Our notation implicitly assumed one more equation, (h) below, now added. Thanks to Peter Selinger for catching this!

Sheaves over such causal coverages now describe how to assign data to your spacetime such that "past data evolves continuously and deterministically into future data". For much more, see Nesta's great PhD thesis, with many fun ideas and pictures: arxiv.org/abs/2406.15406.

That means you need a way to think about paths in terms of open sets rather than points. The idea: approximate a path by finer and finer finite sequences of opens that are causally before each other.

Well, you don't need points to describe that, and can do it purely with open sets. This new paper with Nesta van der Schaaf describes how: arxiv.org/abs/2510.17417. The idea is in the picture: region U is covered by collection A of regions if any causal path that ends up in U must pass through A.

Also about circuits: the physics Nobel prize just announced!

So much material for titles here, too! "If you liked it you shoulda put a rig on it", "The rig is up", "Taking back control". What's your favourite?

Theorem: these two ways to build a theory of controlled circuits are the same! Practically, this means the equations are complete, and we can use them to manipulate and optimise controlled circuits. Foundationally, this pins down what this not-quite-data-or-control-flow control really is about.

Another thing you can do is make a new theory where 'controlled gates' are, roughly, matrices of base gates. Technically, you freely adjoin sums to the base (tensor) prop, universally giving it rig structure.

The equations all have natural interpretations. For example, the complementarity equation (e) says that a gate on the target wire is the same as a positively controlled and then a negatively controlled version of it: the control bit is either on or off.

Start with any base circuit theory, in terms of tensors only; technically, as a prop. Build a new controlled circuit theory that has controlled versions of the base gates, subject to these equations.

The key is that circuits contain controlled gates. It's not quite data flow, nor control flow, in the usual computer science sense. But it is clearly important. Can we take back this control, and separate this control from a base theory of uncontrolled circuits? Yes we can!