categorical pullbackCategoricalPullback2004-02-14 23:21:232009-02-18 15:10:02Definitionpullback squarepushoutcategorical pushoutpushout squaregeneralized pullbackgeneralized pushouthave pullbackshave pushouts% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
\usepackage[curve]{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newcommand{\from}{\leftarrow}\PMlinkescapeword{factor}
\subsection*{Pullbacks}
Let $f:X\to B$ and $g:Y\to B$ be morphisms in a category $\mathcal{C}$. Then a \emph{pullback diagram}, or \emph{pullback square} of $f$ and $g$ is a commutative diagram
\[\xymatrix{
A\ar[d]_p \ar[r]^q & Y\ar[d]^g \\
X\ar[r]_f & B.
}\]
such that if we have another commutative diagram
\[\xymatrix{
Z\ar[d]_r\ar[r]^s & Y\ar[d]^g \\
X\ar[r]_f & B.
}\]
then there is a unique morphism $h:Z\to A$ with the commutative diagram
\[\xymatrix{
Z\ar@/^1ex/[rrd]^s \ar@/_1ex/[rdd]_r \ar@{.>}[rd]|h & & \\
& A\ar[d]_p\ar[r]^q & Y\ar[d]^g \\
& X\ar[r]_f & B.
}\]
A \emph{pullback} of $(f,g)$ is the ordered triple $(A,q,p)$. We also say that $p$ is a \emph{pullback of $g$ along $f$}, and $q$ a \emph{pullback of $f$ along $g$}. When a pullback of $(f,g)$ exists, it is unique up to isomorphism. The object $A$, and the morphisms $p$ and $q$ in the diagram above are often denoted by $$X\times_B Y,\qquad 1_X \times_B g\qquad \mbox{and} \qquad f\times_B 1_Y$$ respectively, and the uniquely determined morphism $h$ by $$\displaystyle{r \choose s}.$$ It is easy to see that $$X\times_B Y\cong Y\times_B X,$$ whenever one (and hence the other) exists.
\textbf{Remarks}.
\begin{itemize}
\item
The pullback of $f$ and $g$ can be equivalently defined as a limiting cone over the diagram $X\to B\from Y$. In other words, a pullback diagram is a terminal object in the category of commutative squares of the form
\[\xymatrix{
Z\ar[d] \ar[r] & Y\ar[d]^g \\
X\ar[r]_f & B.
}\]
\item A category $\mathcal{C}$ is said to \emph{have pullbacks} if every diagram $X \to B \from Y$ can be completed into a pullback diagram.
\item
The notion of pullbacks can be generalized: let $\lbrace x_i: C_i \to C \mid i\in I\rbrace$ be a collection of morphisms indexed by set $I$, considered as a small diagram. The \emph{generalized pullback} of the $x_i$'s is just the limiting cone of the diagram. Using this definition, the generalized pullback of one morphism $f$ is the identity morphism of $\operatorname{dom}(f)$, the domain of $f$, and the generalized pullback of the empty set is a terminal object.
\item A pullback is sometimes known as an \emph{amalgamated sum}.
\end{itemize}
\subsection*{Pushouts}
Dually, given morphisms $f:B\to X$ and $g:B\to Y$, a \emph{pushout square} of $f$ and $g$ is a commutative diagram
\[\xymatrix{
B\ar[d]_f \ar[r]^g & Y\ar[d]^s \\
X\ar[r]_t & A
}\]
such that if we have another commutative diagram
\[\xymatrix{
B\ar[d]_f \ar[r]^g & Y\ar[d]^u \\
X\ar[r]_v & Z
}\]
Then there is a unique morphism $h:A\to Z$ with the following commutative diagram:
\[\xymatrix{
B\ar[d]_f \ar[r]^g & Y\ar[d]^s \ar@/^1ex/[ddr]^u & \\
X\ar[r]_t \ar@/_1ex/[drr]_v & A \ar@{.>}[dr]|-h & \\
& & Z.
}\]
The pair $(s,t)$ of morphisms is a \emph{pushout} of $(f,g)$. $s$ is the the \emph{pushout of $f$ along $g$}, and $t$ the \emph{pushout of $g$ along $f$}. Like pullbacks, pushouts are unique up to unique isomorphism when they exist. The object $A$ and the morphisms $s$ and $t$ are typically written $$X\amalg_B Y,\qquad f\amalg_B 1_Y \qquad \mbox{and} \qquad 1_X \amalg_B g,$$ and the unique morphism $h$ is denoted by $$(f \enspace g).$$
\textbf{Remark}. The pushout of $f$ and $g$ can be thought of as the limiting cocone under the diagram $X \from B \to Y$. Equivalently, they are initial objects in the category of commutative squares whose top edge is $B\to Y$ and left edge is $B\to X$. A category is said to \emph{have pushouts} if every diagram $X \from B \to Y$ can be completed to a pushout diagram. The \emph{generalized pushout} is defined as the limiting cocone under the diagram consisting of morphisms $y_i: B\to B_i$, where $i$ belongs to some set $I$.