%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: categorical sequence
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%%% Owner: bci1
%%% Author(s): bci1
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\begin{document}
\begin{definition}
A \emph{categorical sequence} is a linear `\htmladdnormallink{diagram}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}' of \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, or arrows, in an abstract \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html}.
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:
\[
\cdots \rightarrow
A\buildrel f \over \longrightarrow
B \buildrel \phi \over \longrightarrow
Hom_{Set}(A,B),
\]
where $Hom_{Set}(A,B)$ is the set of \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} from set $A$ to set $B$.
\end{definition}
\subsection{Examples}
\subsubsection{The chain complex is a categorical sequence example:}
Consider a ring $R$ and the chain complex consisting of
a sequence of $R$-modules and \htmladdnormallink{homomorphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}:
\[
\cdots \rightarrow
A_{n+1} \buildrel {d_{n+1}} \over \longrightarrow
A_n \buildrel {d_n} \over \longrightarrow
A_{n-1} \rightarrow
\cdots
\]
(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 $\im d_{n+1} \subseteq \ker d_n$ that needs to be satisfied in order to define this categorical sequence completely as a \emph{chain complex}). Furthermore, a sequence of homomorphisms
$$\cdots \rightarrow
A_{n+1} \buildrel {f_{n+1}} \over \longrightarrow
A_n \buildrel {f_n} \over \longrightarrow
A_{n-1} \rightarrow
\cdots $$
is said to be {\it exact} if each pair of adjacent homomorphisms $(f_{n+1}, f_n)$ is \emph{exact}, that is, if ${\rm im} f_{n+1} = {\rm ker} f_n$ for all $n$. This \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} can be then generalized to
morphisms in a categorical exact sequence, thus leading to the corresponding definition of an
\htmladdnormallink{exact sequence}{http://planetmath.org/encyclopedia/ExactSequence2.html} in an \htmladdnormallink{abelian category}{http://planetphysics.us/encyclopedia/AbelianCategory2.html}.
\begin{remark}
Inasmuch as \htmladdnormallink{categorical diagrams}{http://planetphysics.us/encyclopedia/CategoricalDiagramsDefinedByFunctors.html} can be defined as \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, exact sequences of special \htmladdnormallink{types}{http://planetphysics.us/encyclopedia/Bijective.html} 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 $E$ \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} 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 $\mathcal{B}$).
\end{remark}
\end{document}
