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categorical pullback
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(Definition)
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Let  and  be morphisms. Then a limiting cone over
 , or a pullback square, is a commutative diagram
The object  is called the vertex of the cone. The pullback
 over  , if it exists, is the vertex of a terminal cone over
 . That is, if  is the vertex of a cone over
 , then  must factor uniquely through
 in the commutative diagram:
Dually, given morphisms  and  , a colimiting cone from
 , usually called a pushout square, is a commutative diagram
The object  is called the base of the cone. The pushout
 from  , if it exists, is the base of an initial cone from
 . That is, if  is the base of a cone from
 , then
 must factor uniquely through  in the commutative diagram:
Pullbacks and pushouts are unique up to unique isomorphism when they exist.
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"categorical pullback" is owned by mps.
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(view preamble)
Cross-references: isomorphism, cone over, terminal, object, commutative diagram, morphisms
There are 31 references to this entry.
This is version 2 of categorical pullback, born on 2004-02-14, modified 2004-02-14.
Object id is 5579, canonical name is CategoricalPullback.
Accessed 11269 times total.
Classification:
| AMS MSC: | 18A30 (Category theory; homological algebra :: General theory of categories and functors :: Limits and colimits ) |
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Pending Errata and Addenda
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![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ B\ar[d]\ar[r] & Y\ar[d] \ X\ar[r] & Z, } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/5579/l2h/img17.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ B\ar[d]\ar[r] & Y\ar[d] \ X\ar[r] & Z, } } \end{xy}$")
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