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.
In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterize its n-arrows using a graphical calculus that we develop here. The hom-spaces are appropriately fibrant, indeed are nerves of categories, which indicates that all of the expected coherence equations in each dimension are present. To justify our terminology, we prove that any adjunction of quasi-categories extends to a homotopy coherent adjunction and furthermore that these extensions are homotopically unique in the sense that the relevant spaces of extensions are contr…
Given an adjoint pair of functors $F,G$, the composite $GF$ naturally gets the structure of a monad. The same monad may arise from many such adjoint pairs of functors, however. Can one describe all of the adjunctions giving rise to a given monad? In this paper we single out a class of adjunctions with especially good properties, and we develop methods for computing all such adjunctions, up to natural equivalence, which give rise to a given monad. To demonstrate these methods, we explicitly compute the finitary homological presentations of the free $A$-module monad on the category of sets, for …
We use the general notion of 2-dimensional adjunction with given coherence equations as introduced by MacDonald-Stone, building on earlier work by Gray, to derive coherence equations for a general 2-monad, which we refer to as a lax-Gray monad. The free lax-Gray 2-monad on one object may be regarded as the suspension of a lax 2-dimensional analogue of the simplicial category Delta. We call this analogue Delta LG for lax-Gray Delta. This is analogous to the way that the free 1-monad Mnd (as presented in Schanuel-Street) is a concrete example of the suspension of the simplicial category Delta, d…
We characterize double adjunctions in terms of presheaves and universal squares, and then apply these characterizations to free monads and Eilenberg--Moore objects in double categories. We improve upon our earlier result in "Monads in Double Categories", JPAA 215:6, pages 1174-1197, 2011, to conclude: if a double category with cofolding admits the construction of free monads in its horizontal 2-category, then it also admits the construction of free monads as a double category. We also prove that a double category admits Eilenberg--Moore objects if and only if a certain parameterized presheaf i…
It it shown that geometric morphisms between elementary toposes can be represented as adjunctions between the corresponding categories of locales. These adjunctions are characterised as those that preserve the order enrichment, commute with the double power locale monad and whose right adjoints preserve finite coproduct. They are also characterised as those adjunctions that preserve the order enrichment and commute with both the upper and the lower power locale monads.
Riehl and Verity have introduced an "$\infty$-cosmic" framework in which they redevelop the category theory of $\infty$-categories using 2-categorical arguments. In this paper, we begin with a self-contained review of the parts of their theory needed to discuss adjunctions and monadicity. This is applied in order to extend to the $\infty$-categorical context the classical criterion for fully faithfulness of the comparison functor induced by an adjunction. We discuss the relation with previous work in the literature---which primarily uses model-categorical techniques---and indicate applications…
This paper provides geometrical descriptions of the Frobenius monad freely generated by a single object. These descriptions are related to results connecting Frobenius algebras and topological quantum field theories. In these descriptions, which are based on coherence results for self-adjunctions (adjunctions where an endofunctor is adjoint to itself), ordinals in epsilon zero play a prominent role. The paper ends by considering how the notion of Frobenius algebra induces the collapse of the hierarchy of ordinals in epsilon zero, and by raising the question of the exact categorial abstraction …
In this article, the author analyses distributive and mixed distributive laws and some of their equivalences through the use of 2-adjunctions of the type $\Adj$-$\Mnd$. As far as the distributive laws are concerned, the equivalence between this structures and monads objects in the 2-category $\Adj_{\R}(Cat)$ is analysed, where these monad objects will correspond to liftings to the category of algebras of Eilenberg-Moore. Second, the equivalence between these structures and a pair consisting of a Eilenberg-Moore lifting and a Kleisli extension is analysed too, according to E. Manes and P. Mulry…
This paper introduces lax orthogonal algebraic weak factorisation systems on 2-categories and describes a method of constructing them. This method rests in the notion of simple 2-monad, that is a generalisation of the simple reflections studied by Cassidy, Hébert and Kelly. Each simple 2-monad on a finitely complete 2-category gives rise to a lax orthogonal algebraic weak factorisation system, and an example of a simple 2-monad is given by completion under a class of colimits. The notions of KZ lifting operation, lax natural lifting operation and lax orthogonality between morphisms are studied…
In this article Hopf parametric adjunctions are defined and analysed within the context of the 2-adjunction of the type $\mathbf{Adj}$-$\mathbf{Mnd}$. In order to do so, the definition of adjoint objects in the 2-category of adjunctions and in the 2-category of monads for $Cat$ are revised and characterized. This article finalises with the application of the obtained results on current categorical characterization of Hopf Monads.
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 $L\dashv U$ is a Hopf adjunction and $U\dashv R$ is a coHopf adjunction. We give sufficient conditions for a strong monoidal $U$ which is part of a (left) Hopf adjunction $L\dashv U$, 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,…
Recently, Dotsenko and Tamaroff have shown that a morphism of $T\longrightarrow S$ of monads over a category $\mathscr C$ satisfies the PBW-property if and only if it makes $S$ into a free right $T$-module. We consider an adjunction $Ψ=(G,F)$ between categories $\mathscr C$, $\mathscr D$, a monad $S$ on $\mathscr C$ and a monad $T$ on $\mathscr D$. We show that a morphism $ϕ:(\mathscr C,S)\longrightarrow (\mathscr D,T)$ that is well behaved with respect to the adjunction $Ψ$ has a PBW-property if and only if it makes $S$ satisfy a certain freeness condition with respect to $T$-modules with val…
Given an adjunction connecting reasonable categories with weak equivalences, we define a new derived bar and cobar construction associated to the adjunction. This yields homotopical models of the completion and cocompletion associated to the monad and comonad of the adjunction. We discuss applications of these resolutions to spectral sequences for derived completions and Goodwillie calculus in general model categories.
For a small quantaloid $\mathcal{Q}$, we consider 2-monads on the 2-category $\mathcal{Q}$-$\bf{Cat}$ and their lax extensions to the 2-category $\mathcal{Q}$-$\bf{Dist}$ of small $\mathcal{Q}$-categories and their distributors, in particular those lax extensions that are flat, in the sense that they map identity distributors to identity distributors. In fact, unlike in the discrete case, a 2-monad on $\mathcal{Q}$-$\bf{Cat}$ may admit only one flat lax extension. Every ordinary monad on the comma category $\bf{Set}/{\rm ob}\mathcal{Q}$ with a lax extension to $\mathcal{Q}$-$\bf{Rel}$ gives ri…
We extend the basic concepts of Street's formal theory of monads from the setting of 2-categories to that of double categories. In particular, we introduce the double category Mnd(C) of monads in a double category C and define what it means for a double category to admit the construction of free monads. Our main theorem shows that, under some mild conditions, a double category that is a framed bicategory admits the construction of free monads if its horizontal 2-category does. We apply this result to obtain double adjunctions which extend the adjunction between graphs and categories and the ad…
We show how to "interleave" the monad for operads and the monad for contractions on the category \coll of collections, to construct the monad for the operads-with-contraction of Leinster. We first decompose the adjunction for operads and the adjunction for contractions into a chain of adjunctions each of which acts on only one dimension of the underlying globular sets at a time. We then exhibit mutual stability conditions that enable us to alternate the dimension-by-dimension free functors. Hence we give an explicit construction of a left adjoint for the forgetful functor $\owc \lra \coll$, fr…
This chapter describes interrelations between: (1) algebraic structure on sets of scalars, (2) properties of monads associated with such sets of scalars, and (3) structure in categories (esp. Lawvere theories) associated with these monads. These interrelations will be expressed in terms of "triangles of adjunctions", involving for instance various kinds of monoids (non-commutative, commutative, involutive) and semirings as scalars. It will be shown to which kind of monads and categories these algebraic structures correspond via adjunctions.
In this work, we describe an adjunction between the comma category of SET-based monads under the V-powerset monad and the category of associative lax extensions of SET-based monads to the category of V-relations. In the process, we give a general construction of the Kleisli extension of a monad to the category of V-relations.
Generalized multicategories, also called $T$-monoids, are well known class of mathematical structures, which include diverse set of examples. In this paper we construct a generalization of the adjunction between strict monoidal categories and multicategories, where the latter are replaced by $T$-monoids. To do this we introduce lax monads in a 3-category, and establish their relationship with equipments, which are bicategory like structures appropriate for the generalized multicategory theory.
It is well established that equational algebraic theories, and the monads they generate, can be used to encode computational effects. An important insight of Power and Shkaravska is that comodels of an algebraic theory T -- i.e., models in the opposite category Set^op -- provide a suitable environment for evaluating the computational effects encoded by T. As already noted by Power and Shkaravska, taking comodels yields a functor from accessible monads to accessible comonads on Set. In this paper, we show that this functor is part of an adjunction -- the "costructure-cosemantics adjunction" of …
Generalized operads, also called generalized multicategories and $T$-monoids, are defined as monads within a Kleisli bicategory. With or without emphasizing their monoidal nature, generalized operads have been considered by numerous authors in different contexts, with examples including symmetric multicategories, topological spaces, globular operads and Lawvere theories. In this paper we study functoriality of the Kleisli construction, and correspondingly that of generalized operads. Motivated by this problem we develop a lax version of the formal theory of monads, and study its connection to …
We give another proof of the fact that there is a dual equivalence between the $\infty$-category of monoidal $\infty$-categories with left adjoint oplax monoidal functors and that with right adjoint lax monoidal functors by constructing a perfect pairing between them.
For a quantaloid $\mathcal{Q}$, considered as a bicategory, Walters introduced categories enriched in $\mathcal{Q}$. Here we extend the study of monad-quantale-enriched categories of the past fifteen years by introducing monad-quantaloid-enriched categories. We do so by making lax distributive laws of a monad $\mathbb{T}$ over the discrete presheaf monad of the small quantaloid $\mathcal{Q}$ the primary data of the theory, rather than the lax monad extensions of $\mathbb{T}$ to the category of $\mathcal{Q}$-relations that they equivalently describe. The central piece of the paper establishes a…
Consider a monad on an idempotent complete triangulated category with the property that its Eilenberg-Moore category of modules inherits a triangulation. We show that any other triangulated adjunction realizing this monad is 'essentially monadic', i.e. becomes monadic after performing the two evident necessary operations of taking the Verdier quotient by the kernel of the right adjoint and idempotent completion. In this sense, the monad itself is 'intrinsically monadic'. It follows that for any highly structured ring spectrum, its category of homotopy (a.k.a. naive) modules is triangulated if …
We give a 3-categorical, purely formal argument explaining why on the category of Kleisli algebras for a lax monoidal monad, and dually on the category of Eilenberg-Moore algebras for an oplax monoidal monad, we always have a natural monoidal structures. The key observation is that the 2-category of lax monoidal monads in any 2-category D with finite products is isomorphic to the 2-category of monoidal objects with oplax morphisms in the 2-category of monads with lax morphisms in D. As we explain at the end of the paper a similar phenomenon occurs in many other situations.
Given a monad and a comonad, one obtains a distributive law between them from lifts of one through an adjunction for the other. In particular, this yields for any bialgebroid the Yetter-Drinfel'd distributive law between the comonad given by a module coalgebra and the monad given by a comodule algebra. It is this self-dual setting that reproduces the cyclic homology of associative and of Hopf algebras in the monadic framework of Boehm and Stefan. In fact, their approach generates two duplicial objects and morphisms between them which are mutual inverses if and only if the duplicial objects are…
State monads in cartesian closed categories are those defined by the familiar adjunction between product and exponential. We investigate the structure of their algebras, and show that the exponential functor is monadic provided the base category is sufficiently regular, and the exponent is a non-empty object.
We define Hopf monads on an arbitrary monoidal category, extending the definition given previously for monoidal categories with duals. A Hopf monad is a bimonad (or opmonoidal monad) whose fusion operators are invertible. This definition can be formulated in terms of Hopf adjunctions, which are comonoidal adjunctions with an invertibility condition. On a monoidal category with internal Homs, a Hopf monad is a bimonad admitting a left and a right antipode. Hopf monads generalize Hopf algebras to the non-braided setting. They also generalize Hopf algebroids (which are linear Hopf monads on a cat…
For a small quantaloid $\mathcal{Q}$ we consider four fundamental 2-monads $\mathbb{T}$ on $\mathcal{Q}\text{-}{\bf Cat}$, given by the presheaf 2-monad $\mathbb{P}$ and the copresheaf 2-monad $\mathbb{P}^{\dagger}$, as well as by their two composite 2-monads, and establish that they all laxly distribute over $\mathbb{P}$. These four 2-monads therefore admit lax extensions to the category $\mathcal{Q}\text{-}{\bf Dist}$ of $\mathcal{Q}$-categories and their distributors. We characterize the corresponding $(\mathbb{T},\mathcal{Q})$-categories in each of the four cases, leading us to both known …
Each distributor between categories enriched over a small quantaloid Q gives rise to two adjunctions between the categories of contravariant and covariant presheaves, and hence to two monads. These two adjunctions are respectively generalizations of Isbell adjunctions and Kan extensions in category theory. It is proved that these two processes are functorial with infomorphisms playing as morphisms between distributors; and that the free cocompletion functor of Q-categories factors through both of these functors.
Monads are well known to be equivalent to lax functors out of the terminal category. Morita contexts are here shown to be lax functors out of the chaotic category with two objects. This allows various aspects in the theory of Morita contexts to be seen as special cases of general results about lax functors. The account we give of this could serve as an introduction to lax functors for those familiar with the theory of monads. We also prove some very general results along these lines relative to a given 2-comonad, with the classical case of ordinary monad theory amounting to the case of the ide…
It is known that the so-called monadic decomposition, applied to the adjunction connecting the category of bialgebras to the category of vector spaces via the tensor and the primitive functors, returns the usual adjunction between bialgebras and (restricted) Lie algebras. Moreover, in this framework, the notions of augmented monad and combinatorial rank play a central role. In order to set these results into a wider context, we are led to substitute the monadic decomposition by what we call the adjoint decomposition. This construction has the advantage of reducing the computational complexity …
We extend Bourke and Garner's idempotent adjunction between monads and pretheories to the framework of $\infty$-categories and we use this to prove many classical results about monads in the $\infty$-categorical framework. Amongst other things, we show that the category of algebras for an accessible monads on a locally presentable $\infty$-category $\mathcal{E}$ is again locally presentable, and that a diagram of accessible monads on a locally presentable $\infty$-category admits a colimit. Our results also provide a new and simpler way to construct and describe monads in terms of theories.
Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal. We describe conditions under which one can transport the weak equivalences from one category to another with the same objects and a broader class of maps. Under mild hypotheses this process produces an equivalence of homotopy theories. We describe examples including algebras over an operad, such as symmetric monoidal categories and $n$-fold monoidal categories; and diagram categories, such as $Γ$-categorie…
This dissertation is devoted to a study of adjunctions concerning categories enriched over a quantaloid Q (or Q-categories for short), with the following types of adjunctions involved: (1) adjoint functors between Q-categories; (2) adjoint distributors between Q-categories; (3) adjoint functors between categories consisting of Q-categories. For a small quantaloid Q and a distributor between Q-categories, two adjunctions between the Q-categories of contravariant and covariant presheaves are presented. These adjunctions respectively extend the fundamental construction of Isbell adjunctions and K…
We give a definition of weak morphism of $T$-algebras, for a $2$-monad $T$, with respect to an arbitrary family $Ω$ of $2$-cells of the base $2$-category. By considering particular choices of $Ω$, we recover the concepts of lax, pseudo and strict morphisms of $T$-algebras. We give a general notion of weak limit, and define what it means for such a limit to be compatible with another family of $2$-cells. These concepts allow us to prove a limit lifting theorem which unifies and generalizes three different previously known results of $2$-dimensional monad theory. Explicitly, by considering the t…
We introduce and study Hopf monads on autonomous categories (i.e., monoidal categories with duals). Hopf monads generalize Hopf algebras to a non-braided (and non-linear) setting. Indeed, any monoidal adjunction between autonomous categories gives rise to a Hopf monad. We extend many fundamental results of the theory of Hopf algebras (such as the decomposition of Hopf modules, the existence of integrals, Maschke's criterium of semisimplicity, etc...) to Hopf monads. We also introduce and study quasitriangular and ribbon Hopf monads (again defined in a non-braided setting).
We introduce the notion of a lax monoidal fibration and we show how it can be conveniently used to deal with various algebraic structures that play an important role in some definitions of the opetopic sets (Baez-Dolan, Hermida-Makkai-Power). We present the 'standard' such structures, the exponential fibrations of basic fibrations and three areas of applications. First area is related to the T-categories of A. Burroni. The monoids in the Burroni lax monoidal fibrations form the fibration of T-categories. The construction of the relative Burroni fibrations and free T-categories in this context,…
Generalizing the approach to pseudo monoidal DG-categories as certain colored non-symmetric DG-operads, we introduce a certain relaxed notion of a category enriched in DG-categories. We construct model structures on the category of colored non-symmetric DG-operads and on the category of DGCat-enriched categories with a fixed set of objects. This allows us to talk about strong homotopy maps in both settings. We discuss the notion of a strong homotopy monad in a DG-category and a notion of strong homotopy adjunction data for two DG-functors.
This chapter uses categorical techniques to describe relations between various sets of operators on a Hilbert space, such as self-adjoint, positive, density, effect and projection operators. These relations, including various Hilbert-Schmidt isomorphisms of the form tr(A-), are expressed in terms of dual adjunctions, and maps between them. Of particular interest is the connection with quantum structures, via a dual adjunction between convex sets and effect modules. The approach systematically uses categories of modules, via their description as Eilenberg-Moore algebras of a monad.
We introduce a generalization of monads, called relative monads, allowing for underlying functors between different categories. Examples include finite-dimensional vector spaces, untyped and typed lambda-calculus syntax and indexed containers. We show that the Kleisli and Eilenberg-Moore constructions carry over to relative monads and are related to relative adjunctions. Under reasonable assumptions, relative monads are monoids in the functor category concerned and extend to monads, giving rise to a coreflection between relative monads and monads. Arrows are also an instance of relative monads…
We define Frobenius-Eilenberg-Moore objects for a dagger Frobenius monad in an arbitrary dagger 2-category, and extend to the dagger context a well-known universal property of the formal theory of monads. We show that the free completion of a 2-category under Eilenberg-Moore objects extends to the dagger context, provided one is willing to work with such dagger Frobenius monads whose endofunctor part suitably commutes with their unit. Finally, we define dagger lax functors and dagger lax-limits of such functors, and show that Frobenius-Eilenberg-Moore objects are examples of such limits.
Using the language of double categories we generalise a classical result on finite-product-preserving left Kan extensions, by Adámek and Rosický, to one on left Kan extensions that preserve algebraic structures defined by `suitable' colax-idempotent 2-monads, as well as obtain two related results. To be precise, by `suitable' 2-monads here we mean ones that extend to normal lax double monads. In an appendix we consider induced algebra structures on presheaf objects.
