Injectivity paucity in AB5 categories of oversize chains
Alexandru Chirvasitu
TL;DR
This paper constructs examples of AB5 abelian categories lacking non-zero injective objects, highlighting the scarcity of injectivity in certain categorical contexts.
Contribution
It introduces a novel construction combining Rickard's examples with a 2-functorial approach to produce categories with specific injectivity properties.
Findings
Examples of AB5 categories with no non-zero injectives
Addresses a gap in the literature on injectivity in AB5 categories
Provides a new method to construct such categories
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 that of -objects equipped with set-indexed families of endomorphisms.
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