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 $b\to c\leftarrow a$.

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 $T^I$.

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

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

Last revised on March 14, 2023 at 22:24:16. See the history of this page for a list of all contributions to it.