TY - JOUR
T1 - THE CATEGORY OF EXTENSIONS AND IDEMPOTENT COMPLETION
AU - Bennett-Tennenhaus, R.
AU - Haugland, J.
AU - Sandøy, M. H.
AU - Shah, A.
N1 - Publisher Copyright:
© R. Bennett-Tennenhaus, J. Haugland, M. H. Sandøy and A. Shah, 2024.
PY - 2024
Y1 - 2024
N2 - 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.
AB - 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.
KW - 2-category
KW - 2-functor
KW - additive category
KW - biadditive functor
KW - Category of extensions
KW - exact category
KW - idempotent completion
KW - n-exangulated category
KW - n-exangulated functor
KW - n-exangulated natural transformation
UR - http://www.scopus.com/inward/record.url?scp=85205295627&partnerID=8YFLogxK
M3 - Journal article
AN - SCOPUS:85205295627
SN - 1201-561X
VL - 41
SP - 1077
EP - 1107
JO - Theory and Applications of Categories
JF - Theory and Applications of Categories
M1 - 33
ER -