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.

Operation of pasting appeared in Bénabou (1967) 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 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 strict n-category was proved by Johnson (1989) and Power (1991), 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 $ℰ : g \circ f \to {i d}_{A}$ , $η : {i d}_{B} \to f \circ g$

The entire pasting diagram represents the vertical composite $({i d}_{f} * ℰ) (η * {i d}_{f})$ 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.

Specifically, the codomain of $η * {i d}_{f}$ and domain of ${i d}_{f} * ℰ$ of the vertical composite $({i d}_{f} * ℰ) (η * {i d}_{f})$ do not equal:

${i d}_{B} \circ f \overset{η * {i d}_{f}}{⟶} (f \circ g) \circ f \overset{ρ}{⟶} f \circ (g \circ f) \overset{{i d}_{f} * ℰ}{⟶} f \circ {i d}_{A}$ .

The pasting theorem guarantees that irrespective of how $ρ$ is taken, the vertical composite $({i d}_{f} * ℰ) (η * {i d}_{f})$ is uniquely defined.

2-cell $α : a \to c \circ b$ , $β : b \to e \circ d$ , $γ : f \circ c \to g$ , $δ : h \circ g \circ e \to i$ , $ϵ : k \to j \circ h$

The above pasting diagram is a vertical composite of horizontal composites of 2-cells of the form:

Operation of pasting in a 2-category, another problem is the need to show that composite of vertical and horizontal composite of 2-cells are unique irrespective of the order of composition. This was proven by Power (1990).

Pasting theorem for 2-category

Anchored graph

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

$s_{G} = s_{H}$ ,
$t_{G} = t_{H}$ , and
${c o d}_{G} = {d o m}_{H}$ .

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

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

\frac{G ⊔ H}{{{c o d}_{G} = {d o m}_{H}}}

of the disjoint union of

G

and

H

, with the codomain of

G

identified with the domain of

H

.

(2) The interior faces of $H G$ are the interior faces of $G$ and $H$ , which are already anchored.

(3) The exterior face of $H G$ is the intersection of ${e x t}_{G}$ and ${e x t}_{H}$ , with

source $s_{G} = s_{H}$ ,
sink $t_{G} = t_{H}$ ,
domain ${d o m}_{G}$ , and
codomain ${c o d}_{H}$ .

2-pasting scheme

A 2-pasting scheme is an anchored graph G together with a decomposition

$G = G_{n} \dots G_{1}$

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 $ϕ$ as follows.

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

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 $ϕ_{P} = ϕ_{e_{m}} \dots ϕ_{e_{1}} \in A (ϕ_{v_{0}}, ϕ_{v_{m}})$ .

$ϕ$ assigns to each interior face $F$ of $G$ a 2-cell $ϕ_{F} : ϕ_{{d o m}_{F}} \to ϕ_{{c o d}_{F}}$ in $A (ϕ_{s_{F}}, ϕ_{t_{F}})$ .

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.

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 $A$ 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".

Notes

↑ Johnson & Yau 2019
1 2 3 4 Johnson 1989
1 2 Hackney et al. 2023
↑ Johnson & Yau 2021, Definition 3.2.11.
↑ Johnson & Yau 2021, Definition 3.2.13.
↑ Johnson & Yau 2021, Definition 3.3.1.
↑ Johnson & Yau 2021, Theorem 3.3.7 (2-Categorical Pasting)
↑ Johnson & Yau 2021, Theorem 3.6.6 (Bicategorical Pasting)
↑ Vittorio 2023, 4.24. Theorem.
↑ Power 1991, Theorem 6.10 (A weak n-categorical pasting theorem)
↑ Power 1991, Definition 6.8
↑ Power 1991, Theorem 6.16 (An n-categorical pasting theorem)

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

[1] Johnson & Yau 2019

[Johnson1989-2] 1 2 3 4 Johnson 1989

[HackneyOzornovaRiehlRovelli2023-3] 1 2 Hackney et al. 2023

[4] Johnson & Yau 2021, Definition 3.2.11.

[5] Johnson & Yau 2021, Definition 3.2.13.

[6] Johnson & Yau 2021, Definition 3.3.1.

[7] Johnson & Yau 2021, Theorem 3.3.7 (2-Categorical Pasting)

[8] Johnson & Yau 2021, Theorem 3.6.6 (Bicategorical Pasting)

[9] Vittorio 2023, 4.24. Theorem.

[10] Power 1991, Theorem 6.10 (A weak n-categorical pasting theorem)

[11] Power 1991, Definition 6.8

[12] Power 1991, Theorem 6.16 (An n-categorical pasting theorem)