Pasting theorem

Summarize Timeline Fact Check

For the example, consider pasting diagram D for the triangle identity of an adjunction

2-cell ${\displaystyle {\mathcal {E}}:g\circ f\rightarrow \mathrm {id} _{A}}$, ${\displaystyle \eta$ :\mathrm {id} _{B}\rightarrow f\circ g}

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), this is the right-hand side of the diagram.^[2]

If a diagram in a 2-category were to be a 2-graph morphism from some 2-graph (that is 2-globular set) into the underlying 2-graph of the 2-category, then in the left-hand side of above pasting diagram, it would not be a "diagram" (in the sense of Johnson), that is, a cell drawn on a diagram it is may not defined as a composite of other cells.^[2]

For the example, the codomain of ${\displaystyle \eta *\mathrm {id} _{f}}$ and domain of ${\displaystyle \mathrm {id} _{f}*{\mathcal {E}}}$ of the vertical composite ${\displaystyle (\mathrm {id} _{f}*{\mathcal {E}})(\eta *\mathrm {id} _{f})}$ do not equal:

${\displaystyle \mathrm {id} _{B}\circ f\mathrel {\stackrel {\eta *\mathrm {id} _{f}}{\longrightarrow }} (f\circ g)\circ f\longrightarrow f\circ (g\circ f)\mathrel {\stackrel {\mathrm {id} _{f}*{\mathcal {E}}}{\longrightarrow }} f\circ \mathrm {id} _{A}}$.

The pasting theorem guarantees that the vertical composite ${\displaystyle (\mathrm {id} _{f}*{\mathcal {E}})(\eta *\mathrm {id} _{f})}$ is uniquely defined.

Summarize Timeline Top Qs Fact Check

2-pasting scheme

Anchored graph

Suppose ${\displaystyle G}$ and ${\displaystyle H}$ are anchored graphs such that:^[4]

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

The vertical composite ${\displaystyle HG}$ is the anchored graph defined by the following data:

(1) The connected plane graph of ${\displaystyle HG}$ is the quotient

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

(2) The interior faces of ${\displaystyle HG}$ are the interior faces of ${\displaystyle G}$ and ${\displaystyle H}$, which are already anchored.

(3) The exterior face of ${\displaystyle HG}$ 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 ${\displaystyle G}$ and ${\displaystyle H}$, with the codomain of ${\displaystyle G}$ identified with the domain of ${\displaystyle H}$.

2-pasting scheme

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

• ${\displaystyle \phi }$ assigns to each vertex ${\displaystyle v}$ in ${\displaystyle G}$ an object ${\displaystyle \phi _{v}}$ in ${\displaystyle A}$.
• ${\displaystyle \phi }$ assigns to each edge ${\displaystyle e}$ in ${\displaystyle G}$ with tail ${\displaystyle u}$ and head ${\displaystyle v}$ 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 ${\displaystyle G}$ 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 ${\displaystyle F}$ of ${\displaystyle G}$ 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 ${\displaystyle G}$ admits a pasting scheme presentation, then a ${\displaystyle G}$-diagram is called a 2-pasting diagram in ${\displaystyle A}$ of shape ${\displaystyle G}$.^[6]

Statement

Pasting theorem for strict 2-category: every 2-pasting diagram in an strict 2-category has a unique composite.^[7]

Pasting theorem for weak 2-category: every 2-pasting diagram in an weak 2-category has a unique composite.^[8]

Every 2-dimensional pasting diagram in a Gray-category has a unique composition up to a contractible groupoid of choices.^[9]

Weak version of pasting theorem for strict n-category: for any positive natural number n, every labelled n-pasting scheme in an strict n-category ${\displaystyle A}$ has a unique "strong" composite.^[10]

Pasting theorem for strict n-category: for every positive natural number n, every labelled n-pasting scheme in an strict n-category ${\displaystyle A}$ has a unique n-pasting composite.^[11]