Pasting theorem

In higher category theory in mathematics, the pasting theorem guarantees that each pasting diagram has a uniquely defined composite that independent of the order of the vertical composite, as long as they are defined. Namely, such a cell is well-defined the several different sequences of compositions which the diagram could be explained as representing yield the same cell.Template:R

Operation of pasting appeared in Template:Harvtxt when introducing weak 2-category.Template:R The pasting theorem for strict 2-category guarantees that every 2-categorical pasting scheme defines a unique composite 2-cell in every 2-category, this is proved by Template:Harvtxt. For weak 2-category it is proved in Appendix A of Template:Harvtxt's thesis as a consequence of the coherence theorem for weak 2-category.Template:R The pasting theorem for strict n-category was proved by Template:Harvtxt and Template:Harvtxt, but the definition of the pasting scheme used in that proof is different.

Example for operation of pasting

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

2-cell $ℰ:g\circ f\to {\mathrm{i}\mathrm{d}}_A$, $\eta :{\mathrm{i}\mathrm{d}}_B\to f\circ g$

The entire pasting diagram represents the vertical composite $({\mathrm{i}\mathrm{d}}_f\ast ℰ)(\eta \ast {\mathrm{i}\mathrm{d}}_f)$ which is a 2-cell in D(A, B), this is the right-hand side of the diagram.

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.Template:R

Specifically, the codomain of $\eta \ast {\mathrm{i}\mathrm{d}}_f$ and domain of ${\mathrm{i}\mathrm{d}}_f\ast ℰ$ of the vertical composite $({\mathrm{i}\mathrm{d}}_f\ast ℰ)(\eta \ast {\mathrm{i}\mathrm{d}}_f)$ do not equal:

${\mathrm{i}\mathrm{d}}_B\circ f\stackrel{\eta \ast {\mathrm{i}\mathrm{d}}_f}{⟶}(f\circ g)\circ f⟶f\circ (g\circ f)\stackrel{{\mathrm{i}\mathrm{d}}_f\ast ℰ}{⟶}f\circ {\mathrm{i}\mathrm{d}}_A$.

The pasting theorem guarantees that the vertical composite $({\mathrm{i}\mathrm{d}}_f\ast ℰ)(\eta \ast {\mathrm{i}\mathrm{d}}_f)$ is uniquely defined.

Pasting theorem for 2-category

2-pasting scheme

Anchored graph

Suppose $G$ and $H$ are anchored graphs such that:

• $s_G=s_H$,
• $t_G=t_H$, and
• ${\mathrm{c}\mathrm{o}\mathrm{d}}_G={\mathrm{d}\mathrm{o}\mathrm{m}}_H$.

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

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

$\frac{G\bigsqcup H}{\{{\mathrm{c}\mathrm{o}\mathrm{d}}_G={\mathrm{d}\mathrm{o}\mathrm{m}}_H\}}$

(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 ${\mathrm{e}\mathrm{x}\mathrm{t}}_G$ and ${\mathrm{e}\mathrm{x}\mathrm{t}}_H$, with

• source $s_G=s_H$,
• sink $t_G=t_H$,
• domain ${\mathrm{d}\mathrm{o}\mathrm{m}}_G$, and
• codomain ${\mathrm{c}\mathrm{o}\mathrm{d}}_H$.

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

Definition of 2-pasting scheme

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

$G=G_n\cdots G_1$

into vertical composites of $n\ge 1$ atomic graphs $G_1,\dots ,G_n$.

2-pasting diagram

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

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

For a directed path $P=v_0{e}_1{v}_1\dots e_m{v}_m$ in $G$ with $m\le 1$, define the horizontal composite 1-cell ${\varphi }_P={\varphi }_{e_m}\cdots {\varphi }_{e_1}\in A({\varphi }_{v_0},{\varphi }_{v_m})$.

• $\varphi$ assigns to each interior face $F$ of $G$ a 2-cell ${\varphi }_F:{\varphi }_{{\mathrm{d}\mathrm{o}\mathrm{m}}_F}\to {\varphi }_{{\mathrm{c}\mathrm{o}\mathrm{d}}_F}$ in $A({\varphi }_{s_F},{\varphi }_{t_F})$.

If $G$ admits a pasting scheme presentation, then a $G$-diagram is called a 2-pasting diagram in $A$ of shape $G$.

Statement

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

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

Pasting theorem for Gray-category

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

Pasting theorem for strict n-category (ver. Power (1991))

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 $A$ has a unique "strong composite".

Pasting theorem for strict n-category: for every positive natural number n, every labelled n-pasting scheme in an strict n-category $A$ has a unique "n-pasting composite".

References

• Template:Cite book
• Template:Cite journal
• Template:Cite book
• Template:Cite arXiv
• Template:Cite book
• Template:Cite thesis
• Template:Cite journal
• Template:Cite journal
• Template:Cite journal
• Template:Cite journal
• Template:Cite journal
• Template:Cite journal

External links

• Template:Cite web
• Template:Cite web
• Template:Eom
• Template:Eom