Wiring Diagram

definition annotation

Wiring diagrams (string diagrams) are visual representations for building new relationships from old. Boxes are relationships, wires are objects/resources, and the diagram itself shows how relationships combine. Invented in the context of monoidal categories by Joyal and Street, they have long been used informally by engineers and scientists. Wires and boxes are icons; more icons appear as more structure is assumed.

Sources: 7 Sketches §2.2.2 (Eq. 2.13 “different styles”), §4.4.2, §5.2, §6.3.2, §6.5.1; DaoFP §15.1 (string diagrams for monads and adjunctions); Kittenlab Lecture 6 (directed port graphs, wiring diagrams as ACSets), 15 (undirected wiring diagrams as cospans); CTfS Examples 5.4.2.4, 5.4.2.8, Applications 5.4.2.5–5.4.2.10

Styles, by structure

structure	diagram style	note
Preorder	single-input, single-output boxes in series	chaining $x_0\le x_1\le x_2$
Symmetric Monoidal Preorder	boxes with many inputs/outputs in series and parallel; crossing wires	Wiring Diagrams for Monoidal Preorders
+ discard axiom	wires may terminate	manufacturing
+ copy axiom	wires may split	informatics
Category	boxes with one input and one output in series	morphisms; the Free Category on a graph
Monoidal Category	as for monoidal preorders, but boxes are named morphisms (§4.4.2)	Symmetric Monoidal Category diagrams
Prop	boxes with $m$ inputs and $n$ outputs on the objects $\underline{n}$	port graphs, signal flow graphs
Compact Closed Category	wires may bend backwards (cups and caps)	Feas
Hypergraph Category	wires may split, merge, start and end freely (spiders)	undirected wiring diagrams, cospans
Operad	boxes nested inside boxes, composed by substitution	$\mathbf{Cospan}$-algebras, decorated cospans
2-categories / adjunctions	string diagrams with regions for categories, strings for functors, dots for natural transformations	DaoFP §15.1

Soundness. A wiring diagram is a graphical proof: if all interior boxes are valid, the exterior box is valid (Wiring Diagrams for Monoidal Preorders). In a monoidal category two diagrams that are isotopic denote equal morphisms — the coherence theorem of Joyal–Street; for operads a wiring diagram is literally a morphism of the operad $\mathbf{Cospan}$ or $\mathbf{Set}$.

The operad of wiring diagrams (Category Theory for Scientists §5.4.2)

CTfS’s operad $\mathcal{W}$: an object is a circle with finitely many cables, each carrying a set of values (black wires $\mathbb{Z}$, red wires $\{\text{sweet},\text{sour},\text{salty},\text{bitter},\text{umami}\}$); an operation ”$E$ is composed of $E_1,E_2,E_3$ in the following way” is a picture of circles inside a circle with cables joined, formalized by a jointly surjective cospan of cable sets; composition is substitution, computed by pushout. The semantics sends a circle to the set of value-assignments on its cables, and a wiring diagram to the relation between local and global assignments — an operad morphism $\mathcal{W}\to \mathcal{R}$ into the operad of relations. Applications: an entity survives exactly the phenomena that each of its parts survives; a mind or an economy analyzed through its connectome or supply chain; Radul–Sussman propagator networks (Operad, Operad Algebra).

Wiring diagrams as data (Kittenlab Lecture 6)

A directed port graph is an ACSet on the schema with boxes, input ports, output ports and wires (source: an output port, target: an input port); a directed wiring diagram additionally has outer ports. Catlab’s WiringDiagram and UndirectedWiringDiagram implement these, and oapply evaluates an operad algebra on a diagram.

JuliaHaskell

Docs: Relational programs / UWDs · Wiring diagrams · Vignette: wiring diagram basics — Kittenlab Lecture 6

using Catlab, Catlab.WiringDiagrams
# a directed wiring diagram with two boxes composed in series (Catlab.WiringDiagrams)
f = Box(:f, [:A], [:B]); g = Box(:g, [:B], [:C])
d = WiringDiagram([:A], [:C])
fv, gv = add_box!(d, f), add_box!(d, g)
add_wires!(d, [(input_id(d), 1) => (fv, 1), (fv, 1) => (gv, 1), (gv, 1) => (output_id(d), 1)])
nboxes(d), nwires(d)      # (2, 3)
 
# an undirected wiring diagram via the @relation macro (hypergraph style)
uwd = @relation (x, z) begin
  R(x, y); S(y, z)
end

-- a wiring diagram as a free symmetric monoidal expression (syntax tree)
data WD = Box String [String] [String]   -- name, input wires, output wires
        | Seq WD WD                      -- series composition
        | Par WD WD                      -- parallel composition
        | Id [String]
        | Swap String String