arxiv: 2604.20554 · v1 · submitted 2026-04-22 · 🧮 math.CT · math.RA

Recognition: unknown

Injectivity paucity in AB5 categories of oversize chains

Alexandru Chirvasitu

Pith reviewed 2026-05-09 22:29 UTC · model grok-4.3

classification 🧮 math.CT math.RA

keywords abelian categoriesAB5 conditioninjective objectsprojective objectsGrothendieck categories2-functorial constructionendomorphism families

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The pith

There exist AB5 abelian categories with no non-zero injective or projective objects.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs examples of abelian categories that satisfy Grothendieck's AB5 condition yet contain no non-zero injective objects and no non-zero projective objects. It does this by taking known AB5 categories without products that do have some injectives and applying a new 2-functorial construction which equips objects with set-indexed families of endomorphisms. This construction is designed to preserve the abelian structure and the AB5 property while reducing the injective and projective objects to only the zero object. A sympathetic reader would care because these examples show that the AB5 condition alone does not force the existence of non-trivial injectives or projectives in abelian categories.

Core claim

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 with a 2-functorial construct attaching to any category C that of C-objects equipped with set-indexed families of endomorphisms.

What carries the argument

The 2-functorial construct attaching to any category C the category of C-objects equipped with set-indexed families of endomorphisms.

If this is right

The resulting categories satisfy AB5 but have only the zero object as injective.
The resulting categories satisfy AB5 but have only the zero object as projective.
The construction applies to Rickard's examples to eliminate their non-trivial injectives.
The approach fills an apparent gap in the literature on Rickard's examples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar modifications might control the existence of special objects in other categories with directed colimits.
The examples suggest AB5 alone is insufficient to guarantee non-zero injectives or projectives.
One could test whether the construction preserves other exactness properties beyond AB5.

Load-bearing premise

The 2-functorial construction preserves the AB5 property and abelian structure while ensuring that the only injective and projective objects are zero.

What would settle it

An explicit non-zero injective object in one of the constructed categories, or a directed colimit whose exactness fails in the new category.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper layers a 2-functor of set-indexed endomorphisms onto Rickard's AB5 categories to force only the zero injective and projective, but the preservation of exact filtered colimits is the part that still needs the closest look.

read the letter

The core contribution is a direct construction that starts from Rickard's AB5 abelian categories (which already lack products but have some injectives) and produces new ones where every non-zero object fails to be injective or projective. The method attaches to each object in the base category C a family of endomorphisms indexed by a large set S, then takes morphisms in the new category D to be those that commute with every endomorphism in the family. This is presented as a 2-functorial operation, which keeps the construction clean and avoids self-reference or fitted parameters.

Referee Report

2 major / 2 minor

Summary. The paper constructs examples of abelian AB5 categories with no non-zero injective or projective objects. It starts from Rickard's AB5 categories (without products but with some non-zero injectives) and applies a 2-functorial construction that produces, for a category C and large index set S, a new category D whose objects are pairs (X, (e_s : X → X)_{s∈S}) with morphisms the C-morphisms that commute with all e_s. The claim is that D is abelian, satisfies AB5 (all small coproducts and exact filtered colimits), and has only the zero object as injective or projective.

Significance. If the central claims hold, the construction supplies the first explicit examples of AB5 abelian categories with injectivity and projectivity paucity, extending Rickard's work on categories without products and addressing an apparent gap. The 2-functorial approach is direct and avoids self-referential definitions, which is a strength if the preservation properties are verified.

major comments (2)

[Construction of D and AB5 verification] The verification that D satisfies AB5 (exactness of filtered colimits) is load-bearing for the headline claim but appears to rest on an implicit claim that the pointwise-induced endomorphisms on the colimit in C automatically satisfy the universal property in D. This needs an explicit check that any failure of the e_s to commute with the colimit cocone is ruled out; otherwise AB5 may fail in D even if it holds in C. (Construction section describing the 2-functor and the proof that D is AB5.)
[Proof of injectivity paucity] The argument that every non-zero object of D fails to be injective (by exhibiting a monomorphism that does not lift, using |S| to obstruct lifts) must be shown to survive the passage from C to D and to be exhaustive. The reduction relies on the cardinality of S; it is unclear whether this obstruction remains effective after equipping objects with the endomorphism families and restricting to commuting morphisms. (Section proving absence of non-zero injectives/projectives.)

minor comments (2)

[Notation and definitions] Notation for the 2-functor and the category D could be clarified with an explicit diagram or commutative square showing how morphisms in D are defined.
[Introduction] The paper mentions addressing an apparent gap in Rickard's examples; a brief comparison table or explicit statement of what was missing in the literature would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where additional explicit verification would strengthen the exposition. The central claims hold, and we address each major comment below with clarifications; revisions will be made to improve readability without altering the arguments.

read point-by-point responses

Referee: [Construction of D and AB5 verification] The verification that D satisfies AB5 (exactness of filtered colimits) is load-bearing for the headline claim but appears to rest on an implicit claim that the pointwise-induced endomorphisms on the colimit in C automatically satisfy the universal property in D. This needs an explicit check that any failure of the e_s to commute with the colimit cocone is ruled out; otherwise AB5 may fail in D even if it holds in C. (Construction section describing the 2-functor and the proof that D is AB5.)

Authors: We agree that an explicit check improves clarity. In the 2-functorial construction, filtered colimits in D are formed by taking the filtered colimit of the underlying objects in C (which exists and is exact by the AB5 assumption on C) and equipping the result with the endomorphisms induced pointwise via the colimit cocone. Because every arrow in the diagram is a morphism in D, it commutes with the respective e_s families; filtered colimits in C preserve compositions and commute with the finite diagrams expressing the commuting relations, so the induced endomorphisms on the colimit automatically commute with all cocone maps. Hence the cocone is universal in D. We will add a short lemma in the construction section spelling out this verification. revision: yes
Referee: [Proof of injectivity paucity] The argument that every non-zero object of D fails to be injective (by exhibiting a monomorphism that does not lift, using |S| to obstruct lifts) must be shown to survive the passage from C to D and to be exhaustive. The reduction relies on the cardinality of S; it is unclear whether this obstruction remains effective after equipping objects with the endomorphism families and restricting to commuting morphisms. (Section proving absence of non-zero injectives/projectives.)

Authors: The argument is robust under the passage to D. Given a non-zero object (X, (e_s)_{s∈S}) of D, we apply the underlying monomorphism from Rickard’s category C that fails to lift in C; any candidate lift in D would additionally have to commute with every e_s. The cardinality of S is chosen larger than any relevant cardinal associated to X, producing an overdetermined system of commuting conditions that no non-zero morphism can satisfy. Because the morphisms of D form a full subcategory of those of C, no additional lifts are created by the extra structure. The same cardinality obstruction therefore rules out lifts in D. We will expand the relevant paragraph to record this independence from the specific choice of endomorphisms. revision: partial

Circularity Check

0 steps flagged

Direct 2-functorial construction with no self-referential reductions

full rationale

The paper defines a new category D from a base AB5 category C by equipping objects with an indexed family of endomorphisms and taking morphisms that commute with them. This is an explicit functorial attachment whose preservation of abelian structure, coproducts, and filtered colimits is verified directly from the universal properties of the base category and the pointwise action on endomorphisms. The claim that non-zero objects fail to be injective is shown by exhibiting a specific monomorphism that cannot lift, using the cardinality of the index set to obstruct any putative lift; this argument does not presuppose the conclusion or reduce to a fitted parameter. Citations to Rickard are to independent prior work and are not load-bearing for the new construction. No equation or definition is equivalent to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard definitions of abelian categories, Grothendieck's AB5 axiom, and functorial constructions without introducing fitted parameters or new postulated entities.

axioms (1)

standard math Standard axioms of abelian categories and Grothendieck's AB5 condition
Invoked throughout as the ambient framework for the constructions.

pith-pipeline@v0.9.0 · 5352 in / 1086 out tokens · 39228 ms · 2026-05-09T22:29:14.015294+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references

[1]

Strecker

Jiří Adámek, Horst Herrlich, and George E. Strecker. Abstract and concrete categories: the joy of cats.Repr. Theory Appl. Categ., 2006(17):1–507, 2006. 2

2006
[2]

Stacks project

The Stacks Project Authors. Stacks project. 8
[3]

Volume 1: Basic category theory, volume 50 ofEncycl

Francis Borceux.Handbook of categorical algebra. Volume 1: Basic category theory, volume 50 ofEncycl. Math. Appl.Cambridge: Cambridge Univ. Press, 1994. 2, 6

1994
[4]

Fraenkel, Yehoshua Bar-Hillel, and Azriel Levy.Foundations of set theory, volume Vol

Abraham A. Fraenkel, Yehoshua Bar-Hillel, and Azriel Levy.Foundations of set theory, volume Vol. 67 ofStudies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam-London, revised edition, 1973. With the collaboration of Dirk van Dalen. 2

1973
[5]

Abelian categories.Repr

Peter Freyd. Abelian categories.Repr. Theory Appl. Categ., 2003(3):xxiii + 1–164, 2003. 2, 4, 5, 8, 9

2003
[6]

Oxford: Oxford University Press,

Niles Johnson and Donald Yau.2-dimensional categories. Oxford: Oxford University Press,
[7]

An introduction to independence proofs

Kenneth Kunen.Set theory. An introduction to independence proofs. 2nd print, volume 102 of Stud. Logic Found. Math.Elsevier, Amsterdam, 1983. 2, 3

1983
[8]

Springer-Verlag, New York, second edition, 1998

Saunders Mac Lane.Categories for the working mathematician, volume 5 ofGraduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1998. 1, 7, 8

1998
[9]

Popescu.Abelian categories with applications to rings and modules, volume 3 ofLond

N. Popescu.Abelian categories with applications to rings and modules, volume 3 ofLond. Math. Soc. Monogr.Academic Press, London, 1973. 1, 2, 3, 5, 6

1973
[10]

A cocomplete but not complete abelian category.Bull

Jeremy Rickard. A cocomplete but not complete abelian category.Bull. Aust. Math. Soc., 101(3):442–445, 2020. 1, 2, 3, 4, 5, 8 Department of Mathematics, University at Buff alo Buff alo, NY 14260-2900, USA E-mail address:achirvas@buffalo.edu 9

2020