The tensor embedding for a grothendieck cosmos

TL;DR

This paper explores the tensor embedding in closed symmetric monoidal categories, establishing its connection to geometrically purity and demonstrating how it provides an exact equivalence with categories of cocontinuous functors, with applications to chain complexes and quasi-coherent sheaves.

Contribution

It introduces the tensor embedding for Grothendieck cosmos, linking it to geometrically pure injectives and showing its role in category equivalences, extending understanding in monoidal category theory.

Findings

Geometrically pure exact category has enough injectives.

Tensor embedding yields an exact equivalence with categories of cocontinuous functors.

Identifies geometrically pure injectives with categorical injectives in certain functor categories.

Abstract

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 $X$.