Category of Chain Bundles

• The category of chain bundles is defined as a framework that organizes sequences of objects ending in zero, with each stage equipped with chosen morphism subsets.
• This structure generalizes traditional chains and chain complexes by allowing multiple compatible morphisms, thus inheriting subobjects, factorization, products, and kernels from the base category.
• Chain bundles connect abstract algebraic theory with geometric applications, exemplified by chains of jet bundles, fibre inclusions, and long exact sequences from fibrations.

The category of chain bundles is a categorical construction designed to organize chain-shaped data of objects and morphisms. In its abstract form, it is built from a category $C$ with zero by taking sequences of objects ending at $0$ together with specified morphism data between successive terms; its morphisms are compatible levelwise maps. The construction was introduced as a way to package chains of algebraic objects into a category of its own, then refined to allow explicit subsets of hom-sets at each stage, and later related to concrete geometric categories such as categories of bundles, jet bundles, and long exact sequences arising from fibrations (Romeo et al., 2020, Romeo et al., 2021, Romeo et al., 27 May 2026).

1. Foundational definition

In the 2020 formulation, a chain bundle in a category $C$ with zero is a subcategory of the form

$$\cdots M_3 \rightharpoonup M_2 \rightharpoonup M_1 \rightharpoonup M_0,$$

where the vertices $M_i\in vC$, the arrows are the hom-sets $\operatorname{Hom}(M_{i+1},M_i)$, and the construction includes all identities and all possible composites. A cochain bundle is defined dually (Romeo et al., 2020).

A later formulation makes the hom-set data explicit. If $C$ is a category with zero, a chain bundle $c$ is written

$$\cdots \; M_3 \; S_3 \; M_2 \; S_2 \; M_1 \; S_1 \; M_0 = 0$$

with each $M_i\in vC$ and each $0$0 any subset of $0$1. The subsets $0$2 are allowed to include identity morphisms and all possible composites of morphisms. In this formulation, a chain bundle is not a single sequence of arrows but a sequence of objects together with possibly many arrows between successive objects; the authors interpret this as a “bundle” of arrows at each step (Romeo et al., 2021).

These two descriptions are compatible at the level of intent: both treat a chain bundle as a chain-shaped categorical object richer than an ordinary chain. A plausible implication is that the later formulation isolates the essential datum already implicit in the earlier hom-set-based definition.

2. Morphisms and the category $0$3

A chain bundle map in the 2020 treatment is a functor between chain bundles whose vertex map is a sequence

$0$4

with the corresponding ladder diagram commuting. The objectwise maps and the induced maps on hom-sets are both part of the data (Romeo et al., 2020).

In the later formulation, if

$0$5

are chain bundles, then a morphism $0$6 is a sequence

$0$7

where $0$8 are morphisms in $0$9, $C$0, and $C$1, satisfying

$C$2

Thus a morphism of chain bundles is given by objectwise maps together with chosen compatible arrows from the source and target chain bundles (Romeo et al., 2021).

The category whose objects are chain bundles in $C$3 and whose morphisms are chain bundle maps is denoted $C$4. The 2021 paper explicitly states that all objects and morphisms in $C$5 are also objects and morphisms in $C$6, so $C$7 may be regarded as a subcategory of $C$8. It also records the dual notion of the category of cochain bundles (Romeo et al., 2021).

3. Subobjects, factorization, and inherited categorical structure

A central theme in the literature is that the chain-bundle construction inherits structural properties from the base category. In the 2020 paper, if $C$9 is a category with subobjects, then a subchain bundle is defined by requiring that each object $$\cdots M_3 \rightharpoonup M_2 \rightharpoonup M_1 \rightharpoonup M_0,$$0 be a subobject of $$\cdots M_3 \rightharpoonup M_2 \rightharpoonup M_1 \rightharpoonup M_0,$$1, and that for every

$$\cdots M_3 \rightharpoonup M_2 \rightharpoonup M_1 \rightharpoonup M_0,$$2

there exists

$$\cdots M_3 \rightharpoonup M_2 \rightharpoonup M_1 \rightharpoonup M_0,$$3

such that $$\cdots M_3 \rightharpoonup M_2 \rightharpoonup M_1 \rightharpoonup M_0,$$4 is the corestriction of $$\cdots M_3 \rightharpoonup M_2 \rightharpoonup M_1 \rightharpoonup M_0,$$5. On this basis, the relation “is a subchain bundle” is proved to be a strict preorder, and $$\cdots M_3 \rightharpoonup M_2 \rightharpoonup M_1 \rightharpoonup M_0,$$6 becomes a category with subobjects whenever $$\cdots M_3 \rightharpoonup M_2 \rightharpoonup M_1 \rightharpoonup M_0,$$7 is a category with subobjects (Romeo et al., 2020).

The same paper proves a factorization statement under an additional restriction. If $$\cdots M_3 \rightharpoonup M_2 \rightharpoonup M_1 \rightharpoonup M_0,$$8 has factorization, then the subcategory $$\cdots M_3 \rightharpoonup M_2 \rightharpoonup M_1 \rightharpoonup M_0,$$9, with the same objects as $M_i\in vC$0 but only those chain bundle maps whose morphism map is full, admits factorization: every such map $M_i\in vC$1 factors as

$M_i\in vC$2

with $M_i\in vC$3 an epimorphism and $M_i\in vC$4 an inclusion. The point of the fullness restriction is that it ensures that the intermediate image chain is well defined levelwise (Romeo et al., 2020).

Later work extends the list of inherited constructions. If $M_i\in vC$5 has products, products in $M_i\in vC$6 are defined componentwise; if $M_i\in vC$7 has kernels, kernels in $M_i\in vC$8 are also defined componentwise; cokernels are described similarly. The product of chain bundles

$M_i\in vC$9

is

$\operatorname{Hom}(M_{i+1},M_i)$0

The same work also describes componentwise factorization of chain bundle morphisms and formulates universal arrows for functors between chain-bundle categories (Romeo et al., 2021).

4. Relation to the category of chains

The category of chains is presented as a specialization of the category of chain bundles. A chain in $\operatorname{Hom}(M_{i+1},M_i)$1 is written

$\operatorname{Hom}(M_{i+1},M_i)$2

where each $\operatorname{Hom}(M_{i+1},M_i)$3 is a single morphism in $\operatorname{Hom}(M_{i+1},M_i)$4, together with identities and all possible composites. The decisive statement is that a chain is a chain bundle having exactly one morphism in each hom-set (Romeo et al., 2021).

A chain map between two chains is a functor whose vertex map is a sequence

$\operatorname{Hom}(M_{i+1},M_i)$5

such that the relevant squares commute. Chains and chain maps therefore form a category, usually described as the category of chains. In this sense, the chain category is a subcategory or specialization of the chain-bundle category obtained by selecting one morphism at each stage (Romeo et al., 2021).

The 2020 paper expresses the same passage in a slightly different way. Starting from a chain bundle in a suitable subcategory, one chooses at most one morphism from each hom-set and obtains a chain; the resulting chains and chain maps form a category $\operatorname{Hom}(M_{i+1},M_i)$6. In an abelian category, choosing morphisms $\operatorname{Hom}(M_{i+1},M_i)$7 satisfying

$\operatorname{Hom}(M_{i+1},M_i)$8

recovers the standard notion of a chain complex. This is important for interpretation: a chain bundle is generally more flexible than a chain complex, and a chain complex appears only after an additional selection of arrows satisfying the usual differential condition (Romeo et al., 2020).

5. Chains arising from the category of bundles

A distinct but related development studies chains built from classical bundles. The 2026 paper defines a bundle as a triple $\operatorname{Hom}(M_{i+1},M_i)$9, defines bundle morphisms $C$0 by a commutative square, and forms the category $C$1 whose objects are bundles and whose morphisms are bundle morphisms. With subobjects given by subbundles, $C$2 is proved to be a category with subobjects (Romeo et al., 27 May 2026).

From $C$3, the paper constructs chains of bundles: $C$4 where each $C$5 is a bundle morphism. A morphism between two such chains is a family

$C$6

with

$C$7

satisfying

$C$8

With subchains defined by levelwise subbundle inclusions, the category of chains of bundles is again a category with subobjects (Romeo et al., 27 May 2026).

The same paper develops three concrete classes of examples. Jet bundles give chains

$C$9

Chains of fibres arise from chains of subbundles, producing sequences of fibre inclusions over a fixed base point. Long exact sequences arising from fibrations

$c$0

are treated as another chain category, with subfibrations inducing subchains (Romeo et al., 27 May 2026).

A recurring source of ambiguity is terminological. In the 2020 and 2021 papers, “chain bundle” refers to an abstract categorical object built from objects and morphism sets in a category with zero. In the 2026 paper, the construction is applied to actual bundles in the classical geometric sense. This suggests two related levels of usage: an abstract chain-bundle formalism and a geometric family of chain categories derived from bundle theory.

6. Examples, extensions, and categorical significance

The abstract theory is illustrated by a broad range of examples. The 2020 paper works with $c$1-modules, subgroup categories such as subgroups of $c$2, and a groupoid setting in which a suitable subcategory of chain bundle maps is itself a groupoid. It also notes that products and coproducts of chain bundles are defined termwise, after padding shorter chains with zeros on the left (Romeo et al., 2020).

The 2021 paper adds several further sources of examples. If $c$3 is a directed graph with a loop at each vertex and is regarded as a category whose objects are vertices and whose morphisms are paths, then for a vertex $c$4 any subset of the set $c$5 of paths ending at $c$6 is a chain bundle in $c$7, giving a category $c$8. The augmented simplex category $c$9, consisting of finite ordinals and order-preserving maps, yields a chain-bundle category $$\cdots \; M_3 \; S_3 \; M_2 \; S_2 \; M_1 \; S_1 \; M_0 = 0$$0. The paper also describes a functor between chain-bundle categories, including the forgetful functor from groups to pointed sets, and formulates universal arrows in this setting (Romeo et al., 2021).

Taken together, these results assign the category of chain bundles a specific categorical role. It is a framework for sequences of objects equipped with many morphisms between successive terms, rather than one distinguished arrow. It supports subobjects, and under suitable hypotheses it supports factorization, products, kernels, cokernels, functors, and universal arrows. The category of chains is recovered by enforcing uniqueness of the arrow at each stage, while classical geometric bundle constructions provide a large supply of concrete chain categories. In the language of the 2026 paper, this places chain categories arising from bundles within the broader setting of Nambooripad-style categories with subobjects (Romeo et al., 2021, Romeo et al., 27 May 2026).