Geodesic
The category of extensions and idempotent completion
Theory and applications of categories, Tome 41 (2024), pp. 1077-1107

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Building on previous work, we study the splitting of idempotents in the category of extensions E-Ext(C) associated to a pair (C,E) of an additive category and a biadditive functor to the category of abelian groups. In particular, we show that idempotents split in E-Ext(C) whenever they do so in C, allowing us to prove that idempotent completions and extension categories are compatible constructions in a 2-category-theoretic sense. Furthermore, we show that the exact category obtained by first taking the idempotent completion of an n-exangulated category (C,E,s), in the sense of Klapproth-Msapato-Shah, and then considering its category of extensions is equivalent to the exact category obtained by first passing to the extension category and then taking the idempotent completion. These two different approaches yield a pair of 2-functors each taking small n-exangulated categories to small idempotent complete exact categories. The collection of equivalences that we provide constitutes a 2-natural transformation between these 2-functors. Similar results with no smallness assumptions and regarding weak idempotent completions are also proved.

Publié le :
Classification : 18E05, 18E10, 18G80, 18N10, 18G99
Keywords: Category of extensions, additive category, biadditive functor, exact category, idempotent completion, n-exangulated category, n-exangulated functor, n-exangulated natural transformation, 2-category, 2-functor 
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     author = {R. Bennett-Tennenhaus and J. Haugland and M. H. Sand{\o}y and A. Shah},
     title = {The category of extensions and idempotent completion},
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     pages = {1077--1107},
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     volume = {41},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2024_41_a32/}
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R. Bennett-Tennenhaus; J. Haugland; M. H. Sandøy; A. Shah. The category of extensions and idempotent completion. Theory and applications of categories, Tome 41 (2024), pp. 1077-1107. http://geodesic.mathdoc.fr/item/TAC_2024_41_a32/