Pasting theorem

In mathematics, specifically the 2-category theory, the pasting theorem states that every 2-categorical pasting scheme defines a unique composite 2-cell in every 2-category. The notion of pasting in 2-category and weak 2-category was first introduced by Bénabou (1967). Typically, pasting is used to specify a cell by giving a pasting diagram. The pasting theorem states that such a cell is well-defined the several different sequences of compositions which the diagram could be explained as representing yield the same cell. The pasting theorem for strict 2-category was proved by Power (1990), and for weak 2-category it is proved in Appendix A of Verity (1992)'s thesis. The pasting theorem for n-category version was proved by Power (1991) and Johnson (1989), but the definition of the pasting scheme differs. String diagrams are justified by the pasting theorem.^{[citation needed]}

Pasting diagram

Example

Consider the pasting diagram D for adjunction

2-cell ${\displaystyle {\mathcal {E}}:gf\rightarrow \mathrm {id} _{A}}$, ${\displaystyle \eta$ :\mathrm {id} _{B}\rightarrow fg}

The entire pasting diagram represents the vertical composite ${\displaystyle (\mathrm {id} _{f}*{\mathcal {E}})(\eta *\mathrm {id} _{f})}$ which is a 2-cell in D(A, B), displayed on the right above^[1]

2-categorical pasting theorem

• Every 2-pasting diagram in an strict 2-category A has a unique composite.^[2]
• Every 2-pasting diagram in an weak 2-category A has a unique composite.^[3]

2-pasting scheme

Anchored graph

Suppose G and H are anchored graphs^[4] such that:

• ${\displaystyle s_{G}=s_{H}}$,
• ${\displaystyle t_{G}=t_{H}}$, and
• ${\displaystyle \mathrm {cod} _{G}=\mathrm {dom} _{H}}$.

The vertical composite HG is the anchored graph defined by the following data:

(1) The connected plane graph of HG is the quotient

${\displaystyle {\frac {G\sqcup H}{\{\mathrm {cod} _{G}=\mathrm {dom} _{H}\}}}}$

(2) The interior faces of HG are the interior faces of G and H, which are already anchored.

(3) The exterior face of HG is the intersection of ${\displaystyle \mathrm {ext} _{G}}$ and ${\displaystyle \mathrm {ext} _{H}}$, with

• source ${\displaystyle s_{G}=s_{H}}$,
• sink ${\displaystyle t_{G}=t_{H}}$,
• domain ${\displaystyle \mathrm {dom} _{G}}$, and
• codomain ${\displaystyle \mathrm {cod} _{H}}$.

of the disjoint union of G and H, with the codomain of G identified with the domain of H.

2-pasting scheme in the sense of Johnson & Yau

A 2-pasting scheme is an anchored graph G together with a decomposition

${\displaystyle G=G_{n}\cdots G_{1}}$

into vertical composites of ${\displaystyle n\geq 1}$ atomic graphs ${\displaystyle G_{1},\dots ,G_{n}}$.^[5]

2-pasting diagram

Suppose A is a 2-category, and G is an anchored graph. A G-diagram in A is an assignment ${\displaystyle \phi }$ as follows.

• ${\displaystyle \phi }$ assigns to each vertex v in G an object ${\displaystyle \phi _{v}}$ in A.
• ${\displaystyle \phi }$ assigns to each edge e in G with tail u and head v a 1-cell ${\displaystyle \phi _{e}\in A(\phi _{u},\phi _{v})}$.

For a directed path ${\displaystyle P=v_{0}e_{1}v_{1}\dots e_{m}v_{m}}$ in G with ${\displaystyle m\leq 1}$, define the horizontal composite 1-cell ${\displaystyle \phi _{P}=\phi _{e_{m}}\cdots \phi _{e_{1}}\in A(\phi _{v_{0}},\phi _{v_{m}})}$.

• ${\displaystyle \phi }$ assigns to each interior face F of G a 2-cell ${\displaystyle \phi _{F}:\phi _{\mathrm {dom} _{F}}\rightarrow \phi _{\mathrm {cod} _{F}}}$ in ${\displaystyle A(\phi _{s_{F}},\phi _{t_{F}})}$.

If G admits a pasting scheme presentation, then a G-diagram is called a 2-pasting diagram in A of shape G.^[6]

Gray-categorical pasting theorem

Every 2-dimensional pasting diagram in a Gray-category has a unique composition up to a contractible groupoid of choices.^[7]

Weak version of strict n-categorical pasting theorem

For any positive natural number n, every labelled n-pasting scheme in an strict n-category A has a unique "strong" composite.^[8]

n-categorical pasting theorem

For every positive natural number n, every labelled n-pasting scheme in an strict n-category A has a unique n-pasting composite.^[9]