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Register a new userOn lax transformations, adjunctions, and monads in $(infty,2)$-categories
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Rune Haugseng
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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.
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In this short note we prove that two definitions of (co)ends in $infty$-categories, via twisted arrow $infty$-categories and via $infty$-categories of simplices, are equivalent. We also show that weighted (co)limits, which can be defined as certain (co)ends, can alternatively be described as (co)limits over left and right fibrations, respectively.
We use Luries symmetric monoidal envelope functor to give two new descriptions of $infty$-operads: as certain symmetric monoidal $infty$-categories whose underlying symmetric monoidal $infty$-groupoids are free, and as certain symmetric monoidal $infty$-categories equipped with a symmetric monoidal functor to finite sets (with disjoint union as tensor product). The latter leads to a third description of $infty$-operads, as a localization of a presheaf $infty$-category, and we use this to give a simple proof of the equivalence between Luries and Barwicks models for $infty$-operads.
Let $U$ be a strong monoidal functor between monoidal categories. If it has both a left adjoint $L$ and a right adjoint $R$, we show that the pair $(R,L)$ is a linearly distributive functor and $(U,U)dashv (R,L)$ is a linearly distributive adjunction, if and only if $Ldashv U$ is a Hopf adjunction and $Udashv R$ is a coHopf adjunction. We give sufficient conditions for a strong monoidal $U$ which is part of a (left) Hopf adjunction $Ldashv U$, to have as right adjoint a twisted version of the left adjoint $L$. In particular, the resulting adjunction will be (left) coHopf. One step further, we prove that if $L$ is precomonadic and $Lmathbf I$ is a Frobenius monoid (where $mathbf I$ denotes the unit object of the monoidal category), then $Ldashv Udashv L$ is an ambidextrous adjunction, and $L$ is a Frobenius monoidal functor. We transfer these results to Hopf monads: we show that under suitable exactness assumptions, a Hopf monad $T$ on a monoidal category has a right adjoint which is also a Hopf comonad, if the object $Tmathbf I$ is dualizable as a free $T$-algebra. In particular, if $Tmathbf I$ is a Frobenius monoid in the monoidal category of $T$-algebras and $T$ is of descent type, then $T$ is a Frobenius monad and a Frobenius monoidal functor.
Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint. In this paper we prove general adjoint functor theorems for functors between $infty$-categories. One of our main results is an $infty$-categorical generalization of Freyds classical General Adjoint Functor Theorem. As an application of this result, we recover Luries adjoint functor theorems for presentable $infty$-categories. We also discuss the comparison between adjunctions of $infty$-categories and homotopy adjunctions, and give a treatment of Brown representability for $infty$-categories based on Hellers purely categorical formulation of the classical Brown representability theorem.
We study lax families of adjoints from a fibrational viewpoint, obtaining a version of the mate correspondence for (op)lax natural transformations of functors from an $infty$-category to the $(infty,2)$-category of $infty$-categories. We apply this to show that the left adjoint of a lax symmetric monoidal functor is oplax symmetric monoidal and that the internal Hom in a closed symmetric monoidal $infty$-category is lax symmetric monoidal in both variables. We also consider units and counits of such families of adjoints, and use them to derive the full (twisted) naturality of passing to the dual.
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