Categories for the practising physicist

In this chapter we survey some particular topics in category theory in a somewhat unconventional manner. Our main focus will be on monoidal categories, mostly symmetric ones, for which we propose a physical interpretation. These are particularly relevant for quantum foundations and for quantum informatics. Special attention is given to the category which has finite dimensional Hilbert spaces as objects, linear maps as morphisms, and the tensor product as its monoidal structure (FdHilb). We also provide a detailed discussion of the category which has sets as objects, relations as morphisms, and the cartesian product as its monoidal structure (Rel), and thirdly, categories with manifolds as objects and cobordisms between these as morphisms (2Cob). While sets, Hilbert spaces and manifolds do not share any non-trivial common structure, these three categories are in fact structurally very similar. Shared features are diagrammatic calculus, compact closed structure and particular kinds of internal comonoids which play an important role in each of them. The categories FdHilb and Rel moreover admit a categorical matrix calculus. Together these features guide us towards topological quantum field theories. We also discuss posetal categories, how group representations are in fact categorical constructs, and what strictification and coherence of monoidal categories is all about. In our attempt to complement the existing literature we omitted some very basic topics. For these we refer the reader to other available sources.

💡 Research Summary

The chapter offers a physicist‑oriented tour of selected topics in category theory, concentrating on symmetric monoidal categories and their physical interpretations. It begins with a concise refresher on basic categorical notions before diving into three principal examples that, despite having seemingly unrelated objects—finite‑dimensional Hilbert spaces, sets with relations, and manifolds with cobordisms—share a remarkably similar internal structure.

In the first example, the category FdHilb has objects that are finite‑dimensional Hilbert spaces and morphisms that are linear maps; the tensor product supplies the monoidal operation. The authors emphasize that FdHilb is compact closed: every object possesses a dual, and evaluation and co‑evaluation morphisms exist. This structure mirrors the algebra of quantum entanglement, teleportation protocols, and the diagrammatic calculus used in categorical quantum mechanics. Internal comonoids (copying and deleting maps) are shown to be absent in a genuine quantum setting, highlighting the contrast with classical information processing.

The second example, Rel, treats sets as objects and binary relations as morphisms, with the Cartesian product as the monoidal product. Rel is also compact closed, but the dual of a set is the set itself, and the co‑evaluation/evaluation maps are given by the universal relation. The internal comonoid structure now implements copying and deletion, reflecting the classical ability to duplicate data. The authors point out that Rel and FdHilb are linked by a shared diagrammatic language and by a categorical matrix calculus: relations can be represented by Boolean matrices, linear maps by complex matrices, and tensor product by the Kronecker product. This parallel enables a unified treatment of quantum circuits and classical relational logic.

The third example, 2Cob, consists of 1‑dimensional manifolds (circles) as objects and 2‑dimensional cobordisms as morphisms, with disjoint union as the monoidal operation. Again the category is compact closed: the “cup” and “cap” cobordisms provide the duality structure. This setting is precisely the domain of topological quantum field theory (TQFT); the authors explain how a symmetric monoidal functor from 2Cob to Vect (or Hilb) encodes a TQFT, and they briefly discuss the cobordism hypothesis.

Beyond these case studies, the chapter identifies four structural commonalities: (1) a graphical calculus that turns complex compositions into planar diagrams, (2) compact closure furnishing duals and evaluation/co‑evaluation, (3) the presence of special internal comonoids that model copying/deleting, and (4) a categorical matrix calculus in FdHilb and Rel that translates categorical composition into ordinary matrix multiplication.

The authors also touch on posetal categories, showing how partial orders can be viewed as thin categories useful for modeling hierarchical physical systems (e.g., energy levels). They reinterpret group representations as functors from a one‑object group category, thereby framing symmetry actions categorically. Finally, the discussion of strictification and coherence theorems assures the reader that the often‑cumbersome associators and unitors can be eliminated or made harmless, guaranteeing that the diagrammatic reasoning employed throughout is mathematically sound.

In sum, the chapter builds a bridge from abstract categorical concepts to concrete physical applications, equipping the practising physicist with a toolbox that unifies quantum information, classical relational reasoning, and topological quantum field theory under a single, diagram‑driven categorical framework.