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categorical sequence (Definition)
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:
$\displaystyle \cdots \rightarrow A\buildrel f \over \longrightarrow B \buildrel \phi \over \longrightarrow Hom_{Set}(A,B) , $
where $ Hom_{Set}(A,B)$ is the set of functions from set $ A$ to set $ B$.

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:

$\displaystyle \cdots \rightarrow A_{n+1} \buildrel {d_{n+1}} \over \longrightarrow A_n \buildrel {d_n} \over \longrightarrow A_{n-1} \rightarrow \cdots $
(with the additional condition imposed by $ d_n\circ d_{n+1} = 0$ for each pair of adjacent homomorphisms $ (d_{n+1}, d_n)$; this is equivalent to the condition $ \operatorname{im}d_{n+1} \subseteq \operatorname{ker}d_n$ that needs to be satisfied in order to define this categorical sequence completely as a chain complex). Furthermore, a sequence of homomorphisms
$\displaystyle \cdots \rightarrow A_{n+1} \buildrel {f_{n+1}} \over \longrightarrow A_n \buildrel {f_n} \over \longrightarrow A_{n-1} \rightarrow \cdots $
is said to be exact if each pair of adjacent homomorphisms $ (f_{n+1}, f_n)$ is exact, that is, if $ {\rm im} f_{n+1} = {\rm ker} f_n$ 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.

Remarks 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, $ \mathcal{A}$ and $ \mathcal{B}$, $ F: \mathcal{A} \to \mathcal{B}$, which preserves categorical exact sequences, that is, if $ F$ carries a short exact sequence $ 0 \to C \to D \to E \to 0$ (with $ 0, C, D$ and $ E$ objects in $ \mathcal{A}$) into the corresponding sequence in the Abelian category $ \mathcal{B}$, ( $ 0 \to F(C) \to F(D) \to F(E) \to 0$), which is also exact (in $ \mathcal{B}$).



"categorical sequence" is owned by bci1.
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See Also: chain complex, exact sequence, commutative diagram, abelian category, short exact sequence, exact functor, exact sequence, tangential Cauchy-Riemann complex of $C^{\infty)$-smooth forms, alternative definition of an Abelian category, superdiagrams as heterofunctors, category theory, Grothendieck category, image of a morphism, homological complex of topological vector spaces, cohomological complex of topological vector spaces, categorical diagrams as functors, spin groups, tangential Cauchy-Riemann complex of smooth forms, exact sequence theorem in $C_3$--category, additive quotient category

Other names:  linear diagrams
Also defines:  linear diagram, (linear) sequence of morphisms, exact functor, short exact sequence
Keywords:  categorical sequence, or linear diagram, of sets and set-theoretical mappings, exact functor, short exact sequence, commutative homological algebra
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Cross-references: objects, preserves, algebra, commutative, abelian, types of morphisms, exact sequences, functors, diagrams, categorical, abelian category, order, equivalent, adjacent, homomorphisms, sequence, ring, chain complex, functions, mappings, category of sets, concrete category, abstract category, morphisms
There are 24 references to this entry.

This is version 27 of categorical sequence, born on 2008-08-17, modified 2008-09-18.
Object id is 10951, canonical name is CategoricalSequence.
Accessed 673 times total.

Classification:
AMS MSC: 18-00 (Category theory; homological algebra :: General reference works )
 18E05 (Category theory; homological algebra :: Abelian categories :: Preadditive, additive categories)
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