Category of Chain Bundles

Abstract

In this paper, we describe the category of chain bundles $\mathfrak {CB}_\mathcal {C}$ in a category $\mathcal {C}$ with zero object. The objects of $\mathfrak {CB}_\mathcal {C}$ are

$$\begin{aligned} \cdots M_{i+1} {\Rrightarrow } M_i {\Rrightarrow } M_{i-1} \cdots {\Rrightarrow } M_0 = \mathbf{0} \end{aligned}$$

where $M_i \in \nu \mathcal {C} \quad \forall i$ and ${\mathop {\Rrightarrow }\limits ^{Hom(M_{i+1},M_i)}}$ denotes the homset $Hom(M_i,M_j)$; in particular, the homsets includes sets of the form $Hom(M_{i},M_i)$ and all possible composite of morphisms in $\mathcal {C}$. The morphisms in this category are appropriate maps (functors) between objects of $\mathfrak {CB}_\mathcal {C}$ called chain bundle maps.