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
is the set of functions
from set
to set
.
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

between two Abelian categories,

and

,

, which preserves categorical exact sequences, that is, if

carries a short exact sequence

(with

and

objects in

) into the corresponding sequence in the Abelian category

, (

), which is also exact (in

).