Dependencies

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 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 \(j\).

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.

Let \(\mathsf{C}\) be a single-sorted \(n\)-category (or \(\omega \)-category) and \(m\) be a natural number such that \(m {\lt} n\) ¹ . We define the underlying single-sorted \(m\)-category of \(\mathsf{C}\) as the single-sorted \(m\)-category \(\mathsf{C}_m\) defined on the set of \(m\)-cells of \(\mathsf{C}\), \(C_m\), with the corestriction of the source, target, identity, and composition operations of \(\mathsf{C}\) to \(C_m\).

Note that, by Lemmas 2.1, 2.2, and 2.3, these operations are well-defined. Also, the axioms of single-sorted categories for \(\mathsf{C}_m\) follow directly from the corresponding axioms for \(\mathsf{C}\).

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

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

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

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 \(m \in \mathcal{I}\). We define the underlying single-sorted \(m\)-functor of \(F\) as the single-sorted \(m\)-functor \(F_m \colon \mathsf{C}_m \to \mathsf{D}_m\) defined as the corestriction of \(F\) to the sets of \(m\)-cells \(C_m\) and \(D_m\).

Note that, by Lemma 2.5, this function is well-defined. Also, the axioms of single-sorted functors for \(F_m\) follow directly from the corresponding axioms for \(F\).

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

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 defined analogously to Definition 1.15, but using single-sorted \(\omega \)-categories and single-sorted \(\omega \)-functors instead.

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

Let \(\mathsf{C}\) be a single-sorted \(\mathcal{I}\)-category and \(m \in \mathcal{I}\). Then, for every \(f, g \in C_m\) (i.e., \(f\) and \(g\) are \(m\)-cells in \(\mathsf{C}\)) and every \(k \in \mathcal{I}\) such that \(k {\lt} m\), if the comoposition \(g \operatorname{\# }^k f\) is defined, then \(g \operatorname{\# }^k f\) is a \(m\)-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 \(F(f) \in D_m\), i.e., the image of \(m\)-cells under a single-sorted functor are \(m\)-cells.

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 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.

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 \(\operatorname{id}_C \colon C \to C\) is a single-sorted \(\mathcal{I}\)-functor from \(\mathsf{C}\) to itself.