The tensor embedding for a Grothendieck cosmos (Q6084689)

The Gabriel-Quillen embedding theorem [\textit{R. W. Thomason} and \textit{T. Trobaugh}, Prog. Math. 88, 247--435 (1990; Zbl 0731.14001), Theorem A.7.1] claims that any small category admits an exact full embedding, which also reflects exactness, into some abelian category. meaning that any small exact category is equivalent, as an exact category, to an extension-closed subcategory of an abelian category. By way of example, \(\left( R\mathsf{-Mod},\mathcal{E} _{\mathrm{pure}}\right) \)\ admits two different exact full embeddings into abelian categories, where \(R\mathsf{-Mod}\)\ is the category of left \(R\)-modules equipped with the pure exact structure \(\mathcal{E}_{\mathrm{pure} }\), in which the exact sequences are directed colimits of split exact sequences in \(R\mathsf{-Mod}\). \begin{itemize} \item[1.] One is the \textit{Yoneda embedding} \begin{align*} \left( R\mathsf{-Mod},\mathcal{E}_{\mathrm{pure}}\right) & \rightarrow \left[ \left( R\mathsf{-}\text{mod}\right) ^{\mathrm{op}},\mathsf{Ab} \right] _{0}\\ M & \mapsto\mathrm{Hom}_{R}\left( -,M\right) \mid_{R\mathsf{-}\text{mod}} \end{align*} \item[2.] The other is the so-called \textit{tensor embedding} \begin{align*} \left( R\mathsf{-Mod},\mathcal{E}_{\mathrm{pure}}\right) & \rightarrow \left[ \mathsf{mod-}R,\mathsf{Ab}\right] _{0}\\ M & \mapsto\left( -\otimes_{R}M\right) \mid_{R\mathsf{-}\text{mod}} \end{align*} where ''\(\mathsf{mod}\)'' means finitely presentable modules and \(\left[ \mathcal{X},\mathsf{Ab}\right] _{0}\)\ denotes the category of additive functors from \(\mathcal{X}\)\ to the category \(\mathsf{Ab}\)\ of abelian groups. \end{itemize} This paper investigates a generalization of the tensor embedding, where \(\left( R\mathsf{-Mod},\mathcal{E}_{\mathrm{pure}}\right) \)\ is replaced by another exact category \(\left( \mathcal{V},\mathcal{E}_{\otimes}\right) \), where \(\left( \mathcal{V},\otimes,I,\left[ -,-\right] \right) \) is an \textit{abelian cosmos}, where the exact structure \(\mathcal{E}_{\otimes} \)\ imposed on \(\mathcal{V}\)\ is the so-called \textit{geometrically pure exact structure}, in which the admissible monomorphisms are the geometrically pure monomorphisms introduced by \textit{T. F. Fox} [J. Pure Appl. Algebra 8, 261--265 (1976; Zbl 0381.18013)]. The synopsis of the paper goes as follows. \begin{itemize} \item[\S 2] recalls some definitions and terminologies from the enriched category theory. \item[\S 3] establishes the first main result concerning the geometrically pure exact category. Theorem. The exact category \(\left( \mathcal{V}_{0},\mathcal{E}_{\otimes}\right) \)\ has enough relative injectives. In the language of relative homological algebra, this means that every object in \(\mathcal{V}_{0}\)\ has a geometrically pure injective preenvelope. If \(\mathcal{V}_{0}\)\ is Grothendieck, then every object in \ even has a geometrically pure injective envelope. \item[\S 4] The first part of this result can also be found in [\textit{E. Hosseini} and \textit{A. Zaghian}, J. Algebra Appl. 19, No. 1, Article ID 2050004, 7 p. (2020; Zbl 1444.18003), Theorem 2.6]. \item[\S 5] turns to the construction of the tensor embedding for the exact category \(\left( \mathcal{V}_{0},\mathcal{E}_{\otimes}\right) \), establishing the second main result. Theorem. Let \(\mathcal{A}\)\ be any small full \(\mathcal{V}\)-subcategory of \(\mathcal{V}\)\ containing the unit object \(I\)\ for the tensor product \(\otimes\). The tensor embedding yields a fully faithful exact functor \begin{align*} \Theta_{0} & :\left( \mathcal{V}_{0},\mathcal{E}_{\otimes}\right) \rightarrow\left( \left[ \mathcal{A},\mathcal{V}\right] _{0},\mathcal{E} _{\ast}\right) \\ X & \mapsto\left( X\otimes-\right) \mid_{\mathcal{A}} \end{align*} which induces an equivalence of exact categories \[ \left( \mathcal{V}_{0},\mathcal{E}_{\otimes}\right) \simeq\left( \mathrm{Ess.\operatorname{Im}\,}\Theta,\mathcal{E}_{\ast}\mid _{\mathrm{Ess.\operatorname{Im}\,}\Theta}\right) \] \item[\S 6] requires \(\mathcal{V}\)\ to be a \textit{Grothendieck} cosmos. Proposition 5.2 shows that there exists some regular cardinal \(\lambda\)\ for which \(\mathcal{V}\)\ is a locally \(\lambda\)-presentable base, and the authors focus only on the case where \(\mathcal{A}=\mathrm{Pres}_{\lambda}\left( \mathcal{V}\right) \)\ is the \(\mathcal{V}\)-subcategory of \(\lambda \)-presentable objects in \(\mathcal{V}\). The third main result (Theorem 5.9) goes as follows. Theorem. The essential image of the fully faithful tensor embedding \begin{align*} \Theta_{0} & :\mathcal{V}_{0}\rightarrow\left[ \mathrm{Pres}_{\lambda }\left( \mathcal{V}\right) ,\mathcal{V}\right] _{0}\\ X & \mapsto\left( X\otimes-\right) \mid_{\mathrm{Pres}_{\lambda}\left( \mathcal{V}\right) } \end{align*} is precisely \[ \mathrm{Ess.\operatorname{Im}\,}\Theta=\lambda\mathrm{-Cocont}\left( \mathrm{Pres}_{\lambda}\left( \mathcal{V}\right) ,\mathcal{V}\right) \] that is to say, the subcategory of \(\lambda\)-cocontinuous \(\mathcal{V} \)-functors from \(\mathrm{Pres}_{\lambda}\left( \mathcal{V}\right) \)\ to \(\mathcal{V}\). Furthermore, \(\Theta_{0}\)\ induces an equivalence of exact categories \[ \left( \mathcal{V}_{0},\mathcal{E}_{\otimes}\right) \simeq\lambda \mathrm{-Cocont}\left( \mathrm{Pres}_{\lambda}\left( \mathcal{V}\right) ,\mathcal{V}\right) \] \item[\S 7] specializes the setup even further by requiring \(\mathcal{V}\)\ to be a Grothendieck cosmos generated by a set of \textit{dualizable} objects and assuming that the unit object \(I\)\ is finitely presentable. The last main result (Theorem 6.13) is established. Theorem. The tensor embedding from Theorem 5.9 with \(\lambda=\aleph_{0}\)\ restricts to an equivalence between the geometrically pure injective objects in \(\mathcal{V}_{0}\)\ and the (categorically) injective objects in \ \(\left[ \mathrm{fp}\left( \mathcal{V}\right) ,\mathcal{V}\right] _{0}\). In symbols, \[ \mathrm{PureInj}_{\otimes}\left( \mathcal{V}_{0}\right) \simeq \mathrm{Inj}\left( \left[ \mathrm{fp}\left( \mathcal{V}\right) ,\mathcal{V}\right] _{0}\right) \] \end{itemize}