It was fun talking to Eugenia; she’s now in Ireland doing interviews for her book. She gave a really good talk about distributive laws for Lawvere theories, and if I have enough energy I’ll blog about it on the *n*-Café. It was full of crunchy facts, as irresistibly tasty as peanuts.

I’m not claiming that these rules *have* to be interpreted as 2-morphisms, just that it’s a fun thing to do.

If we impose them as equations, we get an interesting category where morphisms are ‘behaviors’ of planar electrical circuits: two circuits can look different yet behave the same way. More mathematically speaking, if we impose them as equations we get a category that’s the ‘full image’ of the black box functor

■:

which sends circuits to their behaviors.

If we don’t impose them at all, we get the category PlaneCirc.

But if we work 2-categorically and impose them as 2-morphisms we can ‘have our cake and eat it too’, getting a single structure that knows both about circuits and their behaviors.

This is my standard mantra about higher categorical structures: they let you talk about how things are ‘the same in a way, yet not identical’. I’ve been slacking off on higher category theory lately, trying to understand network theory with categories, but the higher categorical approach will eventually be good to try.

And we can do it using your program Globular! I’m eager to see it on a web browser.

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