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'categorical pullback'
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| Title of object: |
categorical pullback |
| Canonical Name: |
CategoricalPullback |
| Type: |
Definition |
| Created on: |
2004-02-14 23:21:23 |
| Modified on: |
2004-02-14 23:24:02 |
| Classification: |
msc:18A30 |
| Defines: |
pullback square, cone, pushout, pushout square, vertex, base |
| Synonyms: |
categorical pullback=pullback |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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Content:
\PMlinkescapeword{factor}
Let $X\to B$ and $Y\to B$ be morphisms. Then a limiting \emph{cone} over
$X\to B\from Y$, or a \emph{pullback square}, is a commutative
diagram
\[\xymatrix{
Z\ar[d]\ar[r] & Y\ar[d] \\
X\ar[r] & B.
}\]
The object $Z$ is called the \emph{vertex} of the cone.
The \emph{pullback} $X\times_B Y$ over $B$, if it exists,
is the vertex of a terminal
cone over $X\to B\from Y$. That is, if $Z$ is the vertex of a cone
over $X\to B\from Y$, then $Z$ must factor uniquely through $X\times_B Y$ in the
commutative diagram:
\[\xymatrix{
Z\ar@/^1ex/[rrd]\ar@/_1ex/[rdd]\ar[rd] & & \\
& X\times_B Y\ar[d]\ar[r] & Y\ar[d] \\
& X\ar[r] & B.
}\]
Dually, given morphisms $B\to X$ and $B\to Y$, a colimiting cone
from $X\from B\to Y$, usually called a \emph{pushout square}, is a
commutative diagram
\[\xymatrix{
B\ar[d]\ar[r] & Y\ar[d] \\
X\ar[r] & Z,
}\]
The object $Z$ is called the \emph{base} of the cone. The \emph{pushout}
$X\amalg_B Y$ from $B$, if it exists, is the base of an initial cone
from $X\from B\to Y$.
That is, if $Z$ is the base of a cone from $X\from B\to Y$, then $X\amalg_B Y$
must factor uniquely through $Z$ in the commutative diagram:
\[\xymatrix{
B\ar[d]\ar[r] & Y\ar[d]\ar@/^1ex/[ddr] & \\
X\ar[r]\ar@/_1ex/[drr] & X\amalg_B Y\ar[dr] & \\
& & Z.
}\]
Pullbacks and pushouts are unique up to unique isomorphism when they exist. |
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