# nLab wide pullback

### Context

#### Limits and colimits

limits and colimits

# Contents

## Definition

A wide pullback in a category $\mathrm{\pi }$ is a product (of arbitrary cardinality) in a slice category $\mathrm{\pi }\beta C$. In terms of $\mathrm{\pi }$, this can be expressed as a limit over a category obtained from a discrete category by adjoining a terminal object.

Yet more explicitly, the wide pullback of a family of coterminal morphisms ${f}_{i}:{A}_{i}\beta C$ is an object $P$ equipped with projection ${p}_{i}:P\beta {A}_{i}$ such that ${f}_{i}{p}_{i}$ is independent of $i$, and which is universal with this property.

Binary wide pullbacks are the same as ordinary pullbacks, a.k.a. fiber products.

Of course, a wide pushout is a wide pullback in the opposite category.

## Properties

On the other hand, together with a terminal object, wide pullbacks generate all limits:

###### Proposition

A category $C$ with all wide pullbacks and a terminal object $1$ is complete. If $C$ is complete and $F:C\beta D$ preserves wide pullbacks and the terminal object, then it preserves all limits.

###### Proof

To build up arbitrary products ${\beta }_{i\beta I}{c}_{i}$ in $C$, take the wide pullback of the family ${c}_{i}\beta 1$. Then to build equalizers of diagrams $f,g:c\stackrel{\beta }{\beta }d$, construct the pullback of the diagram

$\begin{array}{ccc}& & d\\ & & \beta \mathrm{\Xi ΄}\\ c& \underset{\beta ¨f,g\beta ©}{\beta }& d\Gamma d\end{array}$\array{ & & d \\ & & \downarrow \delta \\ c & \underset{\langle f, g \rangle}{\to} & d \times d }

From products and equalizers, we can get arbitrary limits.

## References

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

Revised on January 4, 2013 21:16:50 by Mike Shulman (108.225.239.218)