nLab
pasting law

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. Consider a commuting diagram in 𝒞\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

O 0,1 f 1,1 O 1,1 f 2,1 O 2,1 h 0 h 1 h 2 O 0,0 f 1,0 O 1,0 f 2,0 O 2,0 \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 𝒞\mathcal{C} such that

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

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

Then

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

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

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

O 0,1 O 1,1 0f 1 1f 1 O 2,1 m 0 m 1 m 2 O 0,0 O 1,0 0f 0 1f 0 O 2,0 \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 𝒞\mathcal{C} in which

    • each of m 0m_0,m 1m_1, m 2m_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)