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.

Pasting diagram

Example

Consider the pasting diagram D for adjunction

File:Pasting theorem.svg

2-cell ${ℰ}:gf\to {\mathrm{i}\mathrm{d}}_A$, $\eta :{\mathrm{i}\mathrm{d}}_B\to fg$

The entire pasting diagram represents the vertical composite $({\mathrm{i}\mathrm{d}}_f*{ℰ})(\eta *{\mathrm{i}\mathrm{d}}_f)$ 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:

• $s_G=s_H$,
• $t_G=t_H$, and
• ${\mathrm{c}\mathrm{o}\mathrm{d}}_G={\mathrm{d}\mathrm{o}\mathrm{m}}_H$.

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

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

$\frac{G\bigsqcup H}{\{{\mathrm{c}\mathrm{o}\mathrm{d}}_G={\mathrm{d}\mathrm{o}\mathrm{m}}_H\}}$

(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 ${\mathrm{e}\mathrm{x}\mathrm{t}}_G$ and ${\mathrm{e}\mathrm{x}\mathrm{t}}_H$, with

• source $s_G=s_H$,
• sink $t_G=t_H$,
• domain ${\mathrm{d}\mathrm{o}\mathrm{m}}_G$, and
• codomain ${\mathrm{c}\mathrm{o}\mathrm{d}}_H$.

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

$G=G_n\cdots G_1$

into vertical composites of $n\ge 1$ 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 $\varphi$ as follows.

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

For a directed path $P=v_0{e}_1{v}_1\dots e_m{v}_m$ in G with $m\le 1$, define the horizontal composite 1-cell ${\varphi }_P={\varphi }_{e_m}\cdots {\varphi }_{e_1}\in A({\varphi }_{v_0},{\varphi }_{v_m})$.

• $\varphi$ assigns to each interior face F of G a 2-cell ${\varphi }_F:{\varphi }_{{\mathrm{d}\mathrm{o}\mathrm{m}}_F}\to {\varphi }_{{\mathrm{c}\mathrm{o}\mathrm{d}}_F}$ in $A({\varphi }_{s_F},{\varphi }_{t_F})$.

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.

References

• Bénabou, Jean (1967). "Introduction to bicategories". Reports of the Midwest Category Seminar. Lecture Notes in Mathematics. 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. 1488. pp. 326–358. doi:10.1007/BFb0084230. ISBN 978-3-540-54706-8. https://www.lfcs.inf.ed.ac.uk/reports/91/ECS-LFCS-91-190. 
• 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. 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. doi:10.1090/tran/8783. 
• Yetter, D. N. (2009). "On deformations of pasting diagrams.". Theory and Applications of Categories [electronic only] 22: 24–53. doi:10.70930/tac/cw7uv9mh. ISSN 1201-561X. http://www.tac.mta.ca/tac/volumes/22/2/22-02.pdf. 
• 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 (Thesis). Vol. 20. pp. 1–266.
• Forest, Simon (2022). "Unifying notions of pasting diagrams". Higher Structures 6 (1): 1-79. https://higher-structures.math.cas.cz/api/files/issues/Vol6Iss1/Forest. 

External links

• "pasting diagram". https://ncatlab.org/nlab/show/pasting+diagram. 
• "pasting scheme". https://ncatlab.org/nlab/show/pasting+scheme. 
• Hazewinkel, Michiel, ed. (2001), "Higher-dimensional category", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• Hazewinkel, Michiel, ed. (2001), "Bicategory", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page