The tensor embedding for a grothendieck cosmos

While the Yoneda embedding and its generalizations have been studied extensively in the literature, the so-called tensor embedding has only received little attention. In this paper, we study the tensor embedding for closed symmetric monoidal categories and show how it is connected to the notion of geometrically purity, which has recently been investigated in works of Enochs, Estrada, Gillespie, and Odaba\c{s}{\i}. More precisely, for a Gro\-thendieck cosmos---that is, a bicomplete Grothendick category $\mathcal{V}$ with a closed symmetric monoidal structure---we prove that the geometrically pure exact category $(\mathcal{V},\mathscr{E}_\otimes)$ has enough relative injectives; in fact, every object has a geometrically pure injective envelope. We also show that for some regular cardinal $\lambda$, the tensor embedding yields an exact equivalence between $(\mathcal{V},\mathscr{E}_\otimes)$ and the category of $\lambda$-cocontinuous $\mathcal{V}$-functors from $\mbox{Pres}(\mathcal{V})$ to $\mathcal{V}$, where the former is the full $\mathcal{V}$-subcategory of $\lambda$-presentable objects in $\mathcal{V}$. In many cases of interest, $\lambda$ can be chosen to be $\aleph_0$ and the tensor embedding identifies the geometrically pure injective objects in $\mathcal{V}$ with the (categorically) injective objects in the abelian category of $\mathcal{V}$-functors from $\mathrm{fp}(\mathcal{V})$ to $\mathcal{V}$. As we explain, the developed theory applies e.g.~to the category $\mathsf{Ch}(R)$ of chain complexes of modules over a commutative ring $R$ and to the category $\mathsf{Qcoh}(X)$ of quasi-coherent sheaves over a (suitably nice) scheme $X$.