Space and time in monoidal categories

The use of categorical methods is becoming more prominent and successful in both physics and computer science. The basic idea is that objects of a category can represent systems, and morphisms can model the processes that transform those systems. We can see parts of computational protocols or physical processes as morphisms, which, when appropriately combined using tensor products and categorical composition, model the protocol or process as a whole. However, in doing so, some information about the protocols or processes is forgotten, namely in what location of spacetime did the events involved take place, and what was the causal structure among them. The goal of this thesis is to explore how these categorical models can be enhanced to include information on the spacetime location and causal structure of events. First, we introduce the theory of subunits, which are subobjects of the monoidal unit for which a canonical isomorphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules, and under mild conditions they endow any monoidal category with a topological intuition. We introduce and study well-behaved notions of restriction, localisation, and support. Subunits in general form only a semilattice, but we develop universal constructions completing any monoidal category to one whose subunits universally form a lattice, preframe, or frame.  Afterwards, we introduce a number of constructions to explore how the theory of subunits can be used in practice. Inspired by logical clocks, we define a diagrammatic category where we can capture simple protocols and their causal structure. To progress towards more detailed spacetime and causal information, we define the category of protocols, which formalises the idea of letting a morphism from a category be supported in a different category. This allows us to have one category to model the systems and processes and another one to model spacetime. In particular, we can treat both toy models of spacetime and more realistic ones in the same mathematical footing. A notion of causal structure is defined for monoidal categories, and a generalisation of the usual causal analysis in physics for points to arbitrary regions is provided. We give examples of protocols seen as diagrams and as objects in the category of protocols, both with toy models of spacetime as well as with more realistic ones

Tensor topology

A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We show that under mild conditions subunits endow any monoidal category with a kind of topological intuition: there are well-behaved notions of restriction, localisation, and support, even though the subunits in general only form a semilattice. We develop universal constructions completing any monoidal category to one whose subunits universally form a lattice, preframe, or frame.Comment: 44 page

A characterisation of ordered abstract probabilities

In computer science, especially when dealing with quantum computing or other non-standard models of computation, basic notions in probability theory like "a predicate" vary wildly. There seems to be one constant: the only useful example of an algebra of probabilities is the real unit interval. In this paper we try to explain this phenomenon. We will show that the structure of the real unit interval naturally arises from a few reasonable assumptions. We do this by studying effect monoids, an abstraction of the algebraic structure of the real unit interval: it has an addition $x+y$ which is only defined when $x+y\leq 1$ and an involution $x\mapsto 1-x$ which make it an effect algebra, in combination with an associative (possibly non-commutative) multiplication. Examples include the unit intervals of ordered rings and Boolean algebras. We present a structure theory for effect monoids that are $\omega$-complete, i.e. where every increasing sequence has a supremum. We show that any $\omega$-complete effect monoid embeds into the direct sum of a Boolean algebra and the unit interval of a commutative unital C$^*$-algebra. This gives us from first principles a dichotomy between sharp logic, represented by the Boolean algebra part of the effect monoid, and probabilistic logic, represented by the commutative C$^*$-algebra. Some consequences of this characterisation are that the multiplication must always be commutative, and that the unique $\omega$-complete effect monoid without zero divisors and more than 2 elements must be the real unit interval. Our results give an algebraic characterisation and motivation for why any physical or logical theory would represent probabilities by real numbers.Comment: 12 pages. V2: Minor change

Space in monoidal categories

The category of Hilbert modules may be interpreted as a naive quantum field theory over a base space. Open subsets of the base space are recovered as idempotent subunits, which form a meet-semilattice in any firm braided monoidal category. There is an operation of restriction to an idempotent subunit: it is a graded monad on the category, and has the universal property of algebraic localisation. Spacetime structure on the base space induces a closure operator on the idempotent subunits. Restriction is then interpreted as spacetime propagation. This lets us study relativistic quantum information theory using methods entirely internal to monoidal categories. As a proof of concept, we show that quantum teleportation is only successfully supported on the intersection of Alice and Bob's causal future