Limits and colimits

Equality and Equivalence



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:

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

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

A pullback is therefore the categorical semantics of an equation.

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

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

Note that there are maps p A:XAp_A : X \to A, p B:XBp_B : X \to B sending any (a,b)X(a,b) \in X to aa and bb, respectively. These maps make this square commute:

X p A p B A B f g C \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

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

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

p Ah=q A p_A h = q_A


p Bh=q B. 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.


In category theory

A pullback is a limit of a diagram like this:

a b f g c \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

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

and the object xx 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 SetSet.

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 CC is the same as a pushout in the opposite category C opC^{op}.

In type theory

In type theory a pullback PP in

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


As an equalizer

If products exist in CC, then the pullback

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

is equivalently the equalizer

a× cba×bgp 2fp 1c 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 ff and gg out of the product of aa with bb.

Pasting of pullbacks


(pasting law)

Consider a pasting diagram of the form

a b c d e f. \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.


Pasting a morphism xax \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 xax \to a includes a morphism into bb, which if bb is a pullback induces the same commuting square over defd \to e \to f and cdc \to d. So one square is universal iff the other is.


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


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

a = a = a = i = a i b p a \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 ii was already an isomorphism.


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

Monomorphisms and isomorphisms


(pullback preserves monomorphisms and isomorphisms)

Pullback preserves monomorphisms and isomorphisms:


a b f *g (pb) g c f d \array{ a &\overset{}{\longrightarrow}& b \\ {}^{\mathllap{f^\ast g}}\downarrow &(pb)& \downarrow^{g} \\ c &\underset{f}{\longrightarrow}& d }

is a pullback square in some category then:

  1. if gg is a monomorphism then f *gf^\ast g is a monomorphism;

  2. if gg is an isomorphism then f *gf^\ast g is an isomorphism.

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


The saturation of the class of pullbacks is the class of limits over categories CC whose groupoid reflection Π 1(C)\Pi_1(C) is trivial and such that CC is L-finite.


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

Revised on September 20, 2017 11:29:37 by Anonymous (