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:

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

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

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

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

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

A pre-single-sorted \(\mathcal{I}\)-category is a tuple \(\mathsf{C} = (C, (\operatorname{sc}^i, \operatorname{tg}^i, \operatorname{\# }^i)_{i \in \mathcal{I}})\) satisfying, for each index \(i \in \mathcal{I}\), the following axioms:

• For all \(f \in C\),

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

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

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

• For all \(f \in C\), the morphisms \(f\) and \(\operatorname{sc}^i (f)\) are composable and

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

• For all \(f \in C\), the morphisms \(\operatorname{tg}^i (f)\) and \(f\) are composable and

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

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

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

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

• For all \(f \in C\) and all indices \(j, k \in \mathcal{I}\) with \(j {\lt} k\),

  \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*}

• For all \(f, g \in C\) and all indices \(j, k \in \mathcal{I}\) with \(j {\lt} k\), 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*}

• For all \(f_1, f_2, g_1, g_2 \in C\) and all indices \(j, k \in \mathcal{I}\) with \(j {\lt} k\), 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 \(j\).

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

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 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.

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

• For all \(f \in C\),

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

• 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

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

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

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

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

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

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

• Composition of morphisms is given by composition of single-sorted functors, which is well defined by Theorem 1.11.

• The identity morphism for each object is given by the identity single-sorted functor, which is well defined by Theorem 1.12.

The category of single-sorted \(\omega \)-categories, denoted by \(\omega \mathsf{Cat}\), is defined analogously to Definition 1.15, but using single-sorted \(\omega \)-categories and single-sorted \(\omega \)-functors instead.