The construction of the cotensor coalgebra for an "abelian monoidal" category $\M$ which is also cocomplete, complete and AB5, was performed in [A. Ardizzoni, C. Menini and D. Ştefan, \emph{Cotensor Coalgebras in Monoidal Categories}, Comm. Algebra, to appear]. It was also proved that this coalgebra satisfies a meaningful universal property which resembles the classical one. Here the lack of the coradical filtration for a coalgebra $E$ in $\M$ is filled by considering a direct limit $\widetilde{D}$ of a filtration consisting of wedge products of a subcoalgebra $D$ of $E$. The main aim of this paper is to characterize hereditary coalgebras $\widetilde{D}$, where $D$ is a coseparable coalgebra in $\M$, by means of a cotensor coalgebra: more precisely, we prove that, under suitable assumptions, $\widetilde{D}$ is hereditary if and only if it is formally smooth if and only if it is the cotensor coalgebra $T^c_{D}(D\w D/D)$ if and only if it is a cotensor coalgebra $T^c_{D}(N)$, where $N$ is a certain $D$-bicomodule in $\M$. Because of our choice, even when we apply our results in the category of vector spaces, new results are obtained.
We introduce the concept of cotensor coalgebra for a given bicomodule over a coalgebra in an abelian monoidal category. Under some further conditions we show that such a cotensor coalgebra exists and satisfies a meaningful universal property. We prove that this coalgebra is formally smooth whenever the comodule is relative injective and the coalgebra itself is formally smooth.
In this article we describe properties of the 2-functor from the 2-category of comonads to the 2-category of functors that sends a comonad to its forgetful functor. This allows us to describe contexts where algebras over a monad are enriched tensored and cotensored over coalgebras over a comonad.
We investigate cofree coalgebras, and limits and colimits of coalgebras in some abelian monoidal categories of interest, such as bimodules over a ring, and modules and comodules over a bialgebra or Hopf algebra. We find concrete generators for the categories of coalgebras in these monoidal categories, and explicitly construct cofree coalgebras, products and limits of coalgebras in each case. This answers an open question of A. Agore on the existence of a cofree coring, and constructs the cofree (co)module coalgebra on a $B$-(co)module, for a bialgebra $B$.
We show that weak monoidal Quillen equivalences induce equivalences of symmetric monoidal $\infty$-categories with respect to the Dwyer-Kan localization of the symmetric monoidal model categories. The result will induce a Dold-Kan correspondence of coalgebras in $\infty$-categories. Moreover, it shows that Shipley's zig-zag of Quillen equivalences lifts to an explicit symmetric monoidal equivalence of $\infty$-categories for the stable Dold-Kan correspondence. We study homotopy coherent coalgebras associated to a monoidal monoidal category. We show examples when these coalgebras cannot be rigi…
We prove that given $\mathcal{C}$ a presentably symmetric monoidal $\infty$-category, and any essentially small $\infty$-operad $\mathcal{O}$, the $\infty$-category of $\mathcal{O}$-algebras in $\mathcal{C}$ is enriched, tensored and cotensored over the presentably symmetric monoidal $\infty$-category of $\mathcal{O}$-coalgebras in $\mathcal{C}$. We provide a higher categorical analogue of the universal measuring coalgebra. For categories in the usual sense, the result was proved by Hyland, López Franco, and Vasilakopoulou.
We prove that the category of dg-coalgebras is symmetric monoidal closed and that the category of dg-algebras is enriched, tensored, cotensored and strongly monoidal over that of coalgebras. We apply this formalism to reconstruct several known adjunctions, notably the bar-cobar adjunction.
We study the bialgebra structures on quiver coalgebras and the monoidal structures on the categories of locally nilpotent and locally finite quiver representations. It is shown that the path coalgebra of an arbitrary quiver admits natural bialgebra structures. This endows the category of locally nilpotent and locally finite representations of an arbitrary quiver with natural monoidal structures from bialgebras. We also obtain theorems of Gabriel type for pointed bialgebras and hereditary finite pointed monoidal categories.
Incidence coalgebras of categories in the sense of Joni and Rota are studied, specifically cases where a monoidal product on the category turns these into (weak) bialgebras. The overlap with the theory of combinatorial Hopf algebras and that of Hopf quivers is discussed, and examples including trees, skew shapes, Milner's bigraphs and crossed modules are considered.
Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in the literature recently. We study Hom-structures from the point of view of monoidal categories; in particular, we introduce a symmetric monoidal category such that Hom-algebras coincide with algebras in this monoidal category, and similar properties for coalgebras, Hopf algebras and Lie algebras.
Differential categories were introduced to provide a minimal categorical doctrine for differential linear logic. Here we revisit the formalism and, in particular, examine the two different approaches to defining differentiation which were introduced. The basic approach used a deriving transformation, while a more refined approach, in the presence of a bialgebra modality, used a codereliction. The latter approach is particularly relevant to linear logic settings, where the coalgebra modality is monoidal and the Seely isomorphisms give rise to a bialgebra modality. Here, we prove that these appa…
After recalling the definition of a bicoalgebroid, we define comodules and modules over a bicoalgebroid. We construct the monoidal category of comodules, and define Yetter--Drinfel'd modules over a bicoalgebroid. It is proved that the Yetter--Drinfel'd category is monoidal and pre--braided just as in the case of bialgebroids, and is embedded into the one--sided center of the comodule category. We proceed to define Braided Cocommutative Coalgebras (BCC) over a bicoalgebroid, and dualize the scalar extension construction of Brzezinski and Militaru [2] and Balint and Slachanyi [1], originally app…
We give further insights into the weighted Hurwitz product and the weighted tensor product of Joyal species. Our first group of results relate the Hurwitz product to the pointwise product, including the interaction with Rota--Baxter operators. Our second group of results explain the first in terms of convolution with suitable bialgebras, and show that these bialgebras are in fact obtained in a particularly straightforward way by freely generating from pointed coalgebras. Our third group of results extend this from linear algebra to two-dimensional linear algebra, deriving the existence of weig…
Braided bialgebras of type one in abelian braided monoidal categories are characterized as braided graded bialgebras which are strongly $\mathbb{N}$-graded both as an algebra and as a coalgebra.
Differential categories are now an established abstract setting for differentiation. However not much attention has been given to the process which is inverse to differentiation: integration. This paper presents the parallel development for integration by axiomatizing an integral transformation, $s_A: !A \to !A \otimes A$, in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold (in a suitable sense): a differential category with integration which satisfies these two theorem is c…
This paper determines what structure is needed for internal homs in a monoidal category C to be liftable to the category C^G of Eilenberg-Moore coalgebras for a monoidal comonad G on C. We apply this to lift star-autonomy with the view to recasting the definition of quantum groupoid.
Entwined modules over cowreaths in a monoidal category are introduced. They can be identified to coalgebras in an appropriate monoidal category. It is investigated when such coalgebras are Frobenius (resp. separable), and when the forgetful functor from entwined modules to representations of the underlying algebra is Frobenius (resp. separable). These properties are equivalent when the unit object of the category is a $\otimes$-generator.
Let $B$ be a bialgebra, and $A$ a left $B$-comodule algebra in a braided monoidal category $\Cc$, and assume that $A$ is also a coalgebra, with a not-necessarily associative or unital left $B$-action. Then we can define a right $A$-action on the tensor product of two relative Hopf modules, and this defines a monoidal structure on the category of relative Hopf modules if and only if $A$ is a bialgebra in the category of left Yetter-Drinfeld modules over $B$. Some examples are given.
A topological commutative ring is said to be rigid when for every set $X$, the topological dual of the $X$-fold topological product of the ring is isomorphic to the free module over $X$. Examples are fields with a ring topology, discrete rings, and normed algebras. Rigidity translates into a dual equivalence between categories of free modules and of "topologically-free" modules and, with a suitable topological tensor product for the latter, one proves that it lifts to an equivalence between monoids in this category (some suitably generalized topological algebras) and coalgebras. In particular,…
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…
We give a natural-deduction-style type theory for symmetric monoidal categories whose judgmental structure directly represents morphisms with tensor products in their codomain as well as their domain. The syntax is inspired by Sweedler notation for coalgebras, with variables associated to types in the domain and terms associated to types in the codomain, allowing types to be treated informally like "sets with elements" subject to global linearity-like restrictions. We illustrate the usefulness of this type theory with various applications to duality, traces, Frobenius monoids, and (weak) Hopf …
We show that, under some mild conditions, a bialgebra in an abelian and coabelian braided monoidal category has a weak projection onto a formally smooth (as a coalgebra) sub-bialgebra with antipode; see Theorem 1.12. In the second part of the paper we prove that bialgebras with weak projections are cross product bialgebras; see Theorem 2.12. In the particular case when the bialgebra $A$ is cocommutative and a certain cocycle associated to the weak projection is trivial we prove that $A$ is a double cross product, or biproduct in Madjid's terminology. The last result is based on a universal pro…
The notion of 2--monoidal category used here was introduced by B.~Vallette in 2007 for applications in the operadic context. The starting point for this article was a remark by Yu. Manin that in the category of quadratic algebras (that is, "quantum linear spaces") one can also define 2--monoidal structure(s) with rather unusual properties. Here we give a detailed exposition of these constructions, together with their generalisations to the case of quadratic operads. Their parallel exposition was motivated by the following remark. Several important operads/cooperads such as genus zero quantum c…
The definition of Azumaya algebras over commutative rings $R$ require the tensor product of modules over $R$ and the twist map for the tensor product of any two $R$-modules. Similar constructions are available in braided monoidal categories and Azumaya algebras were defined in these settings. Here we introduce Azumaya monads on any category $\A$ by considering a monad $\bF$ on $\A$ endowed with a distributive law $λ: FF\to FF$ satisfying the Yang-Baxter equation (BD-law). This allows to introduce an {\em opposite monad} $\bF^\la$ and a monad structure on $FF^\la$. For an {\em Azumaya monad} we…
Man-in-the-Middle (MM) is not only a ubiquitous attack pattern in security, but also an important paradigm of network computation and economics. Recognizing ongoing MM-attacks is an important security task; modeling MM-interactions is an interesting task for semantics of computation. Traced monoidal categories are a natural framework for MM-modelling, as the trace structure provides a tool to hide what happens *in the middle*. An effective analysis of what has been traced out seems to require an additional property of traces, called *normality*. We describe a modest model of network computatio…
We introduce the notion of an enriched fibration, i.e. a fibration whose total category and base category are enriched in those of a monoidal fibration in an appropriate way. Furthermore, we provide a way to obtain such a structure, starting from actions of monoidal categories with parameterized adjoints. The motivating goal is to capture certain example cases, like the fibration of modules over algebras enriched in the opfibration of comodules over coalgebras.
It is well-known that the category of small dg categories dgCat, though it is monoidal, does not form a monoidal model category. In this paper we construct a monoidal model structure on the category of pointed curved coalgebras ptdCoa* and show that the Quillen equivalence relating it to dgCat is monoidal. We also show that dgCat is a ptdCoa*-enriched model category. As a consequence, the homotopy category of dgCat is closed monoidal and is equivalent as a closed monoidal category to the homotopy category of ptdCoa*. In particular, this gives a conceptual construction of a derived internal hom…
We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele in [A. Van Daele, Multiplier Hopf algebras, {\em Trans. Amer. Math. Soc.}, {\bf 342}(2), (1994) 917--932.] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly non-unital, idempotent, non-degenerate, $k$-projective) algebra over a commutative ring $k$ is a multip…
In the theory of coalgebras $C$ over a ring $R$, the rational functor relates the category of modules over the algebra $C^*$ (with convolution product) with the category of comodules over $C$. It is based on the pairing of the algebra $C^*$ with the coalgebra $C$ provided by the evaluation map $\ev:C^*\ot_R C\to R$. We generalise this situation by defining a {\em pairing} between endofunctors $T$ and $G$ on any category $\A$ as a map, natural in $a,b\in \A$, $$β_{a,b}:\A(a, G(b)) \to \A(T(a),b),$$ and we call it {\em rational} if these all are injective. In case $\bT=(T,m_T,e_T)$ is a monad an…
Differential categories axiomatize the basics of differentiation and provide categorical models of differential linear logic. A differential category is said to have antiderivatives if a natural transformation $\mathsf{K}$, which all differential categories have, is a natural isomorphism. Differential categories with antiderivatives come equipped with a canonical integration operator such that generalizations of the Fundamental Theorems of Calculus hold. In this paper, we show that Blute, Ehrhard, and Tasson's differential category of convenient vector spaces has antiderivatives. To help prove…
Jacobs' hypernormalisation is a construction on finitely supported discrete probability distributions, obtained by generalising certain patterns occurring in quantitative information theory. In this paper, we generalise Jacobs' notion in turn, by describing a notion of hypernormalisation in the abstract setting of a symmetric monoidal category endowed with a linear exponential monad -- a structure arising in the categorical semantics of linear logic. We show that Jacobs' hypernormalisation arises in this fashion from the finitely supported probability measure monad on the category of sets, whi…
A classical result of Tannaka duality is the fact that a coalgebra over a field can be reconstructed from its category of finite dimensional representations by using the forgetful functor which sends a representation to its underlying vector space. There is also a corresponding recognition result, which characterizes those categories equipped with a functor to finite dimensional vector spaces which are equivalent to the category of finite dimensional representations of a coalgebra. In this paper we study a generalization of these questions to an arbitrary cosmos, that is, a complete and cocomp…
For our concepts of change of base and comonadicity, we work in the general context of the tricategory $\mathrm{Caten}$ whose objects are bicategories $\mathscr{V}$ and whose morphisms are categories enriched on two sides. For example, for any monoidal comonad $G$ on a cocomplete closed monoidal category $\mathscr{C}$, the forgetful functor $U : \mathscr{C}^G\to \mathscr{C}$ is comonadic when regarded as a morphism in $\mathrm{Caten}$ between one-object bicategories. We show that the forgetful pseudofunctor $\mathscr{U}:\mathscr{V}^\mathscr{G}\rightarrow \mathscr{V}$ from the bicategory of Eil…
This paper is the first in a series in which we offer a new framework for hermitian K-theory in the realm of stable $\infty$-categories. Our perspective yields solutions to a variety of classical problems involving Grothendieck-Witt groups of rings and clarifies the behaviour of these invariants when 2 is not invertible. In this article we lay the foundations of our approach by considering Lurie's notion of a Poincaré $\infty$-category, which permits an abstract counterpart of unimodular forms called Poincaré objects. We analyse the special cases of hyperbolic and metabolic Poincaré objects, a…
