Idempotent Completions of n-Exangulated Categories

Carlo Klapproth; Dixy Msapato; Amit Shah

doi:10.1007/s10485-023-09758-5

Idempotent Completions of n-Exangulated Categories

Carlo Klapproth, Dixy Msapato, Amit Shah^*

^*Corresponding author af dette arbejde

Institut for Matematik

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

1 Citationer (Scopus)

Abstract

Suppose (C,E,s) is an n-exangulated category. We show that the idempotent completion and the weak idempotent completion of C are again n-exangulated categories. Furthermore, we also show that the canonical inclusion functor of C into its (resp. weak) idempotent completion is n-exangulated and 2-universal among n-exangulated functors from (C,E,s) to (resp. weakly) idempotent complete n-exangulated categories. Furthermore, we prove that if (C,E,s) is n-exact, then so too is its (resp. weak) idempotent completion. We note that our methods of proof differ substantially from the extriangulated and (n+2)-angulated cases. However, our constructions recover the known structures in the established cases up to n-exangulated isomorphism of n-exangulated categories.

Originalsprog	Engelsk
Artikelnummer	7
Tidsskrift	Applied Categorical Structures
Vol/bind	32
Nummer	1
ISSN	0927-2852
DOI	https://doi.org/10.1007/s10485-023-09758-5
Status	Udgivet - feb. 2024

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title = "Idempotent Completions of n-Exangulated Categories",

abstract = "Suppose (C,E,s) is an n-exangulated category. We show that the idempotent completion and the weak idempotent completion of C are again n-exangulated categories. Furthermore, we also show that the canonical inclusion functor of C into its (resp. weak) idempotent completion is n-exangulated and 2-universal among n-exangulated functors from (C,E,s) to (resp. weakly) idempotent complete n-exangulated categories. Furthermore, we prove that if (C,E,s) is n-exact, then so too is its (resp. weak) idempotent completion. We note that our methods of proof differ substantially from the extriangulated and (n+2)-angulated cases. However, our constructions recover the known structures in the established cases up to n-exangulated isomorphism of n-exangulated categories.",

keywords = "18G15, Additive category, Idempotent completion, Karoubi envelope, n-exangulated category, n-exangulated functor, n-exangulated natural transformation, Primary 18E05, Secondary 16U40, Split idempotents, Weak idempotent completion",

author = "Carlo Klapproth and Dixy Msapato and Amit Shah",

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year = "2024",

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AU - Klapproth, Carlo

AU - Msapato, Dixy

AU - Shah, Amit

N1 - Publisher Copyright: © The Author(s) 2024.

PY - 2024/2

Y1 - 2024/2

N2 - Suppose (C,E,s) is an n-exangulated category. We show that the idempotent completion and the weak idempotent completion of C are again n-exangulated categories. Furthermore, we also show that the canonical inclusion functor of C into its (resp. weak) idempotent completion is n-exangulated and 2-universal among n-exangulated functors from (C,E,s) to (resp. weakly) idempotent complete n-exangulated categories. Furthermore, we prove that if (C,E,s) is n-exact, then so too is its (resp. weak) idempotent completion. We note that our methods of proof differ substantially from the extriangulated and (n+2)-angulated cases. However, our constructions recover the known structures in the established cases up to n-exangulated isomorphism of n-exangulated categories.

AB - Suppose (C,E,s) is an n-exangulated category. We show that the idempotent completion and the weak idempotent completion of C are again n-exangulated categories. Furthermore, we also show that the canonical inclusion functor of C into its (resp. weak) idempotent completion is n-exangulated and 2-universal among n-exangulated functors from (C,E,s) to (resp. weakly) idempotent complete n-exangulated categories. Furthermore, we prove that if (C,E,s) is n-exact, then so too is its (resp. weak) idempotent completion. We note that our methods of proof differ substantially from the extriangulated and (n+2)-angulated cases. However, our constructions recover the known structures in the established cases up to n-exangulated isomorphism of n-exangulated categories.

KW - 18G15

KW - Additive category

KW - Idempotent completion

KW - Karoubi envelope

KW - n-exangulated category

KW - n-exangulated functor

KW - n-exangulated natural transformation

KW - Primary 18E05

KW - Secondary 16U40

KW - Split idempotents

KW - Weak idempotent completion

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DO - 10.1007/s10485-023-09758-5

M3 - Journal article

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SN - 0927-2852

VL - 32

JO - Applied Categorical Structures

JF - Applied Categorical Structures

IS - 1

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ER -

Idempotent Completions of n-Exangulated Categories

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