nLab pasting law for pullbacks

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:

• for category theory: at pullback – pasting law; also e.g. Bauer 2012;

• for (∞,1)-category theory: at (∞,1)-limit – pushout pasting law

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 a regular epimorphism.

Then the 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)

For a more general statement that holds in categories that are not necessarily regular, see Przybylek (2013).

Prop. implies the $\infty$-category theoretic reverse pasting law, 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. .

Further related statements

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.

Proposition

(Pasting Law for Pullbacks in a Cube)
Suppose we have a commutative cube in a category:

Denote the cube faces by F (Front), B (Back), U (Up), D (Down), L (Left), and R (Right) (this is Rubik’s cube Singmaster notation).

a. Assume F and D are pullback squares. Then the following are equivalent:

1. The total rectangle from $W'\to Z'$ to $X\to Y$ is a pullback.

2. T and B are pullback squares.

3. T or B is a pullback square.

b. Assume F, D, R are pullback squares (these are the faces sharing $Y$ as a common vertex). Then all three other faces B, T, L are also pullback squares if at least one of them is.

a’. Assume T and B are pushout squares. Then the following are equivalent:

1. The total rectangle from $W'\to Z'$ to $X\to Y$ is a pushout.

2. F and D are pushout squares.

3. F or D is a pushout square.

b’. Assume B, T, L are pushout squares (these are the faces sharing $W'$ as a common vertex). Then all three other faces F, D, R are also pushout squares if at least one of them is.

Proof

We only show a and b (for a’ and b’ are their duals). Part a follows directly from Proposition . To see part b, consider the subgroup of order three of cube symmetries generated by a 120° rotation around the axis that goes from $W'$ to $Y$. If we let this group act on the statement of part a, we obtain three different versions of part a. Now combine these with the hypotheses of part b to obtain the claim.

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:

• Saunders MacLane, Ex. 8 on p. 72 in: Categories for the Working Mathematician, Graduate texts in mathematics, Springer 1971 (doi:10.1007/978-1-4757-4721-8)

• Jiri Adamek, Horst Herrlich, George Strecker, Prop. 11.10 in: Abstract and Concrete Categories, John Wiley and Sons, New York (1990) reprinted as: Reprints in Theory and Applications of Categories 17 (2006) 1-507 (tac:tr17, book webpage, pdf)

A proof is spelled out in:

• Andrej Bauer, The pullback lemma in gory detail, 2012 (pdf, pdf)

Reverse pasting law

The reverse pasting law is discussed in:

• Aurelio Carboni, George Janelidze, Max Kelly, Robert Paré, Lemma 4.6 of: On localization and stabilization for factorization systems, Appl. Categ. Structures 5 (1997) 1-58 [doi:10.1023/A:1008620404444]

• Richard Garner, Steve Lack, Lemma 2.2 in: On the axioms for adhesive and quasiadhesive categories, Theory and Applications of Categories, 27 3 (2012) 27-46 [arXiv:1108.2934, tac:27-03]

• Marino Gran, An introduction to regular categories, in: New Perspectives in Algebra, Topology and Categories, Coimbra Mathematical Texts, Springer (2021) [arXiv:2004.08964, ISBN:978-3-030-84319-9]

• Michal Przybylek, The other pullback lemma (2013) [arXiv:1311.2974]

cf. MO:143070

In model category theory

Discussion in model category-theory

for right proper model categories:

• Philip Hirschhorn, Prop. 13.3.15 of: Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002) (ISBN:978-0-8218-4917-0, pdf toc, pdf)

and for general model categories:

• Alisa Govzmann, Damjan Pištalo, Norbert Poncin, Prop. 8 in: Indeterminacies and models of homotopy limits [arXiv:2109.12395]

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

Discussion in $(\infty,1)$-category theory

• Jacob Lurie, Lemma 4.4.2.1 in Higher Topos Theory