The tensor embedding for a Grothendieck cosmos
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The tensor embedding for a Grothendieck cosmos. / Holm, Henrik; Odabaşı, Sinem.
In: Science China Mathematics, Vol. 66, No. 11, 2023, p. 2471–2494.Research output: Contribution to journal › Journal article › Research › peer-review
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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 -
ID: 358723015