We set \(\mathrm{Cat}_{(\infty, {0})} \coloneqq \mathcal S\) to be the \(\infty\)-category of small spaces, and equip it with its Cartesian presentably symmetric monoidal structure. We inductively define the Cartesian20 presentably symmetric monoidal \(\infty\)-category of \((\infty, k)\)-categories \(\mathrm{Cat}_{(\infty, {k})}\coloneqq\mathrm{Cat}[\mathrm{Cat}_{(\infty, {k-1})}]\).
5.1 Basic notions in \((\infty,k)\)-category theory[009A]
Throughout, we will use the theory of enriched \(\infty\)-categories developed in [GH15], see also subsection A.10.
In [Hau15, Thm. 1.2], Haugseng showed that the \(\infty\)-category \(\mathrm{Cat}_{(\infty, {k})}\) from definition 5.1.1 satisifes the axioms of Barwick and Schommer-Pries [BS21] and hence is equivalent to most other known models of the \(\infty\)-category of \((\infty,k)\)-categories.
Consider the diagram 
of adjunctions, where \(i\) denotes the fully faithful inclusion of spaces as \(\infty\)-groupoids. Since all these functors preserve finite products21 , they are all symmetric monoidal. By applying \(\mathrm{Cat}[-]\) iteratively, we obtain an analogous diagram 
of symmetric monoidal adjoint functors for any \(k \geq 0\) (with \(i_{k+1}\) fully faithful). Thereafter, for any \(j \geq k \geq 0\) we obtain an analogous diagram 
of symmetric monoidal adjoint functors by composition.
In diagram ([009E]), we refer to \(|-|_k\) as the \((\infty,k)\)-category completion functor and to \(\iota_k\) as the maximal sub-\((\infty,k)\)-category functor.22 For brevity, we may omit the fully faithful inclusion functor \(i_j\) from our notation, implicitly considering an \((\infty,k)\)-category as an \((\infty,j)\)-category with no noninvertible \(i\)-morphisms for any \(i > k\).
For any \(j \geq k \geq 0\), the inclusion \(\mathrm{Cat}_{(\infty, {k})} \xhookrightarrow{i_j} \mathrm{Cat}_{(\infty, {j})}\) identifies \(\mathrm{Cat}_{(\infty, {k})}\) as the full subcategory of \(\mathrm{Cat}_{(\infty, {j})}\) on those \((\infty,j)\)-categories whose \(i\)-morphisms are all invertible for all \(i > k\) [GH15, Prop. 6.1.7(iv)].23 We use this fact without further comment.
The \(n\)-cell (or walking \(n\)-morphism) is the \((\infty,n)\)-category \(c_n \coloneqq \Sigma^n[{\sf pt}] \in \mathrm{Cat}_{(\infty, {n})}\).24 Its boundary (or the walking pair of parallel \((n-1)\)-morphisms25) is the \((\infty,n)\)-category \(\partial c_n \coloneqq \partial \Sigma^n[{\sf pt}] \coloneqq \Sigma^n[\emptyset]\) (which is in fact an \((\infty,n-1)\)-category). We use both notations interchangeably, depending on our desired emphasis. We also introduce the notation \[j_n \colon \partial c_n \coloneqq \Sigma^n[\emptyset] \xrightarrow{\Sigma[\emptyset \longrightarrow{\sf pt}]} \Sigma^n[{\sf pt}] \eqqcolon c_n\] for the inclusion, which corepresents the functor taking an \(n\)-morphism to its source and target (which are parallel \((n-1)\)-morphisms).
For \(n \geq 0\), it follows from [GH15, Lem. 6.1.9] that the map \(|j_n|_0 \colon |\partial c_n|_0 \rightarrow|c_n|_0\) is equivalent to the map \(S^{n-1} \rightarrow{\sf pt}.\) By induction, it follows that for \(k < n\), \(|j_n|_k \colon |\partial c_n|_k \rightarrow|c_n|_k\) is equivalent to the map \(\Sigma^k[S^{n-k-1}] \rightarrow\Sigma^k[{\sf pt}] = c_k\) induced by \(S^{n-k-1} \rightarrow{\sf pt}\).
Let \(\alpha \colon \partial c_k \rightarrow\mathcal C\) be a pair of parallel \((k-1)\)-morphisms in an \((\infty,k)\)-category \(\mathcal C\). The space of \(k\)-morphisms filling \(\alpha\) is \[\mathrm{kHom}_{\mathcal C}(\alpha) \coloneqq \mathrm{Hom}_{\mathrm{Cat}_{(\infty, {k})}}(c_k, \mathcal C) \times_{\mathrm{Hom}_{\mathrm{Cat}_{(\infty, {k})}}(\partial c_k, \mathcal C)} \{\alpha\}.\]
Given a space \(X \in \mathcal S\) and \(k \geq 0\), the map \(\emptyset \rightarrow X\) induces a functor of \((\infty,k)\)-categories \(\partial c_k = \Sigma^k[\emptyset] \rightarrow\Sigma^k [X]\). It then follows from the universal property of \(\Sigma\) that for any \(\mathcal C\in \mathrm{Cat}_{(\infty, {k})}\) and any pair of parallel \((k-1)\)-morphisms \(\alpha\colon \partial c_k \rightarrow\mathcal C\), we obtain an equivalence of spaces \[\mathrm{Hom}_{\mathrm{Cat}_{(\infty, {k})}}(\Sigma^k[X], \mathcal C) \times_{\mathrm{Hom}_{\mathrm{Cat}_{(\infty, {k})}}(\partial c_k, \mathcal C)} \{\alpha\} \simeq \mathrm{Hom}_{\mathcal S}(X,\mathrm{kHom}_{\mathcal C}(\alpha)).\]
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2