### Centres

Craig Pastro and I have a preprint out!

Restricted exchange, braidings and the monoidal centre; Jonathan A. Cohen and Craig A. Pastro

**Abstract:**Braided monoidal categories arise naturally as centres of monoidal categories and have been the focus of much recent attention in both mathematics and physics. By suitably restricting the use of the exchange rule, we obtain a sequent calculus whose categorical semantics may be seen as freely constructing the centre of a monoidal category. This calculus is shown to admit a strongly normalising and confluent cut elimination procedure. The resulting logic fits neatly into the landscape of noncommutative logics and is distinguished by possessing a particularly perspicuous semantics.

## 1 Comments:

Craig and I were briefly chatting about this today. I wondered what you get if you ask for a monoidal comonad together with a braiding for the monoidal category of Eilenberg-Moore coalgebras. It seems to give something different from your approach, since box formulas can only be exchanged with other box formulas, but it might be a way of doing something similar.

(The motivation for this suggestion is by analogy with the ordinary linear logic ! modality: one characterisation of a linear exponential comonad is as a monoidal comonad such that the tensor product on the category of coalgebras is actually a categorical product, i.e. naturally has weakening and contraction.)

We haven't worked out any details though, so it might be totally bogus.

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