nLab pasting diagram

Idea

Given an ordinary category $C$, a pasting diagram in $C$ is a sequence of composable morphisms in $C$

$$a_1 \stackrel{f_1}{\to} a_2 \stackrel{f_2}{\to} \cdots \stackrel{f_n}{\to} a_{n+1} \,.$$

We think of these arrows as not yet composed, but pasted together at their objects, such as to form a composable sequence, and then say the value of the sequence is the composite morphism in $C$. Or, we could say that a pasting diagram is a specified decomposition of whatever morphism $f$ it evaluates to, thus breaking down $f$ into morphisms $f_i$ which, in practice, are usually “more basic” than $f$ relative to some type of structure on $C$. For example, if $C$ is a monoidal category, the $f_i$ might be instances of associativity isomorphisms.

Pasting decompositions become more elaborate in higher categories. An example of a pasting diagram in a (let’s say strict) 2-category is a pasting of two squares

$$\array{ a &\to& b &\to& c \\ \downarrow &\swArrow& \downarrow &\swArrow& \downarrow \\ d &\to& e &\to& f } \,.$$

To evaluate this diagram as a single 2-morphism, we read this diagram explicitly as the vertical composite of the 2-morphism

$$\array{ a &\to& b &\to& c \\ && \downarrow &\swArrow& \downarrow \\ && e &\to& f }$$

with the 2-morphism

$$\array{ a &\to& b && \\ \downarrow &\swArrow& \downarrow && \\ d &\to& e &\to& f } \,.$$

(Notice that the two halves of the boundary of each of these two 2-morphisms are themselves 1-dimensional pasting diagrams.) Similarly, there can be situations where one pastes together other shapes (triangles, pentagons, etc.), and there may be multiple paths on the way to resolving the diagram into a single 2-morphism, but the idea is that all such paths evaluate to the same 2-morphism, at least in a strict 2-category. General theorems which refer to the uniqueness of pastings are called pasting theorems.

On the other hand, the following diagram is not a pasting diagram:

$$\array{ a &\to& b &\to& c \\ \downarrow &\neArrow& \downarrow &\swArrow& \downarrow \\ d &\to& e &\to& f } \,.$$

Thus, formal definitions of pasting diagram include conditions which impose a consistent “directionality” of the cells so they can be pasted together.

In an n-category a pasting diagram is similarly a collection of n-morphisms with a prescribed way how they are to fit together at their boundaries. There are a number of ways of formalizing pasting diagrams, depending partly on the constituent shapes one allows, and also on the technical hypotheses that allow proofs of pasting theorems, which include directionality conditions but usually also “loop-freeness” conditions to ensure there exists a unique way to resolve or evaluate the diagram. (N.B.: usually a completely unambiguous evaluation is only possible in a strict $n$-category; in a weak $n$-category, one allows uniqueness up to a “contractible space of choices”.)

Despite some technical differences among the formalizations, the core idea throughout is that the overall geometric shapes of pasting diagrams should be (contractible) $n$-dimensional polyhedra, broken down into smaller polyhedral cells which come equipped with directionality or orientations, that can be sensibly pasted together as in the above descriptions once the cells have been assigned values in an $n$-category.

Notions of pasting diagrams

Various formalisms for pasting diagrams have been proposed. They include

• pasting schemes (Johnson)

• parity complexes (Street)

• directed complexes? (Steiner)

• generalized parity complexes? (Forest)

• regular directed complexes (Hadzihasanovic)

Each of these formalisms involve graded sets $\{C_n\}_{n \geq 0}$ together with maps $\partial^+_n: C_{n+1} \to P(C_n)$, $\partial^-_n: C_{n+1} \to P(C_n)$. This goes under various names; here we call it a parity structure. It should be thought of as assigning to each “cell” of dimension $n+1$ a collection of positive boundary cells and negative boundary cells in dimension $n$. The formalisms above are distinguished by the choice of axioms on parity structures, but there is definite kinship among them.

Also related are various notions of categories of shapes, including

• test category

• direct category

• compositions in cubical sets

Examples

• Toda bracket

Related concepts

• pasting law for pullbacks/homotopy pullbacks

References

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

• Jean Benabou, Introduction to bicategories, Reports of the Midwest Category Seminar, Lecture Notes in Mathematics, 47, 1967. (doi:10.1007/BFb0074299)

A reasonably self-contained, formal, and illustrated seven-page introduction to pasting diagrams in the context of weak 2-categories, and their relation to string diagrams is given in Section A.4 of:

• Chris Schommer-Pries, The Classification of Two-Dimensional Extended Topological Field Theories, 2011. (arXiv:1112.1000v2)

A survey discussion of pasting in 2-categories is in

• John Power, 2-Categories (pdf)

from definition 2.10 on. Details are in

• John Power, A 2-categorical pasting theorem , Journal of Algebra, 129, 1990. [doi:10.1016/0021-8693(90)90229-H]

Dominic Verity gave a bicategorical pasting theorem in

• Dominic Verity, Enriched categories, internal categories, and change of base, PhD thesis, Cambridge University, 1992. Reprinted in TAC (link).

A definition and discussion of pasting diagrams in strict omega-categories is in

• Sjoerd Crans, Pasting presentation for Omega-categories (web)

• Sjoerd Crans, Pasting schemes for the monoidal biclosed structure on $\omega$-$\mathbf{Cat}$ (web, ps, pdf)

The notion of pasting scheme used by Crans was introduced by Johnson,

• Michael Johnson, The combinatorics of n-categorical pasting, Journal of Pure and Applied Algebra, 62, 1989. [doi:10.1016/0022-4049(89)90136-9]

Other notions of pasting presentations have been given by Street and by Steiner:

• Ross Street, Parity complexes, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 32, 1991. (link)

• Ross Street, Parity complexes : corrigenda, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 35, 1994. (link)

• Richard Steiner, The algebra of directed complexes, Applied Categorical Structures, 1, 1993. (doi:0.1007/BF00873990)

These notions were compared, and a new generalized one introduced, in

• Simon Forest, Unifying notions of pasting diagrams, Higher Structures, 6, 2022 (arxiv:1903.00282, doi:10.21136/HS.2022.01)

Another recent approach is:

• Amar Hadzihasanovic, Combinatorics of higher-categorical diagrams, 2024 (link)

For an online link to the notion of directed complex, see

• Sjoerd Crans and Richard Steiner, Presentations of omega-categories by directed complexes, Journal of the Australian Mathematical Society, 63, 1997. (doi:10.1017/S1446788700000318)

There is also

• Michael Makkai, Computads and 2 dimensional pasting diagrams (pdf)

For a cubical approach to multiple compositions and other references see the paper

• Philip J. Higgins, Thin elements and commutative shells in cubical omega-categories, Theory and Applications of Categories, 14, 2005. (link)

A formal approach to pasting diagrams in Gray categories:

• Nicola Di Vittorio, A Gray-categorical pasting theorem, Theory and Applications of Categories 39 5 (2023) 150-171 [tac:39-05]