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. Pasting was introduced by Bénabou (1967) when treatment of weak 2-categories. 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 Power (1990). For weak 2-category it is proved in Appendix A of Verity (1992)'s thesis as a consequence of the coherence theorem for weak 2-category. The pasting theorem for n-category was proved by Power (1991) and Johnson (1989), but the definition of the pasting scheme used in that proof is different.
Example of a pasting diagram For the example, consider pasting diagram D for the triangle identity of an adjunction
2-cell E : g ∘ f → i d A {\displaystyle {\mathcal {E}}:g\circ f\rightarrow \mathrm {id} _{A}} , η : i d B → f ∘ g {\displaystyle \eta :\mathrm {id} _{B}\rightarrow f\circ g}
The entire pasting diagram represents the vertical composite ( i d f ∗ E ) ( η ∗ i d f ) {\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. 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. For the example, the codomain of η ∗ i d f {\displaystyle \eta *\mathrm {id} _{f}} and domain of i d f ∗ E {\displaystyle \mathrm {id} _{f}*{\mathcal {E}}} of the vertical composite ( i d f ∗ E ) ( η ∗ i d f ) {\displaystyle (\mathrm {id} _{f}*{\mathcal {E}})(\eta *\mathrm {id} _{f})} do not equal:
i d B ∘ f ⟶ η ∗ i d f ( f ∘ g ) ∘ f ⟶ f ∘ ( g ∘ f ) ⟶ i d f ∗ E f ∘ i d A {\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 ( i d f ∗ E ) ( η ∗ i d f ) {\displaystyle (\mathrm {id} _{f}*{\mathcal {E}})(\eta *\mathrm {id} _{f})} is uniquely defined.
2-categorical pasting theorem
2-pasting scheme
Anchored graph Suppose G {\displaystyle G} and H {\displaystyle H} are anchored graphs such that:
s G = s H {\displaystyle s_{G}=s_{H}} ,
t G = t H {\displaystyle t_{G}=t_{H}} , and
c o d G = d o m H {\displaystyle \mathrm {cod} _{G}=\mathrm {dom} _{H}} . The vertical composite H G {\displaystyle HG} is the anchored graph defined by the following data: (1) The connected plane graph of H G {\displaystyle HG} is the quotient
G ⊔ H { c o d G = d o m H } {\displaystyle {\frac {G\sqcup H}{\{\mathrm {cod} _{G}=\mathrm {dom} _{H}\}}}}
(2) The interior faces of H G {\displaystyle HG} are the interior faces of G {\displaystyle G} and H {\displaystyle H} , which are already anchored. (3) The exterior face of H G {\displaystyle HG} is the intersection of e x t G {\displaystyle \mathrm {ext} _{G}} and e x t H {\displaystyle \mathrm {ext} _{H}} , with
source s G = s H {\displaystyle s_{G}=s_{H}} , sink t G = t H {\displaystyle t_{G}=t_{H}} , domain d o m G {\displaystyle \mathrm {dom} _{G}} , and codomain c o d H {\displaystyle \mathrm {cod} _{H}} . of the disjoint union of G {\displaystyle G} and H {\displaystyle H} , with the codomain of G {\displaystyle G} identified with the domain of H {\displaystyle H} .
2-pasting scheme A 2-pasting scheme is an anchored graph G together with a decomposition
G = G n ⋯ G 1 {\displaystyle G=G_{n}\cdots G_{1}}
into vertical composites of n ≥ 1 {\displaystyle n\geq 1} atomic graphs G 1 , … , G n {\displaystyle G_{1},\dots ,G_{n}} .
2-pasting diagram Suppose A {\displaystyle A} is a 2-category, and G {\displaystyle G} is an anchored graph. A G {\displaystyle G} -diagram in A {\displaystyle A} is an assignment ϕ {\displaystyle \phi } as follows.
ϕ {\displaystyle \phi } assigns to each vertex v {\displaystyle v} in G {\displaystyle G} an object ϕ v {\displaystyle \phi _{v}} in A {\displaystyle A} .
ϕ {\displaystyle \phi } assigns to each edge e {\displaystyle e} in G {\displaystyle G} with tail u {\displaystyle u} and head v {\displaystyle v} a 1-cell ϕ e ∈ A ( ϕ u , ϕ v ) {\displaystyle \phi _{e}\in A(\phi _{u},\phi _{v})} . For a directed path P = v 0 e 1 v 1 … e m v m {\displaystyle P=v_{0}e_{1}v_{1}\dots e_{m}v_{m}} in G {\displaystyle G} with m ≤ 1 {\displaystyle m\leq 1} , define the horizontal composite 1-cell ϕ P = ϕ e m ⋯ ϕ e 1 ∈ A ( ϕ v 0 , ϕ v m ) {\displaystyle \phi _{P}=\phi _{e_{m}}\cdots \phi _{e_{1}}\in A(\phi _{v_{0}},\phi _{v_{m}})} .
ϕ {\displaystyle \phi } assigns to each interior face F {\displaystyle F} of G {\displaystyle G} a 2-cell ϕ F : ϕ d o m F → ϕ c o d F {\displaystyle \phi _{F}:\phi _{\mathrm {dom} _{F}}\rightarrow \phi _{\mathrm {cod} _{F}}} in A ( ϕ s F , ϕ t F ) {\displaystyle A(\phi _{s_{F}},\phi _{t_{F}})} . If G {\displaystyle G} admits a pasting scheme presentation, then a G {\displaystyle G} -diagram is called a 2-pasting diagram in A {\displaystyle A} of shape G {\displaystyle 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.
Gray-categorical pasting theorem Every 2-dimensional pasting diagram in a Gray-category has a unique composition up to a contractible groupoid of choices.
n-categorical pasting theorem 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 {\displaystyle 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 {\displaystyle A} has a unique n-pasting composite.
Notes
References Bénabou, Jean (1967). "Introduction to bicategories". Reports of the Midwest Category Seminar. Lecture Notes in Mathematics. Vol. 47. pp. 1–77. doi:10.1007/BFB0074299. ISBN 978-3-540-03918-1. Power, A.J (1990). "A 2-categorical pasting theorem". Journal of Algebra. 129 (2): 439–445. doi:10.1016/0021-8693(90)90229-H. Power, A. J. (1991). "An n-categorical pasting theorem". Category Theory. Lecture Notes in Mathematics. Vol. 1488. pp. 326–358. doi:10.1007/BFb0084230. ISBN 978-3-540-54706-8. Johnson, Niles; Yau, Donald (2019). "A bicategorical pasting theorem". arXiv:1910.01220 [math.CT]. Johnson, Niles; Yau, Donald (2021). "Pasting Diagrams". 2-Dimensional Categories. pp. 99–146. arXiv:2002.06055. doi:10.1093/oso/9780198871378.003.0003. ISBN 978-0-19-887137-8. Johnson, Michael. Pasting Diagrams in n-Categories with Applications to Coherence Theorems and Categories of Paths (PDF) (Thesis). Johnson, Michael (1989). "The combinatorics of n-categorical pasting". Journal of Pure and Applied Algebra. 62 (3): 211–225. doi:10.1016/0022-4049(89)90136-9. Hackney, Philip; Ozornova, Viktoriya; Riehl, Emily; Rovelli, Martina (January 2023). "An (∞,2)-categorical pasting theorem". Transactions of the American Mathematical Society. 376 (1): 555–597. arXiv:2106.03660. doi:10.1090/tran/8783. Yetter, D. N. (2009). "On deformations of pasting diagrams" (PDF). Theory and Applications of Categories. 22: 24–53. doi:10.70930/tac/cw7uv9mh. ISSN 1201-561X. Vittorio, Nicola Di (2023). "A Gray-categorical pasting theorem". Theory and Applications of Categories. 39: 150–171. doi:10.70930/tac/1l9k8c4l. Verity, Dominic (1992). "Enriched categories, internal categories and change of base" (PDF). Reprints in Theory and Applications of Categories. 20: 1–266. Forest, Simon (2022). "Unifying notions of pasting diagrams". Higher Structures. 6 (1): 1–79.
External links "pasting diagram". ncatlab.org. "pasting scheme". ncatlab.org. Street, Ross (2001) [1994], "Higher-dimensional category", Encyclopedia of Mathematics, EMS Press Street, Ross (2001) [1994], "Bicategory", Encyclopedia of Mathematics, EMS Press