Pullback

A universal object representing compatible pairs over a cospan.

Pullback

Definition

Let $\mathcal{C}$ be a category and let

X \xrightarrow{f} Z \xleftarrow{g} Y

be a cospan of morphisms .

A pullback (or fiber product) of $(f,g)$ is an object $P$ together with morphisms

p_X:P\to X,\qquad p_Y:P\to Y

such that

f\circ p_X = g\circ p_Y

(using composition ), and satisfying the following universal property:

For any object $W$ and morphisms $u:W\to X$ , $v:W\to Y$ with $f\circ u = g\circ v$ , there exists a unique morphism $\langle u,v\rangle:W\to P$ such that

p_X\circ \langle u,v\rangle = u,\qquad p_Y\circ \langle u,v\rangle = v.

When it exists, the pullback is unique up to unique isomorphism and is often denoted $X\times_Z Y$ .

Relationship to other constructions

A pullback is a special limit : it is the limit of the diagram $X\to Z \leftarrow Y$ .
In categories with products and equalizers , one can realize a pullback as an equalizer of the two maps $X\times Y \rightrightarrows Z,\quad (x,y)\mapsto f(x),\ (x,y)\mapsto g(y).$

Examples

Example (Set)

In $\mathbf{Set}$ , the pullback is the subset of the cartesian product

X\times Y=\{(x,y)\}

(consisting of ordered pairs ) given by

X\times_Z Y \;=\; \{(x,y)\in X\times Y \mid f(x)=g(y)\}.

The maps $p_X,p_Y$ are the coordinate projections.

Example (Grp)

In $\mathbf{Grp}$ , the pullback of $f:X\to Z$ and $g:Y\to Z$ is the subgroup

X\times_Z Y \;=\; \{(x,y)\in X\times Y \mid f(x)=g(y)\},

with group operation defined componentwise and projections $p_X,p_Y$ .

Example (Top)

In $\mathbf{Top}$ , the pullback is the same set-theoretic fiber product as in $\mathbf{Set}$ , equipped with the subspace topology inherited from $X\times Y$ (with the product topology). The projections are continuous and satisfy the universal property in $\mathbf{Top}$ .