[Paper Review] On lax transformations, adjunctions, and monads in $(\infty,2)$-categories

Rune Haugseng|arXiv (Cornell University)|Feb 3, 2020

Homotopy and Cohomology in Algebraic Topology26 references7 citations

TL;DR

This paper establishes equivalences between three perspectives on monads in $(\infty,2)$-categories: as associative algebras in endofunctor categories, as monadic right adjoints, and as functors from the universal monad 2-category. Using the Gray tensor product and simplicial $\infty$-categories, it identifies lax transformations with components that are equivalences as 'icons'—an $(\infty,2)$-categorical analogue—enabling a full comparison of monad morphisms across all three frameworks.

ABSTRACT

We use the basic expected properties of the Gray tensor product of $(\infty,2)$-categories to study (co)lax natural transformations. Using results of Riehl-Verity and Zaganidis we identify lax transformations between adjunctions and monads with commutative squares of (monadic) right adjoints. We also identify the colax transformations whose components are equivalences (generalizing the "icons" of Lack) with the 2-morphisms that arise from viewing $(\infty,2)$-categories as simplicial $\infty$-categories. Using this characterization we identify the $\infty$-category of monads on a fixed object and colax morphisms between them with the $\infty$-category of associative algebras in endomorphisms.

Motivation & Objective

• To extend the known equivalence of monad definitions in $(\infty,2)$-categories to include morphisms between monads.
• To characterize lax transformations between adjunctions and monads using commutative squares of monadic right adjoints.
• To identify colax transformations with componentwise equivalences as the $(\infty,2)$-categorical analogue of Lack's 'icons'.
• To show that the $\infty$-category of monads with colax morphisms is equivalent to the $\infty$-category of associative algebras in endomorphisms.
• To unify three perspectives on monads: as algebras in endofunctor categories, as monadic right adjoints, and as functors from the universal monad 2-category.

Proposed method

• Assumes basic properties of the $(\infty,2)$-categorical Gray tensor product (Assumption 3.5) to define lax and colax functors.
• Uses Riehl–Verity and Zaganidis’s work on $(\infty,2)$-categories to relate lax morphisms of adjunctions and monads to commutative squares of monadic right adjoints.
• Characterizes colax transformations with componentwise equivalences as arising from simplicial $\infty$-categories via cocartesian fibrations over $\Delta^{\mathrm{op}}$.
• Applies non-symmetric $\infty$-operad theory to relate monads to associative algebras in endomorphism $\infty$-categories.
• Constructs a functor $\mathrm{Nat}(X,Y) \to \mathrm{Fun}^{\mathrm{colax}}(X,Y)$ that identifies the former with the wide subcategory of colax transformations whose components are equivalences.
• Uses pullbacks of cocartesian fibrations to define monoidal structures on endomorphism $\infty$-categories and relate them to algebraic structures.

Experimental results

Research questions

• RQ1 How do lax transformations between adjunctions and monads relate to commutative squares of monadic right adjoints in $(\infty,2)$-categories?
• RQ2 What is the $(\infty,2)$-categorical analogue of Lack's 'icons'—colax transformations with componentwise equivalences?
• RQ3 How do the three definitions of monads (as algebras in endofunctors, as monadic right adjoints, and as functors from the universal monad 2-category) compare when morphisms are included?
• RQ4 What is the relationship between the $\infty$-category of monads with colax morphisms and the $\infty$-category of associative algebras in endomorphisms?
• RQ5 Can the equivalence between monad definitions be extended to include morphisms, using the structure of simplicial $\infty$-categories and cocartesian fibrations?

Key findings

• The $\infty$-category of monads with colax morphisms on a fixed object $X$ is equivalent to the $\infty$-category of associative algebras in the monoidal $\infty$-category $\mathrm{End}(X)$, with the forgetful functor corresponding to the underlying object.
• The $\infty$-category of lax transformations between monads is equivalent to the $\infty$-category of monadic right adjoints in $\mathrm{CAT}_\infty$, via restriction to right adjoints.
• Colax transformations with componentwise equivalences are precisely those that arise from viewing $(\infty,2)$-categories as simplicial $\infty$-categories, providing an $(\infty,2)$-categorical analogue of icons.
• The fiber of $\mathrm{MND}(\mathrm{CAT}_\infty)^{\mathrm{colax}}$ over an $\infty$-category $C$ is equivalent to $\mathrm{Alg}(\mathrm{End}(C))$, the $\infty$-category of associative algebras in endofunctors of $C$.
• The monadic right adjoint associated to a monad $T$ via the Riehl–Verity construction is equivalent to the $\infty$-category of left $T$-modules, confirming consistency with Lurie's monadicity theorem.
• The forgetful functor from associative algebras in $\mathrm{End}(X)$ to $\mathrm{End}(X)$ corresponds to the restriction of the inclusion $\mathrm{Mnd}(X)^{\mathrm{colax}} \to \mathrm{End}(X)^{\mathrm{colax}}$ to fibers at $X$, establishing a coherent algebraic structure.