nLab pullback

Contents

Context

Limits and colimits

limits and colimits

Contents

Idea

In the category Set a ‘pullback’ is a subset of the cartesian product of two sets. Given a diagram of sets and functions like this:

$\array{ && A &&&& B \\ & && {}_{f}\searrow & & \swarrow_g && \\ &&&& C &&&& }$

the ‘pullback’ of this diagram is the subset $X \subseteq A \times B$ consisting of pairs $(a,b)$ such that the equation $f(a) = g(b)$ holds.

A pullback is therefore the categorical semantics of an equation.

This construction comes up, for example, when $A$ and $B$ are fiber bundles over $C$: then $X$ as defined above is the product of $A$ and $B$ in the category of fiber bundles over $C$. For this reason, a pullback is sometimes called a fibered product (or fiber product or fibre product).

In this case, the fiber of $A \times_C B$ over a (generalized) element $x$ of $C$ is the ordinary product of the fibers of $A$ and $B$ over $x$. In other words, the fiber product is the product taken fiber-wise. Of course, the fiber of $A$ at the generalized element $x\colon I \to C$ is itself a fibre product $I \times_C A$; the terminology depends on your point of view.

Note that there are maps $p_A : X \to A$, $p_B : X \to B$ sending any $(a,b) \in X$ to $a$ and $b$, respectively. These maps make this square commute:

$\array{ &&&& X \\& && {}^{p_A}\swarrow && \searrow^{p_B} \\ && A &&&& B \\ & && {}_f\searrow & & \swarrow_{g} && \\ &&&& C &&&& }$

In fact, the pullback is the universal solution to finding a commutative square like this. In other words, given any commutative square

$\array{ &&&& Y \\& && {}^{q_A}\swarrow && \searrow^{q_B} \\ && A &&&& B \\ & && {}_f\searrow & & \swarrow_{g} && \\ &&&& C &&&& }$

there is a unique function $h: Y \to X$ such that

$p_A h = q_A$

and

$p_B h = q_B\,.$

Since this universal property expresses the concept of pullback purely arrow-theoretically, we can formulate it in any category. It is, in fact, a simple special case of a limit.

Definition

In category theory

A pullback is a limit of a diagram like this:

$\array{ && a &&&& b \\ & && {}_{f}\searrow & & \swarrow_g && \\ &&&& c &&&& }$

Such a diagram is also called a pullback diagram or a cospan. If the limit exists, we obtain a commutative square

$\array{ &&&& x \\& && {}^{p_a}\swarrow && \searrow^{p_b} \\ && a &&&& b \\ & && {}_f\searrow & & \swarrow_{g} && \\ &&&& c &&&& }$

and the object $x$ is also called the pullback. It is well defined up to unique isomorphism. It has the universal property already described above in the special case of the category $Set$.

The last commutative square above is called a pullback square.

The concept of pullback is dual to the concept of pushout: that is, a pullback in $C$ is the same as a pushout in the opposite category $C^{op}$.

In type theory

In type theory a pullback $P$ in

$\array{ P &\to& A \\ \downarrow && \downarrow^{\mathrlap{f}} \\ B &\stackrel{g}{\to}& C }$

is given by the dependent sum over the dependent equality type

$P = \sum_{a : A} \sum_{b : B} (f(a) = g(b)) \,.$

Properties

Proposition

(pullbacks as equalizers)

If products exist in $C$, then the pullback

$\array{ a \times_c b &\to& a \\ \downarrow && \downarrow^{\mathrlap{f}} \\ b &\stackrel{g}{\to}& c }$

is equivalently the equalizer

$a \times_c b \to a \times b \stackrel{\overset{f p_1}{\longrightarrow}}{\underset{g p_2}{\longrightarrow}} c$

of the two morphisms induced by $f$ and $g$ out of the product of $a$ with $b$.

Proposition

(pullbacks preserve monomorphisms and isomorphisms)

Pullbacks preserve monomorphisms and isomorphisms:

If

$\array{ a &\overset{}{\longrightarrow}& b \\ {}^{\mathllap{f^\ast g}}\downarrow & & \downarrow^{g} \\ c &\underset{f}{\longrightarrow}& d }$

is a pullback square in some category then:

1. if $g$ is a monomorphism then $f^\ast g$ is a monomorphism;

2. if $g$ is an isomorphism then $f^\ast g$ is an isomorphism.

On the other hand that $f^\ast g$ is a monomorphism does not imply that $g$ is a monomorphism.

Proposition

(pasting law for pullbacks)

In any category consider a diagram of the form

$\array{ a &\to& b &\to& c \\ \downarrow && \downarrow && \downarrow \\ d &\to& e &\to& f }$

There are three commuting squares: the two inner ones and the outer one.

Suppose the right-hand inner square is a pullback, then:

The square on the left is a pullback if and only if the outer square is.

Proof

Pasting a morphism $x \to a$ with the outer square gives rise to a commuting square over the (composite) bottom and right edges of the diagram. The square over the cospan in the left-hand inner square arising from $x \to a$ includes a morphism into $b$, which if $b$ is a pullback induces the same commuting square over $d \to e \to f$ and $c \to d$. So one square is universal iff the other is.

Proposition

The converse implication does not hold: it may happen that the outer and the left square are pullbacks, but not the right square.

Proof

For instance let $i : a \to b$ be a split monomorphism with retract $p : b \to a$ and consider

$\array{ a & \stackrel{=}{\to} & a & \stackrel{=}{\to} & a \\ \downarrow^{\mathrlap{=}} && \downarrow^{\mathrlap{i}} && \downarrow^{\mathrlap{=}} \\ a &\stackrel{i}{\to}& b &\stackrel{p}{\to}& a }$

Then the left square and the outer rectangle are pullbacks but the right square cannot be a pullback unless $i$ was already an isomorphism.

Remark

On the other hand, in the (∞,1)-category of ∞-groupoids, there is a sort of “partial converse”; see homotopy pullback#HomotopyFiberCharacterization.

Saturation

The saturation of the class of pullbacks is the class of limits over categories $C$ whose groupoid reflection $\Pi_1(C)$ is trivial and such that $C$ is L-finite.

Pullback functor

If $f : X \to Y$ is a morphism in a category $C$ with pullbacks, there is an induced pullback functor $f^* : C/Y \to C/X$, sometimes also called base change.

• Robert Paré, Simply connected limits. Can. J. Math., Vol. XLH, No. 4, 1990, pp. 731-746, CMS