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More of a comment than a question: If is a monoidal category, then is lax monoidal (for the cartesian product on ), but it is NOT strong monoidal. That is, there is a non-invertible natural transformation . It takes about 5 seconds thinking to realise this, but I made exactly this error in one of my papers, and peer review didn't catch it either
The failure of this functor from being strong monoidal could be a good description for the "complexity" of things.
Paolo Perrone said:
The failure of this functor from being strong monoidal could a good description for the "complexity" of things.
It would be really interesting if someone could illustrate this with a small example.
For with cartesian product, the elements of in the image of the laxator are exactly those functions that can be factored as . So "complexity" in this example is the ability of every output of a function to depend on every input
Jules Hedges said:
For with cartesian product, the elements of in the image of the laxator are exactly those functions that can be factored as . So "complexity" in this example is the ability of every output of a function to depend on every input
Interesting! This is similar to a situation in engineering where we say "there is no cross-talk". If we view as a message sent across a channel from two sources and , then the condition says that the message from the two sources decomposes into messages generated by each source individually.
So, I suppose the more "complex" case would correspond to when there is cross-talk.
This business about the laxator also shows up in the theory of decorated cospans, where we use it to describe, e.g. how there are more chemical reactions involving molecules A,B,C,D,E than those the consist of a reaction involving A,B,C and a reaction involving D,E.