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5
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'commutative diagram'
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| Title of object: |
commutative diagram |
| Canonical Name: |
CommutativeDiagram |
| Type: |
Definition |
| Created on: |
2003-02-02 20:29:23 |
| Modified on: |
2004-01-24 03:19:30 |
| Classification: |
msc:18A05 |
| Defines: |
commutative diagram |
Revision comment (for changes between this and next version):
| Removed redundant "commutative diagram" synonym |
Preamble:
%\documentclass{amsart}
\usepackage{amsmath}
\usepackage[all,poly,knot,dvips]{xy}
%\usepackage{pstricks,pst-poly,pst-node,pstcol}
\usepackage{amssymb,latexsym}
\usepackage{amsthm,latexsym}
\usepackage{eucal,latexsym}
% THEOREM Environments --------------------------------------------------
\newtheorem{thm}{Theorem}
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\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
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\newtheorem{claim}[thm]{Claim}
\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\theoremstyle{remark}
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\numberwithin{equation}{subsection}
%--------------------- Greek letters, etc -------------------------
\newcommand{\CA}{\mathcal{A}}
\newcommand{\CC}{\mathcal{C}}
\newcommand{\CM}{\mathcal{M}}
\newcommand{\CP}{\mathcal{P}}
\newcommand{\CS}{\mathcal{S}}
\newcommand{\BC}{\mathbb{C}}
\newcommand{\BN}{\mathbb{N}}
\newcommand{\BR}{\mathbb{R}}
\newcommand{\BZ}{\mathbb{Z}}
\newcommand{\FF}{\mathfrak{F}}
\newcommand{\FL}{\mathfrak{L}}
\newcommand{\FM}{\mathfrak{M}}
\newcommand{\Ga}{\alpha}
\newcommand{\Gb}{\beta}
\newcommand{\Gg}{\gamma}
\newcommand{\GG}{\Gamma}
\newcommand{\Gd}{\delta}
\newcommand{\GD}{\Delta}
\newcommand{\Ge}{\varepsilon}
\newcommand{\Gz}{\zeta}
\newcommand{\Gh}{\eta}
\newcommand{\Gq}{\theta}
\newcommand{\GQ}{\Theta}
\newcommand{\Gi}{\iota}
\newcommand{\Gk}{\kappa}
\newcommand{\Gl}{\lambda}
\newcommand{\GL}{\Lamda}
\newcommand{\Gm}{\mu}
\newcommand{\Gn}{\nu}
\newcommand{\Gx}{\xi}
\newcommand{\GX}{\Xi}
\newcommand{\Gp}{\pi}
\newcommand{\GP}{\Pi}
\newcommand{\Gr}{\rho}
\newcommand{\Gs}{\sigma}
\newcommand{\GS}{\Sigma}
\newcommand{\Gt}{\tau}
\newcommand{\Gu}{\upsilon}
\newcommand{\GU}{\Upsilon}
\newcommand{\Gf}{\varphi}
\newcommand{\GF}{\Phi}
\newcommand{\Gc}{\chi}
\newcommand{\Gy}{\psi}
\newcommand{\GY}{\Psi}
\newcommand{\Gw}{\omega}
\newcommand{\GW}{\Omega}
\newcommand{\Gee}{\epsilon}
\newcommand{\Gpp}{\varpi}
\newcommand{\Grr}{\varrho}
\newcommand{\Gff}{\phi}
\newcommand{\Gss}{\varsigma}
\def\co{\colon\thinspace} |
Content:
\begin{defn}
Let $\mathcal{C}$ be a category. A \emph{diagram} in $\CC$ is a
directed graph $\GG$ with vertex set $V$ and edge set $E$, (``loops''
and ``parallel edges'' are allowed) together with two maps
$o\co V\to\mathrm{Obj}(\CC)$, $m\co E\to \mathrm{Morph}(\CC)$ such that
if $e\in E$ has source $s(e)\in V$ and target $t(e)\in V$ then
$m(e) \in \text{Hom}_{\CC}\left(s(e),t(e)\right)$.
\end{defn}
Usually diagrams are denoted by drawing the corresponding graph
and labeling its vertices (respectively edges) with their images under $o$
(respectively $m$), for example if $f\co A\to B$ is a morphism
$$\xymatrix@1{ {A}\ar[r]^f&{B} }$$
is a diagram. Often (as in the previous example) the vertices themselves are
not drawn since their position can be deduced by the position of their
labels.
\begin{defn}
Let $D=(\GG,o,m)$ be a diagram in the category $\CC$ and $\Gg=(e_1,\ldots,e_n)$
be a path in $\GG$. Then the \emph{composition along} $\Gg$ is the following
morphism of $\CC$
$$\circ(\Gg):=m(e_n)\circ\cdots\circ m(e_1)\,.$$
We say that $D$ is
\emph{commutative} or that it \emph{commutes} if for any two objects in
the image of $o$, say $A=o(v_1)$ and $B=o(v_2)$, and any two paths $\Gg_1$
and $\Gg_2$ that connect $v_1$ to $v_2$ we have
$$\circ(\Gg_1)=\circ(\Gg_2)\,.$$
\end{defn}
For example the commutativity of the triangle
$$\xymatrix{
{A}\ar[rr]^{f}\ar[dr]_{h}&&{B}\ar[dl]^{g}\\
&{C}&
}
$$
translates to $h=g\circ f$, while the commutativity of the square
$$\xymatrix{
{A}\ar[r]^{f}\ar[d]_{k}&{B}\ar[d]^{g}\\
{C}\ar[r]_{h}&{D}
}
$$
translates to $g\circ f=h\circ k$. |
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