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 \(\mathsf{C} = (C, (\operatorname{sc}^k, \operatorname{tg}^k, \operatorname{\# }^k)_{k \in \mathcal{I}})\).

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

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

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 \(j\).

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

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.

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 \(\mathsf{C} = (C, (\operatorname{sc}^k, \operatorname{tg}^k, \operatorname{\# }^k)_{k \in \mathcal{I}})\) 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

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

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.