Category Of Algebras Research Articles

Algebraic structures in comodule categories over weak bialgebras

For a bialgebra L coacting on a -algebra A, a classical result states A is a right L-comodule algebra if and only if A is an algebra in the category of right L-comodules. We generalize this to the setting of weak bialgebras H. We prove that there is an isomorphism between the categories of right H-comodule algebras and the category of algebras in . We also recall and introduce the formulaic notion of H coacting on a -coalgebra and on a Frobenius -algebra, respectively, and prove analogous category isomorphisms. Many examples are provided throughout.

Regular evolution algebras are universally finite

In this paper we show that evolution algebras over any given field k \Bbbk are universally finite. In other words, given any finite group G G , there exist infinitely many regular evolution algebras X X such that A u t ( X ) ≅ G Aut(X)\cong G . The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme G \mathbf {G} over k \Bbbk is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra.

When the Schur functor induces a triangle-equivalence between Gorenstein defect categories

Let $R$ be an Artin algebra and $e$ an idempotent of $R$. Assume that ${\rm Tor}_i^{eRe}(Re,G)=0$ for any $G\in{\rm GProj} eRe$ and $i$ sufficiently large. Necessary and sufficient conditions are given for the Schur functor $S_e$ to induce a triangle-equivalence $\mathbb{D}_{def}(R)\simeq\mathbb{D}_{def}(eRe)$. Combine this with a result of Psaroudakis-Skartsaterhagen-Solberg [29], we provide necessary and sufficient conditions for the singular equivalence $\mathbb{D}_{sg}(R)\simeq\mathbb{D}_{sg}(eRe)$ to restrict to a triangle-equivalence $\underline{{\rm GProj} R}\simeq\underline{{\rm GProj} eRe}$. Applying these to the triangular matrix algebra $T=\left( \begin{array}{cc} A & M \quad 0 & B \end{array} \right)$, corresponding results between candidate categories of $T$ and $A$ (resp. $B$) are obtained. As a consequence, we infer Gorensteinness and CM-freeness of $T$ from those of $A$ (resp. $B$). Some concrete examples are given to indicate one can realise the Gorenstein defect category of a triangular matrix algebra as the singularity category of one of its corner algabras.

Rota–Baxter operators on involutive associative algebras

In this paper, we consider Rota-Baxter operators on involutive associative algebras. We define cohomology for Rota-Baxter operators on involutive algebras that governs the formal deformation of the operator. This cohomology can be seen as the Hochschild cohomology of a certain involutive associative algebra with coefficients in a suitable involutive bimodule. We also relate this cohomology with the cohomology of involutive dendriform algebras. Finally, we show that the standard Fard-Guo construction of the functor from the category of dendriform algebras to the category of Rota-Baxter algebras restricts to the involutive case.

The Quantum Group Dual of the First-Row Subcategory for the Generic Virasoro VOA

In several examples it has been observed that a module category of a vertex operator algebra (VOA) is equivalent to a category of representations of some quantum group. The present article is concerned with developing such a duality in the case of the Virasoro VOA at generic central charge; arguably the most rudimentary of all VOAs, yet structurally complicated. We do not address the category of all modules of the generic Virasoro VOA, but we consider the infinitely many modules from the first row of the Kac table. Building on an explicit quantum group method of Coulomb gas integrals, we give a new proof of the fusion rules, we prove the analyticity of compositions of intertwining operators, and we show that the conformal blocks are fully determined by the quantum group method. Crucially, we prove the associativity of the intertwining operators among the first-row modules, and find that the associativity is governed by the 6j-symbols of the quantum group. Our results constitute a concrete duality between a VOA and a quantum group, and they will serve as the key tools to establish the equivalence of the first-row subcategory of modules of the generic Virasoro VOA and the category of (type-1) finite-dimensional representations of {mathcal {U}}_q (mathfrak {sl}_2).

Morita theorem for hereditary Calabi-Yau categories

We give a structure theorem for Calabi-Yau triangulated category with a hereditary cluster tilting object. We prove that an algebraic d-Calabi-Yau triangulated category with a d-cluster tilting object T such that its shifted sum T⊕⋯⊕T[−(d−2)] has hereditary endomorphism algebra H is triangle equivalent to the orbit category Db(modH)/τ−1/(d−1)[1] of the derived category of H for a naturally defined (d−1)-st root τ1/(d−1) of the AR translation, provided H is of non-Dynkin type. We also show that heredity of H follows from that of T when d=3, that of T⊕T[−1] when d=4, and similarly from a smaller endomorphism algebra for higher dimensions under vanishing of some negative self-extensions of T. Our result therefore generalizes the established theorems by Keller–Reiten and Keller–Murfet–Van den Bergh. Furthermore, we show that enhancements of such triangulated categories are unique. Finally we apply our results to Calabi-Yau reductions of a higher cluster category of a finite dimensional algebra and of the singularity category of an invariant subring.

On Auslander–Reiten components of string complexes for a certain class of symmetric special biserial algebras

Let \(\mathbb {k}\) be an algebraically closed field. In this article, inspired by the description of indecomposable objects in the derived category of a gentle algebra obtained by Bekkert and Merklen, we define string complexes for a certain class \({\mathscr {C}}\) of symmetric special biserial algebras, which are indecomposable perfect complexes in the corresponding derived category. We also prove that if \(\Lambda \) is a \(\mathbb {k}\)-algebra in the class \({\mathscr {C}}\) and \(P^\bullet \) is a string complex over \(\Lambda \), then \(P^\bullet \) lies in the rim of its Auslander–Reiten component.

MODAL OPERATORS ON RINGS OF CONTINUOUS FUNCTIONS

AbstractIt is a classic result in modal logic, often referred to as Jónsson-Tarski duality, that the category of modal algebras is dually equivalent to the category of descriptive frames. The latter are Kripke frames equipped with a Stone topology such that the binary relation is continuous. This duality generalizes the celebrated Stone duality for boolean algebras. Our goal is to generalize descriptive frames so that the topology is an arbitrary compact Hausdorff topology. For this, instead of working with the boolean algebra of clopen subsets of a Stone space, we work with the ring of continuous real-valued functions on a compact Hausdorff space. The main novelty is to define a modal operator on such a ring utilizing a continuous relation on a compact Hausdorff space.Our starting point is the well-known Gelfand duality between the category ${\sf KHaus}$ of compact Hausdorff spaces and the category $\boldsymbol {\mathit {uba}\ell }$ of uniformly complete bounded archimedean $\ell $ -algebras. We endow a bounded archimedean $\ell $ -algebra with a modal operator, which results in the category $\boldsymbol {\mathit {mba}\ell }$ of modal bounded archimedean $\ell $ -algebras. Our main result establishes a dual adjunction between $\boldsymbol {\mathit {mba}\ell }$ and the category ${\sf KHF}$ of what we call compact Hausdorff frames; that is, Kripke frames equipped with a compact Hausdorff topology such that the binary relation is continuous. This dual adjunction restricts to a dual equivalence between ${\sf KHF}$ and the reflective subcategory $\boldsymbol {\mathit {muba}\ell }$ of $\boldsymbol {\mathit {mba}\ell }$ consisting of uniformly complete objects of $\boldsymbol {\mathit {mba}\ell }$ . This generalizes both Gelfand duality and Jónsson-Tarski duality.

On generalized algebraic theories and categories with families

AbstractWe give a syntax independent formulation of finitely presented generalized algebraic theories as initial objects in categories of categories with families (cwfs) with extra structure. To this end, we simultaneously define the notion of a presentation Σ of a generalized algebraic theory and the associated category CwFΣ of small cwfs with a Σ-structure and cwf-morphisms that preserve Σ-structure on the nose. Our definition refers to the purely semantic notion of uniform family of contexts, types, and terms in CwFΣ. Furthermore, we show how to syntactically construct an initial cwf with a Σ-structure. This result can be viewed as a generalization of Birkhoff’s completeness theorem for equational logic. It is obtained by extending Castellan, Clairambault, and Dybjer’s construction of an initial cwf. We provide examples of generalized algebraic theories for monoids, categories, categories with families, and categories with families with extra structure for some type formers of Martin-Löf type theory. The models of these are internal monoids, internal categories, and internal categories with families (with extra structure) in a small category with families. Finally, we show how to extend our definition to some generalized algebraic theories that are not finitely presented, such as the theory of contextual cwfs.

Quotient rings of 𝐻𝔽₂∧ℍ𝔽₂

We study modules over the commutative ring spectrum H F 2 ∧ H F 2 H\mathbb F_2\wedge H\mathbb F_2 , whose coefficient groups are quotients of the dual Steenrod algebra by collections of the Milnor generators. We show that very few of these quotients admit algebra structures, but those that do can be constructed simply: killing a generator ξ k \xi _k in the category of associative algebras freely kills the higher generators ξ k + n \xi _{k+n} . Using new information about the conjugation operation in the dual Steenrod algebra, we also consider quotients by families of Milnor generators and their conjugates. This allows us to produce a family of associative H F 2 ∧ H F 2 H\mathbb F_2\wedge H\mathbb F_2 -algebras whose coefficient rings are finite-dimensional and exhibit unexpected duality features. We then use these algebras to give detailed computations of the homotopy groups of several modules over this ring spectrum.

Dualities for subresiduated lattices

A subresiduated lattice is a pair (A, D), where A is a bounded distributive lattice, D is a bounded sublattice of A and for every \(a,b\in A\) there is \(c\in D\) such that for all \(d\in D\), \(d\wedge a\le b\) if and only if \(d\le c\). This c is denoted by \(a\rightarrow b\). This pair can be regarded as an algebra \(\left<A,\wedge ,\vee ,\rightarrow ,0,1\right>\) of type (2, 2, 2, 0, 0) where \(D=\{a\in A\mid 1\rightarrow a=a\}\). The class of subresiduated lattices is a variety which properly contains to the variety of Heyting algebras. In this paper we present dual equivalences for the algebraic category of subresiduated lattices. More precisely, we develop a spectral style duality and a bitopological style duality for this algebraic category. Finally we study the connections of these results with a known Priestley style duality for the algebraic category of subresiduated lattices.

W-types in setoids

We present a construction of W-types in the setoid model of extensional Martin-L\"of type theory using dependent W-types in the underlying intensional theory. More precisely, we prove that the internal category of setoids has initial algebras for polynomial endofunctors. In particular, we characterise the setoid of algebra morphisms from the initial algebra to a given algebra as a setoid on a dependent W-type. We conclude by discussing the case of free setoids. We work in a fully intensional theory and, in fact, we assume identity types only when discussing free setoids. By using dependent W-types we can also avoid elimination into a type universe. The results have been verified in Coq and a formalisation is available on the author's GitHub page.

Dichotomy between Deterministic and Probabilistic Models in Countably Additive Effectus Theory

Effectus theory is a relatively new approach to categorical logic that can be seen as an abstract form of generalized probabilistic theories (GPTs). While the scalars of a GPT are always the real unit interval [0,1], in an effectus they can form any effect monoid. Hence, there are quite exotic effectuses resulting from more pathological effect monoids. In this paper we introduce sigma-effectuses, where certain countable sums of morphisms are defined. We study in particular sigma-effectuses where unnormalized states can be normalized. We show that a non-trivial sigma-effectus with normalization has as scalars either the two-element effect monoid 0,1 or the real unit interval [0,1]. When states and/or predicates separate the morphisms we find that in the 0,1 case the category must embed into the category of sets and partial functions (and hence the category of Boolean algebras), showing that it implements a deterministic model, while in the [0,1] case we find it embeds into the category of Banach order-unit spaces and of Banach pre-base-norm spaces (satisfying additional properties), recovering the structure present in GPTs. Hence, from abstract categorical and operational considerations we find a dichotomy between deterministic and convex probabilistic models of physical theories.

On the behaviour of coalgebras with side effects and algebras with effectful iteration

Abstract For every finitary monad $T$ on sets and every endofunctor $F$ on the category of $T$-algebras, we introduce the concept of an ffg-Elgot algebra for $F$, i.e. an algebra admitting coherent solutions for finite systems of recursive equations with effects represented by the monad $T$. The goal is to study the existence and construction of free ffg-Elgot algebras. To this end, we investigate the locally ffg fixed point $\varphi F$, i.e. the colimit of all $F$-coalgebras with free finitely generated carrier, which is shown to be the initial ffg-Elgot algebra. This is the technical foundation for our main result: the category of ffg-Elgot algebras is monadic over the category of $T$-algebras.

On the Probabilistic Convergence Spaces: Monad and its Eilenberg–Moore Category

Motivated by the category of probabilistic convergence spaces — a supercategory of the category of topological spaces; recently, we brought to light the categories of probabilistic convergence groups, probabilistic metric probabilistic convergence groups, probabilistic convergence transformation groups, along with their underpinning natural examples. The purpose of this paper is, first, to establish a result on the isomorphism between the categories of probabilistic metric groups, and probabilistic metric probabilistic convergence groups. Second, among others, we explore a monad in relation with probabilistic convergence groups, and probabilistic convergence spaces, and their related algebras. In so doing, we consider a product of the categories of probabilistic convergence groups and probabilistic convergence spaces in an attempt to construct a monad on it such that the corresponding category of algebras, the so-called Eilenberg–Moore category, is isomorphic to the category of probabilistic convergence transformation groups. Finally, invoking so-called Beck’s theorem on characterization of algebras, and starting with a particular adjunction, we achieve a monad. Conversely, given a monad, we obtain an adjunction which coincides with the original monad.

Comparing localizations across adjunctions

We show that several apparently unrelated formulas involving left or right Bousfield localizations in homotopy theory are induced by comparison maps associated with pairs of adjoint functors. Such comparison maps are used in the article to discuss the existence of functorial liftings of homotopical localizations and cellularizations to categories of algebras over monads acting on model categories, with emphasis on the cases of module spectra and algebras over simplicial operads. Some of our results hold for algebras up to homotopy as well; for example, if $T$ is the reduced monad associated with a simplicial operad and $f$ is any map of pointed simplicial sets, then $f$-localization coincides with $Tf$-localization on spaces underlying homotopy $T$-algebras, and similarly for cellularizations.

On the Topological Dynamics of Dynamical Manifolds and Their Fundamental Group

The paper aims to deduce the relation between the category of topology and algebra from viewpoint of geometry and dynamical system. We introduce and define a dynamical manifold as a manifold associated with a time parameter. We obtain the induced chain of topological dynamics on the fundamental group from the chain of dynamical maps on a dynamical manifold. For many adjunctions in this context, we deduce the limit topological dynamics and conditional topological dynamics on the fundamental group. We use the category of commutative diagrams as chains of dynamical manifolds to deduce the chains on fundamental groups. Also, we describe how the manifold changes in a dynamical system from the view of the fundamental group.

A–infinity algebras, strand algebras, and contact categories

In previous work we showed that the contact category algebra of a quadrangulated surface is isomorphic to the homology of a strand algebra from bordered sutured Floer theory. Being isomorphic to the homology of a differential graded algebra, this contact category algebra has an A-infinity structure, allowing us to combine contact structures not just by gluing, but also by higher-order operations. In this paper we investigate such A-infinity structures and higher order operations on contact structures. We give explicit constructions of such A-infinity structures, and establish some of their properties, including conditions for the vanishing and nonvanishing of A-infinity operations. Along the way we develop several related notions, including a detailed consideration of tensor products of strand diagrams.

A special issue on categorical algebras and computation in celebration of John Power’s 60th birthday, part I

A special issue on categorical algebras and computation in celebration of John Power’s 60th birthday, part I

A new equivalence between singularity categories of commutative algebras

We construct a triangle equivalence between the singularity categories of two isolated cyclic quotient singularities of Krull dimensions two and three, respectively. This is the first example of a singular equivalence involving connected commutative algebras of odd and even Krull dimension. In combination with Orlov's localization result, this gives further singular equivalences between certain quasi-projective varieties of dimensions two and three, respectively.