[Paper Review] Equivariant monads and equivariant lifts versus a 2-category of distributive laws

Zoran Škoda|ArXiv.org|Jul 11, 2007

Homotopy and Cohomology in Algebraic Topology7 references3 citations

TL;DR

This paper establishes an isomorphism between the 2-category of equivariant monads and colax equivariant functors in C-actegories and a 2-category of distributive laws, generalizing Beck's classical bijection. It shows that monads with C-actions correspond to pairs of monads and distributive laws, with higher coherence structures replaced by multigonal diagrams, and extends the theory to relative distributive laws between pseudoalgebras.

ABSTRACT

Fix a monoidal category C. The 2-category of monads in the 2-category of C-actegories, colax C-equivarant functors, and C-equivariant natural transformations of colax functors, may be recast in terms of pairs consisting of a usual monad and a distributive law between the monad and the action of C, morphisms of monads respecting the distributive law, and transformations of monads satisfying some compatibility with the actions and distributive laws involved. The monads in this picture may be generalized to actions of monoidal categories, and actions of PRO-s in particular. If C is a PRO as well, then in special cases one gets various distributive laws of a given classical type, for example between a comonad and an endofunctor or between a monad and a comonad. The usual pentagons are in general replaced by multigons, and there are also ``mixed'' multigons involving two distinct distributive laws. Beck's bijection between the distributive laws and lifts of one monad to the Eilenberg-Moore category of another monad is here extended to an isomorphism of 2-categories. The lifts of maps of above mentioned pairs are colax C-equivariant. We finish with a short treatment of relative distributive laws between two pseudoalgebra structures which are relative with respect to the distributivity of two pseudomonads involved, what gives a hint toward the generalizations.

Motivation & Objective

• To generalize Beck's classical bijection between distributive laws and monad lifts to a 2-categorical isomorphism.
• To reformulate the 2-category of monads in C-actegories using distributive laws between monads and monoidal actions.
• To extend the theory to actions of PROs and pseudoalgebras, including relative distributive laws.
• To provide a categorical framework for higher distributive laws with multigonal coherence conditions instead of pentagons.
• To formalize equivariant lifts of monads via colax C-equivariant functors and compatible natural transformations.

Proposed method

• Define a C-actegory as a category equipped with a coherent left action of a monoidal category C.
• Introduce colax C-equivariant functors between C-actegories via natural transformations ζ: F(C ⋄ M) ⇒ C ⋄ F(M) satisfying coherence with action isomorphisms.
• Construct a strict 2-category C-act^c of C-actegories, colax C-equivariant functors, and C-equivariant natural transformations.
• Represent monads in this 2-category as pairs (T, λ) where T is a monad and λ is a distributive law between T and the C-action.
• Use pasting diagrams of multigons (generalizing Beck’s pentagons) to encode coherence for distributive laws between pseudoalgebras.
• Define relative distributive laws between pseudoalgebras using a canonical invertible pseudodistributive law can: CD ⇒ DC between pseudomonads.

Experimental results

Research questions

• RQ1 How can Beck’s bijection between distributive laws and monad lifts be extended to a 2-categorical isomorphism?
• RQ2 What is the structure of monads in the 2-category of C-actegories, and how can they be described via distributive laws?
• RQ3 How do coherence conditions for equivariant monads generalize from pentagons to multigons in higher distributive laws?
• RQ4 What is the role of the canonical pseudodistributive law can: CD ⇒ DC in defining relative distributive laws between pseudoalgebras?
• RQ5 How do colax C-equivariant functors and transformations encode equivariant lifts of monads?

Key findings

• The 2-category of monads in C-actegories is isomorphic to a 2-category of distributive laws between monads and C-actions.
• The classical Beck bijection is extended to an isomorphism of 2-categories, with lifts of monads realized as colax C-equivariant functors.
• Coherence for distributive laws is governed by multigonal diagrams instead of pentagons, reflecting higher categorical structure.
• The theory generalizes to actions of PROs, including classical distributive laws between monads and comonads.
• Relative distributive laws between pseudoalgebras are defined using a canonical pseudodistributive law can: CD ⇒ DC, with pasting diagrams involving nontrivial 2-cells.
• The tin can identity and other coherence identities for distributive laws are expressed via pasting diagrams with embedded higher coherence cells.