Definition 0.1. A categorical sequence is a linear ‘diagram’ of morphisms, or arrows, in
an abstract category. In a concrete category, such as the category of sets, the categorical
sequence consists of sets joined by set-theoretical mappings in linear fashion, such as:
where HomSet(A,B) is the set of functions from set A to set B.
0.1 Examples
0.1.1 The chain complex is a categorical sequence example:
Consider a ring R and the chain complex consisting of a sequence of R-modules and
homomorphisms:
(with the additional condition imposed by dn ∘dn+1 = 0 for each pair of adjacent homomorphisms
(dn+1,dn); this is equivalent to the condition im dn+1 ⊆ ker dn that needs to be satisfied in order to
define this categorical sequence completely as a chain complex). Furthermore, a sequence of
homomorphisms
is said to be exact if each pair of adjacent homomorphisms (fn+1,fn) is exact, that is, if
imfn+1 = kerfn for all n. This concept can be then generalized to morphisms in a categorical
exact sequence, thus leading to the corresponding definition of an exact sequence in an Abelian
category.
Remark 0.1. Inasmuch as categorical diagrams can be defined as functors, exact sequences
of special types of morphisms can also be regarded as the corresponding, special functors.
Thus, exact sequences in Abelian categories can be regarded as certain functors of Abelian
categories; the details of such functorial (abelian) constructions are left to the reader as
an exercise. Moreover, in (commutative or Abelian) homological algebra, an exact functor
is simply defined as a functor F between two Abelian categories, 𝒜 and ℬ, F : 𝒜 → ℬ,
which preserves categorical exact sequences, that is, if F carries a short exact sequence
0 → C → D → E → 0 (with 0,C,D and E objects in 𝒜) into the corresponding sequence
in the Abelian category ℬ, (0 → F(C) → F(D) → F(E) → 0), which is also exact (in ℬ).
