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.
Operation of pasting appeared in when introducing weak 2-category. 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 . For weak 2-category it is proved in Appendix A of 's thesis as a consequence of the coherence theorem for weak 2-category. The pasting theorem for strict n-category was proved by and , 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 ,
The entire pasting diagram represents the vertical composite 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.
Specifically, the codomain of and domain of of the vertical composite do not equal:
.
The pasting theorem guarantees that the vertical composite is uniquely defined.
Pasting theorem for 2-category 2-pasting scheme Anchored graph
Suppose and are anchored graphs such that: , , and .
The vertical composite is the anchored graph defined by the following data:
(1) The connected plane graph of is the quotient
(2) The interior faces of are the interior faces of and , which are already anchored.
(3) The exterior face of is the intersection of and , with
source , sink , domain , and codomain .
of the disjoint union of and , with the codomain of identified with the domain of .
Definition of 2-pasting scheme
A 2-pasting scheme is an anchored graph G together with a decomposition
into vertical composites of atomic graphs .
2-pasting diagram Suppose is a 2-category, and is an anchored graph. A -diagram in is an assignment as follows.
assigns to each vertex in an object in . assigns to each edge in with tail and head a 1-cell .
For a directed path in with , define the horizontal composite 1-cell .
assigns to each interior face of a 2-cell in .
If admits a pasting scheme presentation, then a -diagram is called a 2-pasting diagram in of shape .
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 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 has a unique "n-pasting composite".
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