In general, the implications in the above result do require the hypothesis (e.g. in the pullback case that the right square is a pullback). However, in some cases this can be omitted.

Proposition

Suppose we have a diagram of the above shape

$\array{
x & \longrightarrow & y & \longrightarrow & z
\\
\downarrow && \downarrow && \downarrow
\\
u & \longrightarrow & v & \longrightarrow & w
}$

in which the total rectangle (consisting of $x,z,u,w$) is a pullback, and moreover the induced map $y\to v\times z$ is a monomorphism. Then the left-hand square (consisting of $x,y,u,v$) is also a pullback.

Another related statement involves a pair of rectangles and equalizers.

Proposition

Suppose $\mathcal{C}$ is any category with equalizers and that we have a diagram of the following shape:

$\array{
x & \longrightarrow & y & \rightrightarrows & z
\\
\downarrow && \downarrow && \downarrow
\\
u & \longrightarrow & v & \rightrightarrows & w
}$

such that the vertical arrows are all monic, the squares on the right are serially commutative, and the lower row is an equalizer. Then the upper row is an equalizer if and only if the left square is a pullback.

Revised on July 21, 2017 10:36:33
by Mike Shulman
(207.207.127.211)