Traced Monoidal Category

Idea

Traced monoidal categories extend monoidal categories with a trace operator that provides a mathematical abstraction of feedback. In the graphical calculus, a trace operator represents feedback by connecting an output wire back to an input, creating a closed loop¹.

Definition

A monoidal category $𝓒$ is said to be right traced if it is equipped with a right trace:

$${\mathrm{tr}}_R^X:𝓒[A\otimes X,B\otimes X]\to 𝓒[A,B]$$

This is regarded as connecting X in the input to X in the intput the the fuctor, representing a feedback cycle.

Applications

• linear algebra
• braid theory
• fixed-point computations
• programming language semantics.

A fundamental structural result is that every traced monoidal category can be embedded in a compact closed category via the Int construction, which generates the free compact closure of the original category²³. Conversely, every compact closed category naturally admits a trace structure, establishing a deep connection between these two categorical frameworks².

Recent Theoretical Advances

Traced Pseudomonoids in Prof

A significant theoretical development comes from the work of Hu and Vicary, who defined traced pseudomonoids as pseudomonoids in a monoidal bicategory equipped with extra structure⁴. Their research provides a new characterization of Cauchy complete traced monoidal categories as algebraic structures in Prof, the monoidal bicategory of profunctors⁵⁶. This externalization of traced monoidal categories allows researchers to reason about the trace using the graphical calculus for monoidal bicategories, providing powerful new tools for analysis⁴.

One particularly elegant result from this approach concerns braided monoidal categories:

Any Cauchy complete category for which the following isomorphism exists, can
be equipped with a trace:
≅

This confirms the standard result that a braided monoidal category is traced, while providing a more general framework for understanding when trace structures can emerge.

Traced Monads and Hopf Monads

A longstanding question in the field has been how to characterize traced monads without explicitly mentioning their Eilenberg-Moore categories. Hasegawa and Lemay made significant progress by introducing trace-coherent Hopf monads on traced monoidal categories⁷⁸.

Their main theorem establishes that:

A symmetric Hopf monad is a traced monad if and only if it is a trace-coherent Hopf monad⁷.

This provides a concrete characterization of the kinds of monads that preserve traced structure. Their work also reveals interesting special cases:

1. For traced Cartesian monoidal categories, trace-coherent Hopf monads can be expressed using the Conway operator⁷
2. For traced coCartesian monoidal categories, any trace-coherent Hopf monad is an idempotent monad⁷
3. There exist traced monads that are not Hopf monads, as well as symmetric Hopf monads that are not trace-coherent⁷

Traced *-Autonomous Categories

Recent research has also explored the interaction between traced structure and $\ast$-autonomous structure. Hu and Vicary proved an equivalence result between the left ⊗-trace and the right ⊸-trace in *-autonomous categories⁹⁴. They also developed a new condition under which traced *-autonomous categories become autonomous⁴.

This builds on previous work showing that every traced symmetric *-autonomous category is compact closed⁹, extending our understanding of when different categorical structures coincide.

The Uniformity Principle

Hasegawa has developed the uniformity principle for traced monoidal categories as a natural generalization of Plotkin’s principle¹⁰¹¹. This principle provides a way to identify full subcategories that inherit traced monoidal structure in various categorical constructions.

For example, given a traced monoidal category C, one can consider C⊸, a full subcategory of the arrow category C→ whose objects are strict maps with respect to the trace¹⁰¹¹. This approach applies to comma categories, categories of (co)algebras of endofunctors, and other constructions, providing a systematic way to propagate traced structure through categorical operations.

Open Questions and Future Directions

Several important questions remain open in the theory of traced monoidal categories:

1. Characterization of Non-Hopf Traced Monads: Further investigation is needed to fully characterize traced monads that are not Hopf monads, and to understand the structural differences that lead to this separation⁷.

2. Conditions for Autonomy: While some conditions are known for when traced *-autonomous categories become autonomous, developing a complete characterization remains an open problem⁹⁴.

3. Balanced vs. Symmetric Settings: Most current results focus on symmetric traced monoidal categories rather than general balanced (or braided) traced monoidal categories. Extending these results to the balanced case represents an important direction for future work⁷.

4. Composition and Trace Interaction: The exact relationship between composition of morphisms and various trace constructions in traced symmetric monoidal categories needs further elucidation¹².

5. Applications to Programming Language Semantics: While the connection between traces and recursion in programming is established⁷, developing comprehensive semantic models based on traced categories for modern programming paradigms remains an active area of research.

Conclusion

Traced monoidal categories continue to be a vibrant area of research in category theory, with significant recent advances in their characterization, relationships to other structures, and generalizations. The externalization approach through traced pseudomonoids, the connections with Hopf monads, and the applications of the uniformity principle represent major steps forward. However, important questions remain regarding their structure, characterization, and applications, ensuring that this field will remain active for years to come.

References

History

Traced monoidal categories represent a powerful framework in category theory that formalizes the concept of feedback loops. Since their introduction by Joyal, Street, and Verity in the 1990s, significant advances have been made in understanding their structure, applications, and relationships to other categorical constructs.

Footnotes

1. https://ncatlab.org/nlab/show/traced+monoidal+category ↩

2. https://en.wikipedia.org/wiki/Traced_monoidal_category ↩ ↩²

3. https://people.math.rochester.edu/faculty/doug/otherpapers/jsv.pdf ↩

4. https://ora.ox.ac.uk/objects/uuid:4787725e-7e86-45b1-86b2-bc291c478bf7/download_file?safe_filename=Hu_and_Vicary_2021_Traced_monoidal_categories.pdf&type_of_work=Conference+item ↩ ↩² ↩³ ↩⁴ ↩⁵

5. https://arxiv.org/abs/2109.00589 ↩

6. https://arxiv.org/abs/2112.14051 ↩

7. https://compositionality.episciences.org/13529/pdf ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸

8. https://arxiv.org/abs/2208.06529 ↩

9. https://www.coalg.org/calco-mfps2021/files/2021/08/Hu-Vicary.pdf ↩ ↩² ↩³

10. https://www.kurims.kyoto-u.ac.jp/~hassei/papers/prims04.pdf ↩ ↩²

11. https://www.kurims.kyoto-u.ac.jp/~hassei/papers/ctcs02.pdf ↩ ↩²

12. https://mathoverflow.net/questions/155649/does-trace-handle-composition-in-a-traced-symmetric-monoidal-category ↩