TY - JOUR

T1 - The tensor embedding for a Grothendieck cosmos

AU - Holm, Henrik

AU - Odabaşı, Sinem

N1 - Publisher Copyright: © 2023, Science China Press.

PY - 2023

Y1 - 2023

N2 - While the Yoneda embedding and its generalizations have been studied extensively in the literature, the so-called tensor embedding has only received a 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 the works of Enochs et al. (2016) and Estrada et al. (2017). More precisely, for a Grothendieck cosmos, i.e., a bicomplete Grothendieck category V with a closed symmetric monoidal structure, we prove that the geometrically pure exact category (V, ℰ⊗) has enough relative injectives; in fact, every object has a geometrically pure injective envelope. We also show that for some regular cardinal λ, the tensor embedding yields an exact equivalence between (V, ℰ⊗) and the category of λ-cocontinuous V -functors from Presλ(V)to V , where the former is the full V -subcategory of λ-presentable objects in V . In many cases of interest, λ can be chosen to be ℵ and the tensor embedding identifies the geometrically pure injective objects in V with the (categorically) injective objects in the abelian category of V -functors from fp(V)to V . As we explain, the developed theory applies, e.g., to the category Ch(R) of chain complexes of modules over a commutative ring R and to the category Qcoh(X) of quasi-coherent sheaves over a (suitably nice) scheme X.

AB - While the Yoneda embedding and its generalizations have been studied extensively in the literature, the so-called tensor embedding has only received a 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 the works of Enochs et al. (2016) and Estrada et al. (2017). More precisely, for a Grothendieck cosmos, i.e., a bicomplete Grothendieck category V with a closed symmetric monoidal structure, we prove that the geometrically pure exact category (V, ℰ⊗) has enough relative injectives; in fact, every object has a geometrically pure injective envelope. We also show that for some regular cardinal λ, the tensor embedding yields an exact equivalence between (V, ℰ⊗) and the category of λ-cocontinuous V -functors from Presλ(V)to V , where the former is the full V -subcategory of λ-presentable objects in V . In many cases of interest, λ can be chosen to be ℵ and the tensor embedding identifies the geometrically pure injective objects in V with the (categorically) injective objects in the abelian category of V -functors from fp(V)to V . As we explain, the developed theory applies, e.g., to the category Ch(R) of chain complexes of modules over a commutative ring R and to the category Qcoh(X) of quasi-coherent sheaves over a (suitably nice) scheme X.

KW - (pre)envelope

KW - (pure) injective object

KW - 18D15

KW - 18D20

KW - 18E10

KW - 18E20

KW - 18G05

KW - enriched functor

KW - exact category

KW - purity

KW - symmetric monoidal category

KW - tensor embedding

KW - Yoneda embedding

U2 - 10.1007/s11425-021-2046-9

DO - 10.1007/s11425-021-2046-9

M3 - Journal article

AN - SCOPUS:85160242993

VL - 66

SP - 2471

EP - 2494

JO - Science China Mathematics

JF - Science China Mathematics

SN - 1674-7283

IS - 11

ER -