Wide Pullback

A wide pullback is the indexed form of a Pullback.

Explicitly, a wide pullback in a Category $\mathcal{C}$ of a family of maps $f_{i}:{{X_{i}}\to{Y}}$ is a Bundle $\pi:P\to X$ equipped with projections $\pi_{i}:{{P}\to{X_{i}}}$ such that

For all $i:I$ , $f_{i}\circ\pi_{i}=\pi$
For any other $(Q,p:{{Q}\to{X}},p_{i}:{{Q}\to{X_{i}}})$ with $f_{i}\circ p_{i}=p$ , there exists a unique universal map $u:Q\to P$ such that $\pi\circ u=p$ and $\pi_{i}\circ u=p_{i}$ .

Warning

It is very important that we parameterize the data by a bundle, and not an object. If we do not, then a wide pullback of an empty family of morphisms yields a Terminal Object, which is not what we get when we take an Indexed Cartesian Product in the Slice Category.

When the indexing set is Merely Inhabited, we can dispense with the bundle map.

Properties

A wide pullback is a Limit of a Walking Wide Cospan.
A wide pullback is an Indexed Cartesian Product in the Slice Category $\mathcal{C}/Y$ .
A category with all $\kappa$ -small wide pullbacks and a Terminal Object is Complete.

We can construct $\kappa$ -small Indexed Products $\prod_{(i:I)}X_{i}$ by taking the wide pullback of a family $X_{i}\to 1$ . We can then build Equalisers $f,g:X\to Y$ by taking a pullback of the diagram:

From here, we can use the fact that $\kappa$ -small indexed products and equalisers give all Limits.
Wide pullbacks are Connected Limits