• a diagram consists of a number of inner boxes, each with some input ports and some output ports, that are wired together inside an outer box, which also has input ports and ouput ports
  • 4 types of ports: inner iput, inner output, outer input, and outer output
  • 2 ways we can wire:
    • wire outer output port to exactly one innter output port
    • wire inner input port to exactly one inner output port or an outer input port
• the category of wiring diagrams has
  • objects: boxes
  • morphisms: wirding diagrams
  • compose: we can compose wiring diagrams by filling the inner boxes with other wiring diagrams

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• : The idea is roughly to think of objects in a monoidal category as “strings” and of morphisms from one tensor product to another as a node which the source strings enter and the target strings exit.
• Further structure on the monoidal category is encoded in geometrical properties on these strings. For instance:
  • putting strings next to each other denotes the monoidal product, and having no string at all denotes the tensor unit;
  • braiding strings over each other corresponds to – yes, the monoidal braiding (if any);
  • bending strings around corresponds to dualities on dualizable objects (if any).