Slogan: colimits model interconnection. Electric circuits, chemical reaction networks, finite state automata and Markov processes are all described by network diagrams, whose wires are undirected — unlike the morphisms of a category. 7 Sketches Chapter 6 shows how universal constructions capture network compositionality: to install a light switch from open components (power source, switch, lamp + resistor), mark each component’s ports (interface, a finite set — possibly empty), draw witnesses-to-connection pointing at two ports each, and identify the indicated ports. Formally this is a finite Colimit in : the Coequalizer of the two functions from witnesses to ports is the set of terminals after interconnection (Example 6.42), and pushouts glue components along shared ports.
Sources: 7 Sketches §6.1 (“The ubiquity of network languages”, “Connections via colimits”, “Composition operations and wiring diagrams”), §6.2 (Examples 6.25, 6.42), §6.6; Kittenlab Lectures 9, 15.
Colimits “are a kind of epiphenomenon of the category”; to use them as an operation one packages them into the category of cospans , whose composition and monoidal product are the colimits (a category with finite colimits is one with an Initial Object and all pushouts, Proposition 6.32). Network-style wiring diagrams are those of hypergraph categories, whose extra icons are spiders that fuse when they share a leg — the Frobenius structure. Cospans in form “the theory of hypergraph categories” (Theorem 6.58), decorating cospans with data builds hypergraph categories of circuits (Decorated Cospan), and assembling all cospans into an Operad lets one tailor the compositional structure itself (-algebras hypergraph props, Proposition 6.101).
The chapter completes an “informal hierarchy of compositional structures”: preorders → categories → monoidal categories → operads; each with its own Wiring Diagram style (manufacturing, signal flow, co-design, networks).