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A special case of the concept of an (inverse or projective) limit. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f0400501.png" /> be a category and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f0400502.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f0400503.png" /> be given morphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f0400504.png" />. An object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f0400505.png" />, together with morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f0400506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f0400507.png" />, is called a fibre product of the objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f0400508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f0400509.png" /> (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005011.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005012.png" />, and if for any pair of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005014.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005015.png" /> there exists a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005018.png" />. The commutative square
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{{MSC|18}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005019.png" /></td> </tr></table>
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The ''fibre product of objects in a category'' is
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a special case of the concept of an [[Projective limit|(inverse or projective) limit]]. Let $\def\fK{ {\mathfrak K}}\fK$ be a [[category]] and let $\def\a{\alpha}\a : A\to C$ and $\def\b{\beta}\b : B\to C$ be given morphisms in $\fK$. An object $D$, together with morphisms $\def\phi{\varphi}\phi:D\to A$, $\psi:D\to B$, is called a fibre product of the objects $A$ and $B$ (over $\a$ and $\b$) if $\phi\a=\psi\b$, and if for any pair of morphisms $\def\g{\gamma}\g:X\to A$, $\def\d{\delta}\d:X\to B$ for which $\g\a =\d\b$ there exists a unique morphism $\xi:X\to D$ such that $\xi\phi = \g$, $\xi\psi = \d$. The commutative square
  
is often called a universal or Cartesian square. The object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005020.png" />, together with the morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005022.png" />, is a limit of the diagram
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$$\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}
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\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex}
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\newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}}
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\begin{array}{ccc} D & \ra{\psi} & B\\ \da{\phi} & & \da{\b}\\A&\ra{\a}& C\end{array}$$
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is often called a universal or Cartesian square. The object $D$, together with the morphisms $\phi$ and $\psi$, is a limit of the diagram
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005023.png" /></td> </tr></table>
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$$\begin{array}{ccc} &  & B\\  & & \da{\b}\\A&\ra{\a}& C.\end{array}$$
 
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The fibre product of $A$ and $B$ over $\a$ and $\b$ is written as
The fibre product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005025.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005027.png" /> is written as
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005028.png" /></td> </tr></table>
 
  
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$$A\times_C B,\quad A\times_{\a,\b}B, \quad \textrm{ or }\quad A\prod_{\a,\b} B.$$
 
If it exists, the fibre product is uniquely defined up to an isomorphism.
 
If it exists, the fibre product is uniquely defined up to an isomorphism.
  
In a category with finite products and kernels of pairs of morphisms, the fibre product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005030.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005032.png" /> is formed as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005033.png" /> be the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005035.png" /> with projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005037.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005038.png" /> be the kernel of the pair of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005039.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005040.png" />, together with the morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005042.png" />, is a fibre product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005044.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005046.png" />. In many categories of structured sets, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005047.png" /> is the subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005048.png" /> consisting of all those pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005049.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005050.png" />.
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In a category with finite products and [[Kernel of a morphism in a category|kernels of pairs of morphisms]], the fibre product of $A$ and $B$ over $\a$ and $\b$ is formed as follows. Let $P=A\times B$ be the product of $A$ and $B$ with projections $\pi_1$ and $\pi_2$ and let $(D,\mu)$ be the kernel of the pair of morphisms $\pi_1\a,\pi_2\b:P\to C$. Then $D$, together with the morphisms $\mu\pi_1=\phi$ and $\mu\pi_2 = \psi$, is a fibre product of $A$ and $B$ over $\a$ and $\b$. In many categories of structured sets, $D$ is the subset of $A\times B$ consisting of all those pairs $(a,b)$ for which $a\a = b\b$.
  
  
  
====Comments====
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====Pullback, Pushout====
In the literature on category theory, fibre products are most commonly called pullbacks, and examples of the dual notion (i.e. fibre products in the opposite of the category under consideration) are called pushouts. The name  "fibre product"  derives from the fact that, in the category of sets (and hence, in any concrete category whose underlying-set functor preserves pullbacks), the fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005051.png" /> over an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005052.png" /> (i.e. the inverse image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005053.png" /> under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005054.png" />) is the Cartesian product of the fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005056.png" />. Note also that (binary) products (cf. [[Product of a family of objects in a category|Product of a family of objects in a category]]) are a special case of pullbacks, in which the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040050/f04005057.png" /> is taken to be a [[Final object|final object]] of the category.
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In the literature on category theory, fibre products are most commonly called [[pullback|pullbacks]], and examples of the dual notion (i.e. fibre products in the opposite of the category under consideration) are called [[pushout|pushouts]]. The name  "fibre product"  derives from the fact that, in the category of sets (and hence, in any concrete category whose underlying-set functor preserves pullbacks), the fibre of $A\times_C B$ over an element $c\in C$ (i.e. the inverse image of $c$ under the mapping $\a\phi$) is the Cartesian product of the fibres $\a^{-1}(c)\subseteq A$ and $\b^{-1}(c)\subseteq B$. Note also that (binary) products (cf.
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[[Product of a family of objects in a category|Product of a family of objects in a category]]) are a special case of pullbacks, in which the object $C$ is taken to be a
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[[Final object|final object]] of the category.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Mitchell,  "Theory of categories" , Acad. Press (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Adámek,  "Theory of mathematical structures" , Reidel (1983)</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|Ad}}||valign="top"| J. Adámek,  "Theory of mathematical structures", Reidel (1983)   {{MR|0735079}}  {{ZBL|0523.18001}} 
 +
|-
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|valign="top"|{{Ref|Mi}}||valign="top"| B. Mitchell,  "Theory of categories", Acad. Press (1965)   {{MR|0202787}}  {{ZBL|0136.00604}} 
 +
|-
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|}

Latest revision as of 10:30, 23 November 2013

2020 Mathematics Subject Classification: Primary: 18-XX [MSN][ZBL]

The fibre product of objects in a category is a special case of the concept of an (inverse or projective) limit. Let $\def\fK{ {\mathfrak K}}\fK$ be a category and let $\def\a{\alpha}\a : A\to C$ and $\def\b{\beta}\b : B\to C$ be given morphisms in $\fK$. An object $D$, together with morphisms $\def\phi{\varphi}\phi:D\to A$, $\psi:D\to B$, is called a fibre product of the objects $A$ and $B$ (over $\a$ and $\b$) if $\phi\a=\psi\b$, and if for any pair of morphisms $\def\g{\gamma}\g:X\to A$, $\def\d{\delta}\d:X\to B$ for which $\g\a =\d\b$ there exists a unique morphism $\xi:X\to D$ such that $\xi\phi = \g$, $\xi\psi = \d$. The commutative square

$$\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{ccc} D & \ra{\psi} & B\\ \da{\phi} & & \da{\b}\\A&\ra{\a}& C\end{array}$$ is often called a universal or Cartesian square. The object $D$, together with the morphisms $\phi$ and $\psi$, is a limit of the diagram

$$\begin{array}{ccc} & & B\\ & & \da{\b}\\A&\ra{\a}& C.\end{array}$$ The fibre product of $A$ and $B$ over $\a$ and $\b$ is written as

$$A\times_C B,\quad A\times_{\a,\b}B, \quad \textrm{ or }\quad A\prod_{\a,\b} B.$$ If it exists, the fibre product is uniquely defined up to an isomorphism.

In a category with finite products and kernels of pairs of morphisms, the fibre product of $A$ and $B$ over $\a$ and $\b$ is formed as follows. Let $P=A\times B$ be the product of $A$ and $B$ with projections $\pi_1$ and $\pi_2$ and let $(D,\mu)$ be the kernel of the pair of morphisms $\pi_1\a,\pi_2\b:P\to C$. Then $D$, together with the morphisms $\mu\pi_1=\phi$ and $\mu\pi_2 = \psi$, is a fibre product of $A$ and $B$ over $\a$ and $\b$. In many categories of structured sets, $D$ is the subset of $A\times B$ consisting of all those pairs $(a,b)$ for which $a\a = b\b$.


Pullback, Pushout

In the literature on category theory, fibre products are most commonly called pullbacks, and examples of the dual notion (i.e. fibre products in the opposite of the category under consideration) are called pushouts. The name "fibre product" derives from the fact that, in the category of sets (and hence, in any concrete category whose underlying-set functor preserves pullbacks), the fibre of $A\times_C B$ over an element $c\in C$ (i.e. the inverse image of $c$ under the mapping $\a\phi$) is the Cartesian product of the fibres $\a^{-1}(c)\subseteq A$ and $\b^{-1}(c)\subseteq B$. Note also that (binary) products (cf. Product of a family of objects in a category) are a special case of pullbacks, in which the object $C$ is taken to be a final object of the category.

References

[Ad] J. Adámek, "Theory of mathematical structures", Reidel (1983) MR0735079 Zbl 0523.18001
[Mi] B. Mitchell, "Theory of categories", Acad. Press (1965) MR0202787 Zbl 0136.00604
How to Cite This Entry:
Fibre product of objects in a category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fibre_product_of_objects_in_a_category&oldid=13216
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article