Injectivity paucity in AB5 categories of oversize chains

Alexandru Chirvasitu

Injectivity paucity in AB5 categories of oversize chains

Category Theory 2026-04-23 v1 Rings and Algebras

Authors: Alexandru Chirvasitu

Abstract

We construct examples of abelian categories with no non-zero injective (or projective) objects satisfying Grothendieck's AB5 condition. The procedure combines Rickard's examples of AB5 categories without products but some non-trivial injectives (also addressing an apparent gap in the literature) with a 2-functorial construct attaching to any category $\mathcal{C}$ that of $\mathcal{C}$ -objects equipped with set-indexed families of endomorphisms.

Keywords

abelian categories AB5 condition injective objects projective objects Grothendieck categories 2-functorial construction oversize chains

Cite

@article{arxiv.2604.20554,
  title  = {Injectivity paucity in AB5 categories of oversize chains},
  author = {Alexandru Chirvasitu},
  journal= {arXiv preprint arXiv:2604.20554},
  year   = {2026}
}

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9 pages

Injectivity paucity in AB5 categories of oversize chains

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