Layered Monoidal Theories

In the first part, we develop layered monoidal theories - a generalisation of monoidal theories combining descriptions of a system at several levels. Via their representation as string diagrams, monoidal theories provide a graphical syntax with a visually intuitive notions of information flow and composition. Layered monoidal theories allow mixing several monoidal theories (together with translations between them) within the same string diagram, while retaining mathematical precision and semantic interpretability. We define three flavours of layered monoidal theories, provide a recursively generated syntax for each, and construct a free-forgetful adjunction with respect to three closely related semantics: opfibrations, fibrations and deflations. We motivate the general theory by providing several examples from existing literature. In the second part, we develop a formal approach to retrosynthesis - the process of backwards reaction search in synthetic chemistry. Chemical processes are treated at three levels of abstraction: (1) (formal) reactions encode all chemically feasible combinatorial rearrangements of molecules, (2) reaction schemes encode transformations applicable to ‘patches’ of molecules (including the functional groups), and (3) disconnection rules encode local chemical rewrite rules applicable to a single bond or atom at a time. We show that the three levels are tightly linked: the reactions are generated by the reaction schemes, while there is a functorial translation from the disconnection rules to the reactions. Moreover, the translation from the disconnection rules to the reactions is shown to be sound, complete and universal - allowing one to treat the disconnection rules as a formal syntax with the semantics provided by the reactions. We tie together the two parts by providing a formalisation of retrosynthesis within a certain layered monoidal theory.

💡 Research Summary

The thesis is divided into two major parts. The first part introduces Layered Monoidal Theories (LMTs), a categorical framework that extends ordinary monoidal theories by allowing several monoidal signatures to coexist in a single diagram together with explicit translations between them. Traditional monoidal theories are presented as string diagrams: wires denote objects (or “colours”), boxes denote generators, and parallel and sequential composition correspond to horizontal and vertical juxtaposition. While powerful, this approach can only capture a single level of abstraction.

LMTs solve this limitation by stacking multiple monoidal theories as layers. Each layer has its own signature and equations, and a translation (a functor) connects the objects and morphisms of one layer to those of another. Three flavours of translation are defined:

1. Opfibration‑based layers – the upper layer “chooses” objects in the lower layer; the free‑forgetful adjunction adds a layer (free functor) and removes it (forgetful functor).
2. Fibration‑based layers – the lower layer “pulls back” structure from the upper layer, mirroring indexed categories.
3. Deflation‑based layers – translations are essentially isomorphisms that collapse unnecessary detail, providing a systematic way to forget information.

For each flavour the author constructs a free‑forgetful adjunction and shows how the three flavours fit together in a triangular relationship, establishing that LMTs are mathematically equivalent to appropriate (op)fibrations or deflations.

The theory is then instantiated on a wide range of concrete domains, demonstrating its expressive power:

• Digital circuits – logical gates form one layer, voltage levels another, allowing simultaneous reasoning about Boolean functionality and physical constraints.
• Electrical circuits – Kirchhoff’s laws and Ohm’s law are placed in separate layers, making the composition of current‑ and voltage‑based diagrams transparent.
• ZX‑calculus – quantum gates and measurement‑based quantum computing are linked across layers, enabling diagrammatic extraction of quantum circuits from high‑level ZX‑diagrams.
• Calculus of Communicating Systems (CCS) – processes and communication channels are treated as distinct layers, illustrating concurrent composition in a monoidal setting.
• Probabilistic channels – probability distributions and sampling operations are layered, so information‑theoretic flow and stochastic transformation coexist in a single picture.

The second part applies LMTs to retrosynthetic analysis in synthetic chemistry. The author proposes a three‑level categorical model of chemical transformation:

1. Reactions – the most abstract layer, consisting of all chemically feasible rearrangements of molecular graphs. Formally, reactions are modeled by double‑pushout (DPO) graph rewriting, which guarantees that every rewrite respects the underlying graph‑theoretic constraints of chemistry.
2. Reaction schemes – a middle layer that operates on “patches” (functional groups) of molecules. Schemes are collections of pattern‑matching rewrite rules that act on sub‑graphs, thus generating families of reactions.
3. Disconnection rules – the finest layer, describing local bond‑breaking or atom‑level rewrites. These are the elementary steps chemists use when planning a synthesis.

Two functors are defined: one from disconnection rules to reaction schemes, and another from reaction schemes to full reactions. The author proves that both functors are sound (every derived reaction is chemically valid), complete (every valid reaction can be obtained from some scheme and rule), and universal (any other formal syntax for retrosynthesis factors uniquely through these functors). Consequently, the whole retrosynthetic process can be expressed as a layered string diagram where the top‑level object is the target molecule, each intermediate step is a morphism between layers, and the entire diagram encodes a concrete synthetic route.

By embedding the three chemical layers into a specific LMT, the thesis shows that retrosynthesis is not just a heuristic search but a mathematically rigorous composition of morphisms in a layered monoidal category. This perspective opens the door to categorical algorithms for automated synthesis planning, where finding a synthetic route becomes a problem of constructing a suitable morphism in the layered diagram.

The thesis concludes with a discussion of limitations and future work. The current framework handles discrete, static systems; extending it to dynamical settings (e.g., thermodynamics ↔ statistical mechanics) would require handling many‑to‑one translations and stochastic state spaces. Moreover, richer forms of inter‑layer interaction—such as non‑linear or many‑to‑many translations—remain to be explored. Nonetheless, the work establishes a powerful new bridge between category theory, graphical rewriting, and practical chemistry, offering a unified language for multi‑level compositional reasoning across diverse scientific domains.