Final pullback complements
# Final pullback complements

## Idea

A *final pullback complement* is a “universal completion of a pair of arrows to a pullback”. Intuitively, it is a way to “remove part of an object of a category” while retaining information about how that part is “connected” to other parts. Final pullback complements play an important role in some kinds of span rewriting.

## Definition

Given morphisms $m:C\to A$ and $g:A\to D$ in a category, a **pullback complement** is a pair of arrows $f:C\to B$ and $n:B\to D$ such that the square

$\array{ C & \to & A \\ \downarrow & & \downarrow \\ B & \to & D }$

commutes and is a pullback. This is of course the dual of a pushout complement.

A morphism of pullback complements is a map $B\to B'$ making the obvious diagrams commute. A **final pullback complement** is a terminal object in the category of pullback complements. Since they have a universal property, final pullback complements are unique up to unique isomorphism when they exist.

## Relation to exponentials

Recall from distributivity pullback that a **pullback around $(g,m)$** is a diagram

$\array{ X & \xrightarrow{p} & C & \xrightarrow{m} & A\\
^q\downarrow &&&& \downarrow^g\\
Y && \xrightarrow{r} && D}$

in which the outer rectangle is a pullback, and that a **distributivity pullback** around $(g,m)$ is a terminal object in the category of pullbacks aroung $(g,m)$. This makes the following evident:

###### Proposition

If a distributivity pullback around $(g,m)$ exists and has the property that $p$ is an isomorphism, then it is also a final pullback complement.

A distributivity pullback has precisely the universal property of an exponential object of $m$ along $g$. Thus, whenever such an exponential $\Pi_g m$ exists and the counit $g^* \Pi_g m \to m$ is an isomorphism, a final pullback complement exists. This is the case in particular in a locally cartesian closed category if $g$ and $m$ are both monomorphisms.

## Relation to pushout complements

In an adhesive category, any pushout complement of a monomorphism is also a final pullback complement.

## Related pages

## References

- Andrea Corradini, Tobias Heindel, Frank Hermann, and Barbara König,
*Sesqui-pushout rewriting*, 2006, springerlink, pdf.