Sequence category

category of sequences

A particular case of the general construction of functor categories or diagram categories. Let $\mathbb{Z}$ be the set of integers equipped with the usual order relation. Then $\mathbb{Z}$ can be considered as a small category with integers as objects and all possible pairs $(i,j)$, where $i,j \in \mathbb{Z}$ and $i \le j$, as morphisms. The pair $(i,j)$ is the unique morphism from the object $i$ to the object $j$. Composition of morphisms is defined as follows: $(j,k)(i,j) = (i,k)$.

For an arbitrary category $\mathfrak{K}$, the category of functors from $\mathbb{Z}$ to $\mathfrak{K}$ is called the category of sequences in $\mathfrak{K}$. To define a functor $F : \mathbb{Z} \rightarrow \mathfrak{K}$, it is sufficient to indicate a family of objects from $\mathfrak{K}$, indexed by the integers, and for each integer $i$ to choose a morphism $\alpha_{i,i+1} : A_i \rightarrow A_{i+1}$. Then the assignment $F(i) = A_i$, $F((i,j)) = \alpha_{i,i+1}$ extends uniquely to a functor $F : \mathbb{Z} \rightarrow \mathfrak{K}$. A natural transformation $\phi$ from the functor $F : \mathbb{Z} \rightarrow \mathfrak{K}$ to a functor $G : \mathbb{Z} \rightarrow \mathfrak{K}$, i.e. a morphism in the category of sequences, is defined by a family of morphisms $\phi_i : F(i) \rightarrow G(i)$ such that $\phi_i.G((i,i+1)) = F((i,i+1)).\phi_{i+1}$ for any $i \in \mathbb{Z}$.

If $\mathfrak{K}$ is a category with null morphisms, then in the category of sequences in $\mathfrak{K}$ one can isolate the full subcategory of complexes, i.e. functors $F : \mathbb{Z} \rightarrow \mathfrak{K}$ such that $F((i+1,i+2)).F((i,i+1)) = 0$ for any $i \in \mathbb{Z}$, cf Complex (in homological algebra). For any Abelian category $\mathfrak{A}$ the category of sequences and the subcategory of complexes are Abelian categories.

Instead of the category $\mathbb{Z}$ one can consider its subcategories of non-negative or non-positive numbers. The corresponding diagram categories are also called categories of sequences.

References

[a1]	S. MacLane, "Homology" , Springer (1963) pp. Chapt. IX, §3