# nLab pasting scheme

Contents

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

Pasting schemes are one possibility to give a rigorous treatment of the (older) notational device of pasting diagrams.

## Notions of pasting scheme

More than one definition of pasting scheme has been used to give a justification to the practice of pasting diagrams.

For example, there are pasting schemes in the sense of Johnson 1987, and there are pasting schemes in the sense of Power 1990.

### Power’s definition

Power’s notion is based on planar embeddings of quivers $Q$ that satisfy the following conditions:

• $Q$ is finite and connected;

• For the free category $F(Q)$ generated by $Q$, every endomorphism is an identity ($Q$ has no directed cycles);

• There are vertices $s, t$ of $Q$ (the source and sink, respectively) such that for every vertex $v$, the hom-sets $F(Q)(s, v)$ and $F(Q)(v, t)$ are inhabited (there exist directed paths in $Q$ from $s$ to $v$ and from $v$ to $t$).

We will call such quivers progressive, in rough analogy with progressive string diagrams.

There is a restricted geometric realization functor

$Quiv = Set^{(0 \rightrightarrows 1)^{op}} \stackrel{skel}{\to} Set^{\Delta^{op}} \stackrel{R}{\to} Space$

(into a convenient category of topological spaces $Space$) which we again denote as $R: Quiv \to Space$.

###### Definition

(after Power) A pasting scheme consists of a progressive quiver $Q$ together with an subspace embedding $i: R(Q) \hookrightarrow \mathbb{R}^2$, such that $i$ is a $C^1$ embedding on each edge interior.

To each pasting scheme $(Q, i)$ we may associate a 2-computad $C = Comp(Q, i)$. Part of the computad structure is easily described:

• The underlying quiver of $C$ is $Q$;

• The elements of $C_2$ (the 2-cells) are the bounded connected components of $\mathbb{R}^2 \setminus i(R(Q))$.

To complete the description of the computad, we need to define the source and target $\sigma(c), \tau(c) \in Mor(F(Q))$ of a 2-cell $c$. For each edge $e \in Q_1$ such that $i(e)$ lies in the boundary of $c$, and each interior point $x \in e$, we let $t(x) = i'(x)$ be the tangent vector, and $n(x)$ the inward-pointing normal (pointing into $c$); the $C^1$ embedding condition ensures that these make sense. Then say that $e$ is a negative edge of $c$ if

$\det \binom{t(x)}{n(x)} \lt 0$

(the cell $c$ is “to the right” as $i$ is traversed along $e$) and a positive edge if

$\det \binom{t(x)}{n(x)} \gt 0$

(the cell $c$ is to the left).

Power shows that under the progressivity conditions, for each $c \in C_2$ there are unique distinct vertices $u, v$ such that the negative edges of $c$ form a directed path $\alpha \in F(Q)(u, v)$ from $u$ to $v$, and the positive edges of $c$ form a directed path $\beta \in F(Q)(u, v)$. We then define $\sigma(c) = \alpha$ and $\tau(c) = \beta$.

Power’s pasting theorem may be stated in the following form:

###### Theorem

For every pasting scheme $(Q, i)$, the free 2-category on the 2-computad $Comp(Q, i)$ has exactly one 2-morphism $\gamma$ whose $0$-domain is the source of $Q$, whose $0$-codomain is the sink of $Q$, whose $1$-domain of $\gamma$ is the directed path consisting of positive edges of the unique unbounded component of $\mathbb{R}^2 \setminus i(R(Q))$, and whose $1$-codomain is the directed path consisting of negative edges of that component.

The notion of pasting in a 2-category was introduced in

• Jean Bénabou, Introduction to bicategories, in Lecture Notes in Mathematics Vol. 47, pp. l-77, Springer-Verlag, New York/Berlin, (1967)

• Michael Johnson?: Pasting Diagrams in $n$-Categories with Applications to Coherence Theorems and Categories of Paths, Doctoral Thesis, University of Sydney, 1987

• Michael Johnson?: The Combinatorics of $n$-Categorical Pasting, Journal of Pure and Applied Algebra 62 (1989)
• John Power: A 2-Categorical Pasting Theorem, Journal of Algebra 129 (1990)

Other notions of pasting presentations have been given by Street (parity complexes) and by Steiner,

• Ross Street, Parity complexes, Cahiers Top. Géom Diff. Catégoriques 32 (1991), 315-343. (link) Corrigenda, Cahiers Top. Géom Diff. Catégoriques 35 (1994), 359-361. (link)

• Richard Steiner, The algebra of directed complexes, Appl. Cat. Struct. 1 (1993), 247-284.