局部可表现线性范畴、Grothendieck范畴及其张量积的滤过双余极限表示
Filtered bicolimit presentations of locally presentable linear categories, Grothendieck categories and their tensor products
AI总结：
本文提出两种将Grothendieck范畴恢复为滤过双余极限的方法，证明第一种与张量积相容而第二种一般不相容，并将Kelly张量积的性质推广到局部可表现线性范畴。
摘要：
我们研究了两种不同的方法，将Grothendieck范畴恢复为小范畴的滤过双余极限，并考察这两种方法与Grothendieck范畴张量积的相容性。首先，我们证明任何局部可表现线性范畴（特别是任何Grothendieck范畴）都可以恢复为其α-可表现对象子范畴的滤过双余极限，其中α在小正则基数的族中变化。然后我们证明，局部可表现线性范畴的张量积（特别是Grothendieck范畴的张量积）可以恢复为相应α-可表现对象子范畴的α-余完备线性范畴的Kelly张量积的滤过双余极限。其次，我们证明任何Grothendieck范畴都可以恢复为其线性site表示的滤过双余极限。然后我们证明，与第一种情况相反，Grothendieck范畴的张量积一般不能恢复为相应线性site的张量积的滤过双余极限。最后，作为第一种表示的直接应用，我们将α-余完备线性范畴的Kelly张量积的函子性、结合性和对称性平移到局部可表现线性范畴的张量积上。
英文摘要：
We investigate two different ways of recovering a Grothendieck category as a filtered bicolimit of small categories and the compatibility of both with the tensor product of Grothendieck categories. Firstly, we show that any locally presentable linear category (and in particular any Grothendieck category) can be recovered as the filtered bicolimit of its subcategories of $α$-presentable objects, with $α$ varying in the family of small regular cardinals. We then prove that the tensor product of locally presentable linear categories (and in particular the tensor product of Grothendieck categories) can be recovered as a filtered bicolimit of the Kelly tensor product of $α$-cocomplete linear categories of the corresponding subcategories of $α$-presentable objects. Secondly, we show that one can recover any Grothendieck category as a filtered bicolimit of its linear site presentations. We then prove that the tensor product of Grothendieck categories, in contrast with the first case, cannot be recovered in general as a filtered bicolimit of the tensor product of the corresponding linear sites. Finally, as a direct application of the first presentation, we translate the functoriality, associativity and symmetry of the Kelly tensor product of $α$-cocomplete linear categories to the tensor product of locally presentable linear categories.
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