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.

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 ${\mathcal {E}}:g\circ f\rightarrow \mathrm {id} _{A}$, ${\displaystyle \eta$ :\mathrm {id} _{B}\rightarrow f\circ g}

The entire pasting diagram represents the vertical composite $(\mathrm {id} _{f}*{\mathcal {E}})(\eta *\mathrm {id} _{f})$ 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 $\eta *\mathrm {id} _{f}$ and domain of $\mathrm {id} _{f}*{\mathcal {E}}$ of the vertical composite $(\mathrm {id} _{f}*{\mathcal {E}})(\eta *\mathrm {id} _{f})$ do not equal:

$\mathrm {id} _{B}\circ f\mathrel {\stackrel {\eta *\mathrm {id} _{f}}{\longrightarrow }} (f\circ g)\circ f\longrightarrow f\circ (g\circ f)\mathrel {\stackrel {\mathrm {id} _{f}*{\mathcal {E}}}{\longrightarrow }} f\circ \mathrm {id} _{A}$.

The pasting theorem guarantees that the vertical composite $(\mathrm {id} _{f}*{\mathcal {E}})(\eta *\mathrm {id} _{f})$ is uniquely defined.

2-categorical pasting theorem

2-pasting scheme

Anchored graph

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

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

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

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

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

(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 $\mathrm {ext} _{G}$ and $\mathrm {ext} _{H}$, with

• source $s_{G}=s_{H}$,
• sink $t_{G}=t_{H}$,
• domain $\mathrm {dom} _{G}$, and
• codomain $\mathrm {cod} _{H}$.

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

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 $n\geq 1$ 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 $\phi$ as follows.

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

For a directed path $P=v_{0}e_{1}v_{1}\dots e_{m}v_{m}$ in $G$ with $m\leq 1$, define the horizontal composite 1-cell $\phi _{P}=\phi _{e_{m}}\cdots \phi _{e_{1}}\in A(\phi _{v_{0}},\phi _{v_{m}})$.

• $\phi$ assigns to each interior face $F$ of $G$ a 2-cell $\phi _{F}:\phi _{\mathrm {dom} _{F}}\rightarrow \phi _{\mathrm {cod} _{F}}$ in $A(\phi _{s_{F}},\phi _{t_{F}})$.

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

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 $A$ 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

• 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