# nLab pasting law

### Context

#### Limits and colimits

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

$\array{ & \longrightarrow && \longrightarrow \\ \downarrow && \downarrow && \downarrow \\ & \longrightarrow && \longrightarrow }$

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

###### Proposition

Outer square being a pullback implies left-hand square being a pullback, in the presence of a jointly monic span.

If

$\array{ O_{0,1} & \overset{f_{1,1}}{\to} & O_{1,1} & \overset{f_{2,1}}{\to} & O_{2,1} & & & & \\ \downarrow h_0 &&\downarrow h_1 &&\downarrow h_2 && && \\ O_{0,0} & \overset{f_{1,0}}{\to} & O_{1,0} & \overset{f_{2,0}}{\to} & O_{2,0} & & & }$

is a commutative diagram in any category $\mathcal{C}$ such that

• the square consisting of $h_0$, $h_2$, $f_{2,1}\circ f_{1,1}$, $f_{2,0}\circ f_{1,0}$ is a pullback in $\mathcal{C}$
• for all morphisms $g_1$ and $g_2$ with $cod(g_i)=O_{1,1}$:

${}\qquad$ ( $f_{2,1}\circ g_1 = f_{2,1}\circ g_2$ and $h_1\circ g_1 = h_1 \circ g_2$) implies ($g_1=g_2$)

Then

• the square consisting of $h_0$, $h_1$, $f_{1,0}$, $f_{1,1}$ is a pullback in $\mathcal{C}$
###### Proposition

An equalizer diagram implying an equalizer diagram, via monos and a pullback square.

Suppose $\mathcal{C}$ is any category with equalizers. Suppose

$\array{ O_{0,1} & \to & O_{1,1} & \underoverset{\quad {}_0f_{1} \quad}{{}_1f_{1}}{\rightrightarrows} & O_{2,1} & & & & \\ \downarrow m_0 &&\downarrow m_1 &&\downarrow m_2 && && \\ O_{0,0} & \to & O_{1,0} & \underoverset{\quad {}_0f_{0} \quad}{ {}_1f_{0} }{\rightrightarrows} & O_{2,0} & & & }$

is a diagram in $\mathcal{C}$ in which

• each of $m_0$,$m_1$, $m_2$ is monic
• the squares on the right are serially commutative
• the lower row is an equalizer

Then

• the upper row is an equalizer iff the left square is a pullback.
Revised on June 20, 2017 01:08:13 by Peter Heinig (2003:58:aa24:9a00:cf7:a27a:814:2924)