Traced Monads and Hopf Monads
Masahito Hasegawa, Jean-Simon Pacaud Lemay

TL;DR
This paper characterizes traced monads using trace-coherent Hopf monads on traced monoidal categories, establishing a precise relationship between these structures without relying on the Eilenberg-Moore category, and provides numerous examples and distinctions.
Contribution
It introduces trace-coherent Hopf monads and proves they are equivalent to traced monads in symmetric Hopf monads, offering a new characterization without Eilenberg-Moore categories.
Findings
Trace-coherent Hopf monads are equivalent to traced monads.
Examples include those induced by cocommutative Hopf algebras.
Characterizations vary for Cartesian and coCartesian traced categories.
Abstract
A traced monad is a monad on a traced symmetric monoidal category that lifts the traced symmetric monoidal structure to its Eilenberg-Moore category. A long-standing question has been to provide a characterization of traced monads without explicitly mentioning the Eilenberg-Moore category. On the other hand, a symmetric Hopf monad is a symmetric bimonad whose fusion operators are invertible. For compact closed categories, symmetric Hopf monads are precisely the kind of monads that lift the compact closed structure to their Eilenberg-Moore categories. Since compact closed categories and traced symmetric monoidal categories are closely related, it is a natural question to ask what is the relationship between Hopf monads and traced monads. In this paper, we introduce trace-coherent Hopf monads on traced monoidal categories, which can be characterized without mentioning the Eilenberg-Moore…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAlgebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology · Advanced Topics in Algebra
