nLab pasting law for pullbacks

Contents

Context

Limits and colimits

limits and colimits

category theory

Contents

Idea

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

Statement

General pasting law

Proposition

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

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

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

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.

Reverse pasting law

In a regular category this also works in the other direction, if the bottom left morphisms is a regular epimorphism:

Proposition

(reverse pasting law)
In a regular category, consider a commuting diagram of the form where

1. the left square is a pullback;

2. the bottom left morphism is an regular epimorphism.

Then the right right square is a pullback iff the total rectangle is.

(e.g. Gran 2021, Lem. 1.15, see also Carboni, Janelidze, Kelly and Paré 1997, Lemma 4.6, Garner and Lack 2012, Lemma 2.2)

This implies the $\infty$-category theoretic statement at least in good cases:

Proposition

(reverse pasting law for $\infty$-pullbacks of $\infty$-groupoids)
The reverse pasting law (Prop. ) holds also for homotopy pullbacks of $\infty$-groupoids as soon as the bottom left morphism is an effective epimorphism in this $\infty$-category.

Proof

By the discussion at homotopy pullback and using the classical model structure on simplicial sets, we may model the situation by a diagram of simplicial sets where both bottom morphisms are Kan fibrations, and then need to show that the ordinary reverse pasting law applies.

Now observe that:

1. an effective epimorphism of $\infty$-groupoids is a surjection on connected components (by this Prop.);

2. a Kan fibration which resolves a surjection on connected components is degreewise surjective (by this Prop.);

3. a degreewise surjection of simplicial sets is an epimorphism (by this Prop.),

hence is a regular epimorphism in a regular category (since sSet is a topos).

Hence the bottom left morphism in our diagram of simplicial sets is a regular epimorphism and the claim follows by Prop. .

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 monomorphisms, 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.

References

In 1-category theory

Discussion in 1-category theory:

Ordinary pasting law

Statements of the pasting law in textbooks, typically leaving the proof to the reader:

A proof is spelled out in:

Reverse pasting law

The reverse pasting law is discussed in:

In model category theory

Discussion in model category-theory

and for general model categories:

In $(\infty,1)$-category theory

Last revised on July 7, 2022 at 11:29:17. See the history of this page for a list of all contributions to it.