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 . 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 , and for weak 2-category it is proved in Appendix A of 's thesis. The pasting theorem for n-category version was proved by and , but the definition of the pasting scheme differs. String diagrams are justified by the pasting theorem.

Pasting diagram

Example

Consider the pasting diagram D for adjunction

2-cell ,

The entire pasting diagram represents the vertical composite which is a 2-cell in D(A, B), displayed on the right above

2-categorical pasting theorem

Every 2-pasting diagram in an strict 2-category A has a unique composite.
Every 2-pasting diagram in an weak 2-category A has a unique composite.

2-pasting scheme

Anchored graph

Suppose G and H are anchored graphs such that:

,
, and
.

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

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

(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 and , with

source ,
sink ,
domain , and
codomain .

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

into vertical composites of atomic graphs .

2-pasting diagram

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

assigns to each vertex v in G an object in A.
assigns to each edge e in G with tail u and head v a 1-cell .

For a directed path in G with , define the horizontal composite 1-cell .

assigns to each interior face F of G a 2-cell in .

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

Gray-categorical pasting theorem

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

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.

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.