Geodesic
Idempotent completion of pretriangulated categories
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 477-494
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A pretriangulated category is an additive category with left and right triangulations such that these two triangulations are compatible. In this paper, we first show that the idempotent completion of a left triangulated category admits a unique structure of left triangulated category and dually this is true for a right triangulated category. We then prove that the idempotent completion of a pretriangulated category has a natural structure of pretriangulated category. As an application, we show that a torsion pair in a pretriangulated category extends uniquely to a torsion pair in the idempotent completion.
A pretriangulated category is an additive category with left and right triangulations such that these two triangulations are compatible. In this paper, we first show that the idempotent completion of a left triangulated category admits a unique structure of left triangulated category and dually this is true for a right triangulated category. We then prove that the idempotent completion of a pretriangulated category has a natural structure of pretriangulated category. As an application, we show that a torsion pair in a pretriangulated category extends uniquely to a torsion pair in the idempotent completion.
DOI : 10.1007/s10587-014-0114-9
Classification : 16B50, 18E05, 18E30, 18E40
Keywords: idempotent completion; pretriangulated category; torsion pair 
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Liu, Jichun; Sun, Longgang. Idempotent completion of pretriangulated categories. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 477-494. doi: 10.1007/s10587-014-0114-9

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