Filtered bicolimit presentations of locally presentable linear categories, Grothendieck categories and their tensor products (Q6606303)

A \textit{Grothendieck category} is a cocomplete abelian category with a generator and exact filtered colimits. The combination of the Gabriel-Popescu theorem [\textit{N. Popesco} and \textit{P. Gabriel}, C. R. Acad. Sci., Paris 258, 4188--4190 (1964; Zbl 0126.03304)] with the theory of enriched sheaves of \textit{F. Borceux} and \textit{C. Quinteiro} [Cah. Topologie Géom. Différ. Catégoriques 37, No. 2, 145--162 (1996; Zbl 0883.18006)] shows that Grothendieck categories are precisely the \textsf{Ab}-enriched topoi [\textit{W. Lowen}, J. Pure Appl. Algebra 190, No. 1--3, 197--211 (2004; Zbl 1051.18007)], where \textsf{Ab}\ denotes the category of abelian groups. We know [\textit{H. Krause}, Doc. Math. 20, 669--688 (2015; Zbl 1348.18018), Corollary 5.2; \textit{G. M. Kelly}, Cah. Topologie Géom. Différ. Catégoriques 23, 3--42 (1982; Zbl 0538.18006), \S 9] that any Grothendieck category can be written as the \(\mathsf{Ind}_{\alpha}\)-completion of its full subcategory of \(\alpha \)-presentable objects for a big enough regular cardinal \(\alpha\).\N\NThe tensor product \(\boxtimes_{\mathsf{G}}\)\ of Grothendieck categories was defined in terms of the presentations of Grothendieck categories as categories of \textsf{Ab}-enriched sheaves. On the other hand, we also have a tensor product of locally presentable categories at our disposal [\url{https://hdl.handle.net/2123/28961}], which is easily translated to the \textsf{Ab}-enriched setup, that is to say, to a tensor product of locally presentable \textsf{Ab}-enriched categories. It was shown in [\textit{W. Lowen} et al., Int. Math. Res. Not. 2018, No. 21, 6698--6736 (2018; Zbl 1408.18024), Theorem 5.4] that the tensor product of Grothendieck categories \(\boxtimes_{\mathsf{G}}\)\ is an instance of the tensor product \(\boxtimes_{\mathsf{LP}}\)\ of locally presentable \textsf{Ab}-enriched categories.\N\NThis paper aims to investigate two different ways of recovering of a Grothendieck category as a filtered bicolimit of small categories and the compatibility of both with the tensor product of Grothendieck categories.\N\NThe synopsis of the paper goes as follows:\N\N\begin{itemize}\N\item[\S 2] fixes the notation and conventions.\N\N\item[\S 3] revises the main aspects of the theory of locally presentable linear categories and their presentations as \(\mathsf{Ind}_{\alpha} \)-completions, particularly reviewing the relation between the Kelly tensor product of \(\alpha\)-cocomplete linear categories and the tensor product of locally presentable linear categories, as well as studying the behvior of this relation when one raises the cardinality \(\alpha\).\N\N\item[\S 4] revises the basic background on the theory of linear sites and their tensor product, as well as how they present, respectively, Grothendieck categories and their tensor product.\N\N\item[\S 5] gives a bald review of the general theory of 2-filtered bicolimits [\textit{E. J. Dubuc} and \textit{R. Street}, Cah. Topol. Géom. Différ. Catég. 47, No. 2, 83--106 (2006; Zbl 1110.18004); ibid. 62, No. 2, 239--242 (2021; Zbl 1469.18029)], particularly showing (Proposition 5.5) that the instance of 2-filtered bicolimits in which the author is interested is well behaved with respect to the llinear enrichment.\N\N\item[\S 6] provides two different ways of representing a Grothendieck category as a filtered bicolimit of small linear categories. The first one shows (Corollary 6.5) that a locally presentable linear category \(\mathcal{C}\) is the linear filtered bicolimit of its family of subcategories of \(\alpha \)-presentable objects \(\left( \mathcal{C}^{\alpha}\right) _{\alpha}\), where \(\alpha\)\ varies in the totally ordered class of small regular cardinals. The second one makes use of the topos-theoretical nature of Grothendieck categories, considering the category \(\mathcal{J}_{\mathcal{C}}\)\ of all site presentations of \(\mathcal{C}\), which is shown to be filtered, and considering the functor\N\[\NG_{\mathcal{C}}:\mathcal{J}_{\mathcal{C}}\rightarrow\mathsf{Cat}\N\]\N\ assigning to each LC morphism its domain. It is shown (Theorem 6.8) that\N\NTheorem. A Grothendieck category \(\mathcal{C}\)\ is the linear filtered bicolimit of \(\mathcal{J}_{\mathcal{C}}\).\N\N\item[\S 7] analyzes the behavior of the Kelly tensor product of \(\alpha \)-cocomplete linear categories and the tensor product of linear sites with respect to the corresponding filtered bicolimit presentations provided in \S 6. It is shown (Theorem 7.3) that\N\NTheorem. The tensor \(\mathcal{C}\boxtimes_{\mathsf{LP}}\mathcal{D}\)\ of two locally \(\kappa\)-presentable linear categories \(\mathcal{C}\)\ and \(\mathcal{D}\)\ can be expressed as the \(\kappa\)-linear filtered bicolimit of the tensor products \(\left( \mathcal{C}^{\alpha}\otimes_{\alpha}\mathcal{D}^{\alpha}\right) \)\ of categories of \(\alpha\)-presentable objects, where \ takes values in the totally ordered set of small regular cardinals \(\geq\kappa\), and the transition maps\N\[\N\mathcal{C}^{\alpha}\otimes_{\alpha}\mathcal{D}^{\alpha}\rightarrow \mathcal{C}^{\beta}\otimes_{\beta}\mathcal{D}^{\beta}\N\]\Nare those induced through the universal property of the Kelly tensor product.\N\N\item[\S 8] describes the functoriality of the tensor product of locally presentable linear categories via the functoriality of the Kelly tensor product of \(\alpha\)-cocomplete linear categories (Proposition 8.8). The associativity and symmetry of the tensor product of locally presentable linear categories are described making use of the same properties of the Kelly tensor product of \(\alpha\)-cocomplete linear categories (Propositions 8.10 and 8.11).\N\end{itemize}