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 Bénabou (1967). 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 Power (1990), and for weak 2-category it is proved in Appendix A of Verity (1992)'s thesis. The pasting theorem for n-category version was proved by Power (1991) and Johnson (1989), but the definition of the pasting scheme differs. String diagrams are justified by the pasting theorem.

Consider the pasting diagram D for adjunction

2-cell ${\displaystyle {\mathcal {E}}:gf\rightarrow \mathrm {id} _{A}}$, ${\displaystyle \eta :\mathrm {id} _{B}\rightarrow fg}$

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), displayed on the right above^[1]

• Every 2-pasting diagram in an strict 2-category A has a unique composite.^[2]
• Every 2-pasting diagram in an weak 2-category A has a unique composite.^[3]

Suppose G and H are anchored graphs^[4] such that:

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

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

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

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

(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 ${\displaystyle \mathrm {ext} _{G}}$ 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 G and H, with the codomain of G identified with the domain of H.

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]

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

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

Every 2-dimensional pasting diagram in a Gray-category has a unique composition up to a contractible groupoid of choices.^[7]

For any positive natural number n, every labelled n-pasting scheme in an strict n-category A has a unique "strong" composite.^[8]

For every positive natural number n, every labelled n-pasting scheme in an strict n-category A has a unique n-pasting composite.^[9]

• 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.

• "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