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Pullback . category theory


context $F:({a\rightarrow z\leftarrow b})\longrightarrow{\bf C}$
definition $\langle Fa\times_{Fz} Fb, \pi\rangle := \mathrm{lim}\,F$

Here we consider a functor $F$ from the category ${a\rightarrow z\leftarrow b}$, consisting of three object and two non-identity arrows $f_a$ and $f_b$, to a category ${\bf C}$.

Universal property

For readability, let's write $A\equiv{Fa}, B\equiv{Fb}, Z\equiv{Fz}, \alpha\equiv{f_a}$ and $\beta\equiv{f_b}$.

(In the picture we have $X\equiv{Fa}, Y\equiv{Fb}, Z\equiv{Fz}, f\equiv{f_a}, g\equiv{f_b}$ and the pullback object is $P\equiv X\times_Z Y$.)

Consider two arrows $\gamma:{\bf C}[X,A]$ and $\delta:{\bf C}[X,B]$, which fulfill the structural condition $\alpha\circ\gamma=\beta\circ\delta$. I.e. when forwarded to $Z$ via $\alpha$, resp. $\beta$, they collapse into a single arrow.

Such two arrows $\gamma,\delta$ can be partially glued together, in the sense that they can both be written as a unique arrow $u$ (with codomain $A\times_Z B$) followed by the projections $\pi_a,\pi_b$.


The pullback object $A\times_Z B$ is the full solution to the equation posed by $\alpha$ and $\beta$. In ${\bf{Set}}$, it's literally the set of pairs $\langle x,y\rangle\in A\times_Z B\subseteq A\times B$, for which $\alpha(x)=\beta(y)$.

When the category contains a terminal object $1$ (where $\alpha$ and $\beta$ are trivial arrows and form a trivial condition), we have $A\times_1 B\cong A\times B$.

The universal property says that all other solution embed in this object, in this is what is meant by full solution.

Special cases
  • If $\pi_a$ is an iso, then $A\times_Z B\cong A$. As $A$ is already the pullback, it alone fully determines the “full solution”.
  • If moreover $\pi_b$ is an iso too, the projections vanish from the diagram and the universal property says that arrows $\gamma,\delta$ (see above) can be wholly glued together, i.e., up to iso, $\alpha\circ\gamma=\beta\circ\delta\implies\gamma=\delta$.
  • In ${\bf{Set}}$, if $\alpha=\beta$, the pullback definition says its elements $\langle x,y\rangle$ fulfill $\alpha(x)=\alpha(y)$, i.e. here the pullback object is the full collection of pairs of term with give the same $\alpha$ value. If moreover $\pi_a$ is iso, any $x$ determines an $\langle x,y\rangle$ and hence a $y$ and the universal property says $\alpha(x)=\alpha(y)\implies x=y$. This is just the definition of an injection.
  • Back to a general category, consider the case where $\pi_a$ is iso AND $\alpha=\beta$. The condition is $\alpha\circ\gamma=\alpha\circ\delta\implies\gamma=\delta$ and we call such an $\alpha$ a monomorphism

(In the picture, $f$ is $\alpha$ and $g,f$ are our $\gamma, \delta$.)


A finite pullback in ${\bf{Set}}$ that I just made up:

  • Generally: If $F(f_b)$ is the inclusion of a subset $Fb\subseteq{Fz}$ in ${Fz}$, the pullback is iso to (i.e. in bijection with) $F(f_a^{-1})Fb$. Further, if $F(f_a)$ is an inclusion too, this is in bijection with $Fa\cap{Fb}$. If the subset-interpretation doesn't apply, the function $F(f_b)$ from $Fb$ to $Fz$ should be viewed as defining fibre bundle over $Fz$ and the pullback gives a fibre bundle from $Fa\times_{Fz} Fb$ to $Fa$.
  • A concrete example: Let

$Fa=\{2,4,6,8\},\ Fb=\{10,20\},\ Fz=\{77,88,99\}$

$F(f_a)(2)=77,\ F(f_a)(4)=77,\ F(f_a)(6)=88,\ F(f_a)(8)=99$

$F(f_b)(10)=88,\ F(f_b)(20)=77$


$Fa\times_{Fz} Fb=\{\langle 2,20\rangle,\langle 4,20\rangle,\langle 6,10\rangle\}$

and $\pi$ are projections like for the product.

Two more prominent examples:

  • If $M$ is a manifold, $p:B\to M$ is a fibre bundle over $M$ and $f:X\to Y$ is an embedding of another manifold $X$ into $Y$, the the pullback object is a fibre bundle over $X$, namely the differential geometric pullback.
  • In $\bf{Set}$, if the “central object” is $\Omega=\{0,1\}$, the right function comes from a singleton $1$ and if the left function $\chi:X\to \Omega$ is a characteristic function, then a pullback object is a subset of $X$: It's defined as collection of arguments where the characteristic function agrees that the value is $1$. This works also for more general “fuzzy” $\Omega$. These $\Omega$ are the “objects of truth values” and are called subobject classifiers.

Digression: The exponential object $B^A$ is a prominent example of an object which isn't a limit, but it can be specifies via universal morphism construction. For sets or types, that's the function space $A\to B$ and for propositions it's the implication. If a category has product, exponential object and a terminal object, then it's called Cartesian closed. A Cartesian closed category with subobject classifier is a topos. We see now how a topos is a general kind of set theory, and simultaneously defines an internal logic.



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