With the advent of linear logic and its geometry of interaction interpretation, *-autonomy and monoidal trace became central concepts in categorical linear logic, with compact closed categories as canonical examples of both structures. Motivated by this perspective, Hajgato and Hasegawa began investigating the relationship between the two in 2013, asking: are there traced *-autonomous categories beyond compact closed categories?
They successfully answered this question by establishing a correspondence between traced *-autonomous categories and compact closed categories equipped with dualizing objects. This result subsequently motivated Hasegawa to seek an analogous characterization in the non-symmetric setting. Although significant progress has been made, and there are compelling reasons to believe such a generalization should hold, the problem remains open.
In this talk, we will review the classical symmetric result, explore the motivations for extending it to the non-symmetric setting, and introduce Hasegawa’s conjecture. Finally, we will discuss the main obstacles to proving the conjecture and outline my current approach to overcoming them.