nLab quasi-pullback

Definition

In category theory, a commutative square

$$\array{ q & \longrightarrow & a \\ \big\downarrow && \big\downarrow \\ b & \longrightarrow & c }$$

in a category with finite limits is called a quasi-pullback if the canonical morphism $q\to a\times_c b$ to the fiber product (induced by its universal property) is an epimorphism.

Specifically, the object $q$ is then called a quasi-pullback of the span $b\to c\leftarrow a$.

A more descriptive terminology would be something like epi-pullback, but “quasi-pullback” is standard in the literature.

Examples

Proposition

For a topos $T$ and $T^I$ its arrow category which is a topos, quasi-pullback squares (in $T$) form a class of open morphisms in $T^I$.

Recall that, for every regular category $A$, there is a double category Rel$(A)$ of relations.

Proposition

Functors $Rel(A) \to Rel(B)$ between double categories of relations are equivalent to functors $A \to B$ preserving quasi-pullbacks.

Related pages

• pullback

References

• André Joyal and Ieke Moerdijk, A completeness theorem for open maps, Annals of Pure and Applied Logic 70.1 (1994): 51-86.

• Robert Paré, Some things about double categories, Talk at Virtual Double Categories Workshop 2022, pdf