Partially traced categories

This paper deals with questions relating to Haghverdi and Scott’s notion of partially traced categories. The main result is a representation theorem for such categories: we prove that every partially traced category can be faithfully embedded in a totally traced category. Also conversely, every symmetric monoidal subcategory of a totally traced category is partially traced, so this characterizes the partially traced categories completely. The main technique we use is based on Freyd’s paracategories, along with a partial version of Joyal, Street, and Verity’s Int-construction.

💡 Research Summary

The paper investigates the notion of partially traced categories introduced by Haghverdi and Scott, and establishes a precise correspondence between these categories and totally traced categories. The authors prove a representation theorem: every partially traced symmetric monoidal category can be faithfully embedded into a totally traced symmetric monoidal category. Conversely, any symmetric monoidal subcategory of a totally traced category inherits a partial trace, making it a partially traced category. Together these results characterize partially traced categories exactly as the symmetric monoidal subcategories of totally traced categories.

The technical development begins with a review of symmetric monoidal categories, monoidal functors, and the axioms of traced monoidal categories (naturality, dinaturality, strength, vanishing I, vanishing II, and yanking). The authors then adapt these axioms to a partial setting by allowing the trace operation Tr_U : C(A⊗U, B⊗U) ⇀ C(A, B) to be a partial function. To handle undefinedness they introduce two notions of Kleene equality: the ordinary Kleene equality (✄  ✂ ✁) and directed Kleene equality (✄ ✂). The partial trace must satisfy the six axioms, interpreted with Kleene equality where appropriate.

The core of the representation theorem uses Freyd’s notion of a paracategory, which is a category‑like structure where composition may be undefined. Starting from a partially traced category C, the authors construct a paracategory Par(C) whose objects are those of C and whose morphisms are the partially defined arrows of C. They then apply a partial version of the Int‑construction of Joyal, Street, and Verity to Par(C). The Int‑construction normally yields a compact closed (hence totally traced) category; the authors modify it so that the trace is defined exactly on the trace class of the original partial trace. The resulting category D = Int(Par(C)) is totally traced, and the canonical embedding F : C → D sends each object A to the pair (A, I) and each morphism f to (f, I). This functor is strong monoidal, faithful, and preserves the partial trace via the total trace in D.

For the converse direction, let D be a totally traced category and C ⊆ D a symmetric monoidal subcategory. Restricting the total trace of D to the morphisms that lie in C yields a partial trace on C. All six axioms are inherited directly, because the total trace already satisfies them. Hence C becomes a partially traced category.

The paper also provides concrete examples. In the category of finite‑dimensional vector spaces with the biproduct ⊕ as monoidal product, the usual total trace does not exist. Two candidate partial traces—the Kleene trace and the sum trace—are defined by infinite series of matrix blocks, but both fail the vanishing II axiom, as shown by an explicit counter‑example. In contrast, the Haghverdi‑Scott “feedback” partial trace, motivated by Geometry of Interaction, satisfies all axioms and thus serves as a genuine partially traced structure on (Vect, ⊕).

The authors discuss the significance of their results. By combining paracategories with a partial Int‑construction, they provide a systematic method to “complete” any partially traced category into a totally traced one while preserving the original partial trace. This technique could be applied to other settings where operations are only partially defined, such as partial functional programming, partial differential operators, or restricted feedback systems in quantum protocols.

In conclusion, the paper delivers a clean categorical characterisation of partially traced categories: they are exactly the symmetric monoidal subcategories of totally traced categories. The representation theorem and its converse give both a constructive embedding and a method to recover the partial trace from a total one, thereby unifying the two notions and opening avenues for further applications in semantics of linear logic, Geometry of Interaction, and beyond.