Colimits and cocompletions in internal higher category theory (Q6617078)

This paper, as the second in a series to develop categorical tools in the context of higher categories to an \(\infty\)-topos \(\mathcal{B}\), aims to provide a systematic framework for this area of ideas in order to open the way for future applications. The first [\textit{L. Martini}, ``Yoneda's lemma for internal higher categories'', Preprint, \url{arXiv:2103.17141}] by the first-named author established Yoneda's lemma for internal higher categories. This paper gives an extensive discussion of adjunctions, (co)limits, Kan extensions and free (co)completions in the internal setting.\N\NThe theory of synthetic higher categories in [\textit{E. Riehl} and \textit{M. Shulman}, High. Struct. 1, No. 1, 147--224 (2017; Zbl 1437.18016)] is closely related to this paper, because Shulman's \(\infty\)-topos semantics [\textit{M. Shulman}, ``All $(\infty,1)$-toposes have strict univalent universes'', Preprint, \url{arXiv:1904.07004}] implies that synthetic higher category theory is to be interpreted in simplicial objects in any \(\infty\)-topos. Many concepts of internal higher category theory have been developed from this standpoint in [\textit{U. Buchholtz} and \textit{J. Weinberger}, High. Struct. 7, No. 1, 74--165 (2023; Zbl 1535.18039); \textit{J. Weinberger}, J. Pure Appl. Algebra 228, No. 9, Article ID 107659, 52 p. (2024; Zbl 07842164); J. Homotopy Relat. Struct. 19, No. 3, 297--378 (2024; Zbl 07919890)].\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] recalls the basic framework of higher category theory internal to an \(\infty\)-topos [\textit{L. Martini}, ``Yoneda's lemma for internal higher categories'', Preprint, \url{arXiv:2103.17141}].\N\N\item[\S 3] studies adjunctions between \(\mathcal{B}\)-categories.\N\N\item[\S 4] addresses limits and colimits in a \(\mathcal{B}\)-category.\N\N\item[\S 5] is dedicated to a more global study of (co)limits in a \(\mathcal{B}\)-category. More precisely, if \textsf{U}\ is an internal class of \(\mathcal{B}\)-categories, the authors define and study what it means for a \(\mathcal{B}\)-category to be \textit{U-(co)complete} and for a functor \(f:\mathsf{C}\rightarrow\mathsf{D}\)\ between \(\mathcal{B}\)-categories to be \textit{U-(co)continuous}.\N\N\item[\S 6] aims to develop the theory of Kan extensions of functors between \(\mathcal{B}\)-categories, establishing the following theorem.\N\NTheorem. Let \(\mathsf{E}\)\ cocomplete (large) \(\mathcal{B}\)-category and let \(f:\mathsf{C}\rightarrow\mathsf{D}\)\ be a functor of small \(\mathcal{B} \)-categories. The the functor\N\[\Nf^{\ast}:\underline{\mathsf{Fun}}_{\mathcal{B}}\left( \mathsf{D} ,\mathsf{E}\right) \rightarrow\underline{\mathsf{Fun}}_{\mathcal{B}}\left( \mathsf{C},\mathsf{E}\right)\N\]\Nhas a left adjoint \(f_{!}\)\ which is fully faithful whenever \(f\)\ is fully faithful.\N\N\item[\S 7] aims to construct and study the free cocompletion by \textsf{U}-colimits of an arbitrary \(\mathcal{B}\)-category for any\ internal class \textsf{U}\ of \(\mathcal{B}\)-categories, establishing the following theorem.\N\NTheorem. For any small \(\mathcal{B}\)-category \(\mathsf{C}\)\ and any cocomplete large \(\mathcal{B}\)-category \(\mathsf{E}\), the functoror of left Kan extension \(h_{!}\)\ along the Yoneda embedding \(h:\mathsf{C}\rightarrow\underline {\mathsf{Fun}}_{\mathcal{B}}\left( \mathsf{C}^{\mathrm{op}},\Omega\right) \)\ induces an equivalence\N\[\Nh_{!}:\underline{\mathsf{Fun}}_{\mathcal{B}}^{\mathrm{cc}}\left( \mathsf{C},\mathsf{E}\right) \rightarrow\underline{\mathsf{Fun}} _{\mathcal{B}}\left( \underline{\mathsf{PSh}}_{\mathcal{B}}\left( \mathsf{C}\right) ,\mathsf{E}\right)\N\]\Nwhere \(\underline{\mathsf{Fun}}_{\mathcal{B}}^{\mathrm{cc}}\left( \underline{\mathsf{PSh}}_{\mathcal{B}}\left( \mathsf{C}\right) ,\mathsf{E}\right) \)\ denotes the full subcategory of \(\underline {\mathsf{Fun}}_{\mathcal{B}}\left( \underline{\mathsf{PSh}}_{\mathcal{B} }\left( \mathsf{C}\right) ,\mathsf{E}\right) \)\ that is spanned by the cocontinuous functor.\N\N\item[Appendices] are concerned with the large \(\mathcal{B}\)-category of \(\mathcal{B}\)-categories (Appendix A), monomorphisms and subcategories of \(\mathcal{B}\)-categories (Appendix B) and localizations of \(\mathcal{B} \)-categories (Appendix C).\N\end{itemize}