nLab
pasting law for pullbacks

Context

Limits and colimits

Category theory

Contents

Idea

In category theory, the pasting law is a statement about (de-)composition of pullback/pushout diagrams.

Statement

Proposition

Let 𝒞\mathcal{C} be a category or more generally an (∞,1)-category or derivator. Consider a commuting diagram in 𝒞\mathcal{C} of the following shape:

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

Then:

  1. if the right square is a pullback, then the total rectangle is a pullback precisely if the left square is a pullback.

  2. if the left square is a pushout, then the total rectangle is a pushout precisely if the right square is a pushout.

For proof see

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

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

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

Another related statement involves a pair of rectangles and equalizers.

Proposition

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

x y z u v w \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)