PlanetPhysics/Categorical Sequence

A categorical sequence is a linear `[[../TrivialGroupoid/|diagram]]' of [[../TrivialGroupoid/|morphisms]], or arrows, in an abstract [[../Cod/|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:

Failed to parse (unknown function "\buildrel"): {\displaystyle \cdots \rightarrow A\buildrel f \over \longrightarrow B \buildrel \phi \over \longrightarrow Hom_{Set}(A,B), }

where $Ho{m}_{Set}(A,B)$ is the set of [[../Bijective/|functions]] from set $A$ to set $B$.

Examples

The chain complex is a categorical sequence example:

Consider a ring $R$ and the chain complex consisting of a sequence of $R$-modules and [[../TrivialGroupoid/|homomorphisms]]:

Failed to parse (unknown function "\buildrel"): {\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 Failed to parse (unknown function "\im"): {\displaystyle \im d_{n+1} \subseteq \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 Failed to parse (unknown function "\buildrel"): {\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 {\it exact} if each pair of adjacent homomorphisms $(f_{n+1},f_n)$ is exact , that is, if $\mathrm{i}\mathrm{m}{f}_{n+1}=\mathrm{k}\mathrm{e}\mathrm{r}{f}_n$ for all $n$. This [[../PreciseIdea/|concept]] can be then generalized to morphisms in a categorical exact sequence, thus leading to the corresponding definition of an exact sequence in an [[../AbelianCategory2/|abelian category]].

Inasmuch as [[../CategoricalDiagramsDefinedByFunctors/|categorical diagrams]] can be defined as [[../TrivialGroupoid/|functors]], exact sequences of special [[../Bijective/|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 $ℬ$, $F:𝒜\to ℬ$, 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 $E$ [[../TrivialGroupoid/|objects]] in $𝒜$) into the corresponding sequence in the Abelian category $ℬ$, ($0\to F(C)\to F(D)\to F(E)\to 0$), which is also exact (in $ℬ$).