The UTxO (unspent transaction output) model is the underlying data structure of Bitcoin, which has since been extended to the Extended UTxO model.

It exists in code, but what does it mean? I will give a novel mathematical model based on some strikingly simple type equations which -- for me at least -- make it easier to see what is structurally going on. I will describe how the equations can be used to obtain two further models, one which is more abstract and one which is more concrete:

• A universal algebra axiomatisation, exhibiting Blockchain as an algebraic structure and so implying a general mathematics and specific testable properties against which to verify an EUTxO implementation, and

• An executable Haskell reference implementation

In the real world, people join humanity, grow up trusting very different quorums from one another, and they may change their quorums with time. In technical terms we could call this an open permissionless system with mutable local quorums.

One might expect such a system to be a jumbled disorder — and yet, somehow, it self-organises into uniform areas of fairly stable consensus. One might almost suspect there could be some deep mathematical principles involved!

In this talk I will discuss consensus in a permissionless open system with mutable local quorums, and show how a topological view of the system gives a clean mathematical analysis with surprisingly good explanatory power.