nLab epi-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 (sometimes) called an epi-pullback (or quasi-pullback or epi cartesian-square) if the canonical morphism $q\to a\times_c b$ to the fiber product (induced by its universal property) is an epimorphism.

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

Examples

Proposition

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

Proposition

Lax double functors $Rel(A) \to Rel(B)$ are equivalent to functors $A \to B$ preserving quasi-pullbacks.

Related pages

• pullback

References

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