nLab pullback

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:

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

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

there is a unique function $h\colon Y \to X$ 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

As a limit

A pullback is a limit of a diagram like this:

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

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 $C^{op}$.

Nuts and bolts

Let $\mathcal{C}$ be a category, with $f\colon a\to c$ and $g\colon b\to c$ coterminal arrows in $\mathcal{C}$ as below

A pullback of $f$ and $g$ consists of an object $x$ together with arrows $p_a\colon x\to a$ and $p_b\colon x\to b$ such that the following diagram commutes universally

This means that for any other object $x'$ with arrows $p'_a\colon x'\to a$ and $p'_b\colon x'\to b$ such that

commutes, there exists a unique arrow $u\colon x'\to x$ such that

commutes.

In type theory

In type theory a pullback $P$ in

is given by the dependent sum over the dependent equality type

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

Properties

Proposition

(pullbacks as equalizers)

If products exist in $C$, then the pullback

is equivalently the equalizer

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

As a consequence of the above, any category with binary products and equalizers has pullbacks. Conversely any category with binary products and pullbacks has equalizers: the equalizer of $f,g:A \to B$ is the pullback of $(1,f) \maps A \to A \times B$ and $(1,g) \maps A \to A \times B$.

Proposition

(pullbacks preserve monomorphisms and isomorphisms)

Pullbacks preserve monomorphisms and isomorphisms:

If

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

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 $i\colon a \to b$ be a split monomorphism with retract $p\colon b \to a$ and consider

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

Related concepts

• pushout

Notions of pullback:

• pullback, fiber product (limit over a cospan)

• wide pullback

• lax pullback, comma object (lax limit over a cospan)

• (∞,1)-pullback, homotopy pullback, ((∞,1)-limit over a cospan)

• base change, context extension

• pullback bundle

• contravariant functor

  • pullback in cohomology, d-invariant

  • pullback of differential forms

  • pullback of a distribution

References

Early use of the terminology “pullback”:

• Barry Mitchell, Section I.7 of: Theory of categories, Pure and Applied Mathematics 17, Academic Press (1965) [ISBN:978-0-12-499250-4]

Textbook accounts:

• Saunders MacLane, p. 71 in: Categories for the Working Mathematician, Springer (1971) [doi:10.1007/978-1-4757-4721-8]

• Francis Borceux, Section 2.5 in Vol. 1: Basic Category Theory of: Handbook of Categorical Algebra, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) (doi:10.1017/CBO9780511525858)

See also:

• André Joyal, Cartesian squares

In view of more general finite limits and L-finite limits:

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

Morphisms admitting pullbacks along them are called carrable in some places, for instance, the following:

• Bertrand Toen, A master course on algebraic stacks (Toulouse 2005) cours 2 (web)
• André Joyal, Notes on Clans and Tribes, arxiv:1710.10238

They are called quarrable in:

• Robin Cockett, Introduction to distributive categories, Mathematical Structures in Computer Science 3 (2009) 277-307 [doi:10.101/S0960129500000232]