-sorted \(n\)-categories.

Let \(\mathsf{C}\) and \(\mathsf{D}\) be single-sorted \(n\)-categories, and let \(F \colon \mathsf{C} \to \mathsf{D}\) be a single-sorted \(n\)-functor. For any \(m \in \mathbb {N}\) with \(n {\lt} m\), the same underlying map \(F\) defines a single-sorted \(m\)-functor \(F^{{\gt}m} \colon \mathsf{C}^{{\gt}m} \to \mathsf{D}^{{\gt}m}\).

Let \(\mathsf{C}\) and \(\mathsf{D}\) be single-sorted \(n\)-categories, and let \(F \colon \mathsf{C} \to \mathsf{D}\) be a single-sorted \(n\)-functor. The same underlying map \(F\) defines a single-sorted \(\omega \)-functor \(F^{{\gt}\omega } \colon \mathsf{C}^{{\gt}\omega } \to \mathsf{D}^{{\gt}\omega }\).

Let \(\mathsf{C}\) and \(\mathsf{D}\) be single-sorted \(n\)-categories, and let \(F \colon \mathsf{C} \to \mathsf{D}\) be a single-sorted \(n\)-functor. For any \(m \in \mathbb {N}\) with \(m {\lt} n\), the birestriction of \(F\) to the sets of \(m\)-cells defines a single-sorted \(m\)-functor \(F^{{\lt}m} \colon \mathsf{C}^{{\lt}m} \to \mathsf{D}^{{\lt}m}\).

A single-sorted \(\omega \)-category is a single-sorted category indexed by the set of natural numbers \(\mathbb {N}\), satisfying and additional axiom: for all \(f \in C\), there exists a dimension \(k \in \mathbb {N}\) such that \(f\) is a \(k\)-cell.

The category of single-sorted \(\omega \)-categories, denoted by \(\omega \mathsf{Cat}\), is the particular case ¹ of Definition 2.15 where \(\mathcal{I}\) is the set of natural numbers \(\mathbb {N}\). Its objects are single-sorted \(\omega \)-categories and its morphisms are single-sorted \(\omega \)-functors.

Let \(\mathsf{C}\) be a single-sorted \(\omega \)-category, and let \(m \in \mathbb {N}\). The underlying \(m\)-category of \(\mathsf{C}\) is the single-sorted \(m\)-category \(\mathsf{C}^{{\lt}m}\) defined by the same construction as in Definition 4.7.

A single-sorted \(\omega \)-functor is a single-sorted functor between single-sorted \(\omega \)-categories.

Let \(\mathsf{C}\) and \(\mathsf{D}\) be single-sorted \(\omega \)-categories, and let \(F \colon \mathsf{C} \to \mathsf{D}\) be a single-sorted \(\omega \)-functor. For any \(m \in \mathbb {N}\), the birestriction of \(F\) to the sets of \(m\)-cells defines a single-sorted \(m\)-functor \(F^{{\lt}m} \colon \mathsf{C}^{{\lt}m} \to \mathsf{D}^{{\lt}m}\).

Let \(n, m \in \overline{\mathbb {N}}\) with \(m \leq n\). The underlying functor \(\operatorname{\mathcal{U}}^{(n, m)}\) from \(n\mathsf{Cat}\) to \(m\mathsf{Cat}\) is the functor that maps every single-sorted \(n\)-category \(\mathsf{C}\) to its underlying \(m\)-category \(\mathsf{C}^{{\lt}m}\) and every single-sorted \(n\)-functor \(F \colon \mathsf{C} \to \mathsf{D}\) to its underlying \(m\)-functor \(F^{{\lt}m} \colon \mathsf{C}^{{\lt}m} \to \mathsf{D}^{{\lt}m}\). If \(n = m\), then \(\operatorname{\mathcal{U}}^{(n, n)}\) stands for the identity functor on \(n\mathsf{Cat}\).

Let \(\mathsf{C}\) be a single-sorted \(\mathcal{I}\)-category and \(m \in \mathcal{I}\). Then, for every \(f, g \in C_m\) and every \(k \in \mathcal{I}\) such that \(k {\lt} m\), if \(f\) and \(g\) are composable at dimension \(k\), then \(g \operatorname{\# }^k f\) is an \(m\)-cell in \(\mathsf{C}\).

Let \(\mathsf{C}\) and \(\mathsf{D}\) be single-sorted \(\mathcal{I}\)-categories, \(F \colon \mathsf{C} \to \mathsf{D}\) be a single-sorted functor, and \(m \in \mathcal{I}\). Then, for every \(f \in C_m\) we have that \(F(f) \in D_m\).

Let \(\mathsf{C}\) be a single-sorted \(\mathcal{I}\)-category and \(m \in \mathcal{I}\). Then, for every \(f \in \mathsf{C}\) and every \(k \in \mathcal{I}\) such that \(k {\lt} m\), we have that \(\operatorname{sc}^k (f)\) is a \(m\)-cell in \(\mathsf{C}\).

Let \(\mathsf{C}\) be a single-sorted \(\mathcal{I}\)-category and \(m \in \mathcal{I}\). Then, for every \(f \in \mathsf{C}\) and every \(k \in \mathcal{I}\) such that \(k {\lt} m\), we have that \(\operatorname{tg}^k (f)\) is a \(m\)-cell in \(\mathsf{C}\).

Let \(\mathsf{C}\) be a pre-many-sorted \(\mathcal{I}\)-category and let \(j {\lt} k\) in \(\mathcal{I}\). Then the identity map \(\operatorname{i}^{(k,j)} \colon C_j \to C_k\) is injective.

Let \(\mathsf{C}\) be a single-sorted \(\mathcal{I}\)-category and \(k \in \mathcal{I}\). A morphism \(f \in C\) is a \(k\)-cell if and only if it is a target-based \(k\)-cell.

It follows that the set of \(k\)-cells \(C_k\) is equal to the set of target-based \(k\)-cells.