Fibre product of objects in a category

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