Morphisms Of Category Research Articles

Tor-tilting and weak silting theories in comodule categories

In this paper, we introduce and investigate Tor-tilting comodules and weak silting comodules, providing a generalization of the established tilting and silting theories in the comodule category. Our exploration yields a comprehensive understanding of their fundamental properties, including their associated torsion theories. We delve into not only the relations between Tor-tilting and weak silting comodules but also the connections among them and (co) tilting as well as (co) silting comodules. Additionally, we define Tor-tilting coflat comodules and Tor-tilting coflat dimensions. Finally, we explore Tor-tilting and weak silting objects in the morphism category, demonstrating their equivalence.

Internalizations of Decorated Bicategories via π2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi _2$$\\end{document}-Indexings

We treat the problem of lifting bicategories into double categories through categories of vertical morphisms. We consider structures on decorated 2-categories allowing us to formally implement arguments of sliding certain squares along vertical subdivisions in double categories. We call these structures π2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi _2$$\\end{document}-indexings. We present a construction associating, to every π2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi _2$$\\end{document}-indexing on a decorated 2-category, a length 1 double internalization.

КАТЕГОРИИ МУЛЬТИМНОЖЕСТВ И СООТВЕТСТВИЙ МЕЖДУ НИМИ

The article studies the category of multisets. The stability of strict epimorphisms in Cartesian squares and strict monomorphisms in coCartesian squares is proved. These results allow us to construct categories of correspondences and cocorrespondences of multisets in which the category of functional morphisms is equivalent to the category of multisets. Within the framework of the correspondence categories, the description of equivalence relations and congruencies on multisets is given. The category of multisets is not exact: according to some equivalence relations, a multiset cannot be factorized. Using the technique of ordered categories with involution, the authors construct a minimal embedding of the category of multisets to the exact category, which is universal. Depending on whether correspondences or noncorrespondences are used, this embedding preserves finite limits or co limits.

Lax pullback complements in partial morphism categories

The goal of this article is to characterize the lax pullback complement of a given partial morphism along a total morphism, in an -partial morphism category, where is an exponentiable stable system. To achieve this, we first show that if a lax pullback complement of a partial morphism exists in the partial morphism category, then a lax pullback complement of its total part exists in the base category. For the converse, we consider two cases. In the first case the base category is assumed to be adhesive and the morphism along which we take the lax pullback complement is admissible (i.e., its pushout along a monomorphism forms a pullback complement square), while in the second case the base category is -cohesive, however the morphism along which we take the lax pullback complement is an arbitrary total morphism. Finally as a byproduct we provide the connection between exponentials in the partial morphism category and its base category.

-PERPENDICULAR WIDE SUBCATEGORIES

Abstract Let $\Lambda $ be a finite-dimensional algebra. A wide subcategory of $\mathsf {mod}\Lambda $ is called left finite if the smallest torsion class containing it is functorially finite. In this article, we prove that the wide subcategories of $\mathsf {mod}\Lambda $ arising from $\tau $ -tilting reduction are precisely the Serre subcategories of left-finite wide subcategories. As a consequence, we show that the class of such subcategories is closed under further $\tau $ -tilting reduction. This leads to a natural way to extend the definition of the “ $\tau $ -cluster morphism category” of $\Lambda $ to arbitrary finite-dimensional algebras. This category was recently constructed by Buan–Marsh in the $\tau $ -tilting finite case and by Igusa–Todorov in the hereditary case.

Determination of some almost split sequences in morphism categories

Almost split sequences lie in the heart of Auslander-Reiten theory. This paper deals with the structure of almost split sequences with certain ending terms in the morphism category of an Artin algebra Λ. Firstly we try to interpret the Auslander-Reiten translates of particular objects in the morphism category in terms of the Auslander-Reiten translations within the category of Λ-modules, and then use them to calculate almost split sequences. In classical representation theory of algebras, it is quite important to recognize the middle term of almost split sequences. As such, another part of the paper is devoted to discuss the middle term of certain almost split sequences in the morphism category of Λ. As an application, we restrict in the last part of the paper to self-injective algebras and present a structural theorem that illuminates a link between representation-finite morphism categories and Dynkin diagrams.

Internalization and enrichment via spans and matrices in a tricategory

We introduce categories $$\mathcal {M}$$ and $$\mathcal {S}$$ internal in the tricategory $$\textrm{Bicat}_3$$ of bicategories, pseudofunctors, pseudonatural transformations and modifications, for matrices and spans in a 1-strict tricategory V. Their horizontal tricategories are the tricategories of matrices and spans in V. Both the internal and the enriched constructions are tricategorifications of the corresponding constructions in 1-categories. Following Fiore et al. (J Pure Appl Algebra 215(6):1174–1197, 2011), we introduce monads and their vertical morphisms in categories internal in tricategories. We prove an equivalent condition for when the internal categories for matrices $$\mathcal {M}$$ and spans $$\mathcal {S}$$ in a 1-strict tricategory V are equivalent, and deduce that in that case their corresponding categories of (strict) monads and vertical monad morphisms are equivalent, too. We prove that the latter categories are isomorphic to those of categories enriched and discretely internal in V, respectively. As a by-product of our tricategorical constructions, we recover some results from Femić (Enrichment and internalization in tricategories, the case of tensor categories and alternative notion to intercategories. arXiv:2101.01460v2 ). Truncating to 1-categories, we recover results from Cottrell et al. (Tbilisi Math J 10(3):239–254, 2017) and Ehresmann and Ehresmann (Cah Topol Géom Differ Catég 19/4:387–443, 1978) on the equivalence of enriched and discretely internal 1-categories.

An orthogonal approach to algebraic weak factorisation systems

We describe an equivalent formulation of algebraic weak factorisation systems, not involving monads and comonads, but involving double categories of morphisms equipped with a lifting operation satisfying lifting and factorisation axioms.

Quantum cluster characters of Hall algebras revisited

Let Q be a finite acyclic valued quiver. We define a bialgebra structure and an integration map on the Hall algebra associated to the morphism category of projective representations of Q. As an application, we recover the surjective homomorphism defined in [12], which realizes the principal coefficient quantum cluster algebra \(\mathcal {A}\,_q(Q)\) as a sub-quotient of the Hall algebra of morphisms. Moreover, we also recover the quantum Caldero–Chapoton formula, as well as some multiplication formulas between quantum Caldero–Chapoton characters.

Categories and semigroups

The aim of this paper is to consider the correspondence between the classification of morphisms in categories and the classes of semigroups with idempotents, in particular, we establish a mutual corresponding theorem of three classes of categories and the classes of left (right) abundant, two-sided abundant semigroups and regular semigroups. We first apply the nine axioms (P1)–(P9) which characterize cancellation and split properties of morphisms in a category to classify categories into three subclasses: idempotent ample ([Formula: see text]-, for short) categories, balanced categories and normal categories. The intrinsic relationship between these categories and their cone semigroups is investigated. It is proved that the cone semigroup of an [Formula: see text]-category is left abundant and vice versa, the category of principal left ∗-ideals of a left abundant (not necessarily right abundant) semigroup is an [Formula: see text]-category. Similar results for balanced categories and abundant semigroups are reproved and strengthened. As a consequence, we employ categorical terms to characterize those abundant semigroups in which the regular elements form a regular subsemigroup. Therefore, the significant theory of normal categories and regular semigroups devoted by K. S. S. Nambooripad is generalized to arbitrary abundant semigroups and even to left and right abundant semigroups, respectively.

Functoriality of the Schmidt construction

Abstract After proving, in a purely categorial way, that the inclusion functor $\textrm {In}_{\textbf {Alg}(\varSigma )}$ from $\textbf {Alg}(\varSigma )$, the category of many-sorted $\varSigma $-algebras, to $\textbf {PAlg}(\varSigma )$, the category of many-sorted partial $\varSigma $-algebras, has a left adjoint $\textbf {F}_{\varSigma }$, the (absolutely) free completion functor, we recall, in connection with the functor $\textbf {F}_{\varSigma }$, the generalized recursion theorem of Schmidt, which we will also call the Schmidt construction. Next, we define a category $\textbf {Cmpl}(\varSigma )$, of $\varSigma $-completions, and prove that $\textbf {F}_{\varSigma }$, labelled with its domain category and the unit of the adjunction of which it is a part, is a weakly initial object in it. Following this, we associate to an ordered pair $(\boldsymbol {\alpha },f)$, where $\boldsymbol {\alpha }=(K,\gamma ,\alpha )$ is a morphism of $\varSigma $-completions from ${{\mathscr {F}}}=(\textbf {C},F,\eta )$ to $\mathscr {G}= (\textbf {D},G,\rho )$ and $f$ a homomorphism of $\textbf {D}$ from the partial $\varSigma $-algebra $\textbf {A}$ to the partial $\varSigma $-algebra $\textbf {B}$, a homomorphism $\varUpsilon ^{\mathscr {G},0}_{\boldsymbol {\alpha }}(f)\colon \textbf {Sch}_{\boldsymbol {\alpha }}(f)\longrightarrow \textbf {B}$. We then prove that there exists an endofunctor, $\varUpsilon ^{\mathscr {G},0}_{\boldsymbol {\alpha }}$, of $\textbf {Mor}_{\textrm {tw}}(\textbf {D})$, the twisted morphism category of $\textbf {D}$, thus showing the naturalness of the previous construction. Afterwards, we prove that, for every $\varSigma $-completion $\mathscr {G}=(\textbf {D},G,\rho )$, there exists a functor $\varUpsilon ^{\mathscr {G}}$ from the comma category $(\textbf {Cmpl}(\varSigma )\!\downarrow \!\mathscr {G})$ to $\textbf {End}(\textbf {Mor}_{\textrm {tw}}(\textbf {D}))$, the category of endofunctors of $\textbf {Mor}_{\textrm {tw}}(\textbf {D})$, such that $\varUpsilon ^{\mathscr {G},0}$, the object mapping of $\varUpsilon ^{\mathscr {G}}$, sends a morphism of $\varSigma $-completion of $\textbf {Cmpl}(\varSigma )$ with codomain $\mathscr {G}$, to the endofunctor $\varUpsilon ^{\mathscr {G},0}_{\boldsymbol {\alpha }}$.

Pullback dan Pushout di Kategori Modul Topologis

A pullback of two morphisms with a common codomain $f\colon A\to C$ and $g\colon B\to C$ is the limit of a diagram consisting $f$ and $g$. The dual notion of a pullback is called a pushout. A pushout of two morphisms with a common domain $k\colon A\to B$ and $l\colon A\to C$ is the colimit of a diagram consisting $k$ and $l$. The pullback and the pushout of two morphisms need not exists. In this paper, we constructed a pullback and a pushout of two morphism in category of topological modules. A pullback of two continuous homomorphisms $f\colon A\to C$ and $g\colon B\to C$ in category of topological modules is a diagram that contains $A\times _{C} B=\{(a,b)\in A\times B \mid f(a)=g(b)\}\subset A\times B$ with the subspace topology on $A\times _{C} B$. Furthermore, the pushout of two continuous homomorphisms $k\colon A\to B$ and $l\colon A\to C$ in category of topological modules is a diagram that contains $B\bigoplus_{A} C=(B\bigoplus C)/\sim$ where $\sim$ is the smallest equivalence relation containing the pairs $(k(a),l(a))$ for all $a\in A$ and topology on $B\bigoplus C$ is coproduct topology $\tau_{coprod}$

Partial pullback complement rewriting along admissible matches

In this article, we first give two equivalent criteria for the existence of pullback complements along a universal morphism in a category in terms of the existence of pullbacks and co-universality of certain morphisms. Then we introduce the notion of admissible morphism and we characterize the existence of pullback complements in a partial morphism category along an admissible morphism in terms of the existence of pullback complements in its base category; when the base category is a certain admissible category. At the end we make a comparison between partial pullback complement rewriting and sesqui-pushout, double-pushout and single pushout rewritings, for an admissible match.

Abelian Quotients Arising from Extriangulated Categories via Morphism Categories

We investigate abelian quotients arising from extriangulated categories via morphism categories, which is a unified treatment for both exact categories and triangulated categories. Let \((\mathcal {C},\mathbb {E},\mathfrak {s})\) be an extriangulated category with enough projectives \(\mathcal {P}\) and \({\mathscr{M}}\) be a full subcategory of \(\mathcal {C}\) containing \(\mathcal {P}\). We show that a certain quotient category of \(\mathfrak {s}\textup {-def}({\mathscr{M}})\), the category of \(\mathfrak {s}\)-deflations \(f {:} M_{1} {\rightarrow } M_{2}\) with \(M_{1},M_{2} {\in } {\mathscr{M}}\), is abelian. Our main theorem has two applications. If \({\mathscr{M}} {=} \mathcal {C}\), we obtain that a certain ideal quotient category \(\mathfrak {s}\textup {-tri}(\mathcal {C})/\mathcal {R}_{2}\) is equivalent to the category of finitely presented modules \(\textup {mod-}(\mathcal {C}/[\mathcal {P}])\), where \(\mathfrak {s}\)-tri\((\mathcal {C})\) is the category of all \(\mathfrak {s}\)-triangles. If \({\mathscr{M}}\) is a rigid subcategory, we show that \({\mathscr{M}}_{L}/[{\mathscr{M}}]\cong \textup {mod-}({\mathscr{M}}/[\mathcal {P}])\) and \({\mathscr{M}}_{L}/[{\Omega }{\mathscr{M}}]\cong (\textup {mod-}({\mathscr{M}}/[\mathcal {P}])^{\textup {op}})^{\textup {op}}\), where \({\mathscr{M}}_{L}\) (resp. \({\Omega }{\mathscr{M}}\)) is the full subcategory of \(\mathcal {C}\) of objects X admitting an \(\mathfrak {s}\)-triangle with \(M_{1}, M_{2}\in {\mathscr{M}}\) (resp. \(M\in {\mathscr{M}}\) and \(P\in \mathcal {P}\)). In particular, we have \(\mathcal {C}/[{\mathscr{M}}]\cong \textup {mod-}({\mathscr{M}}/[\mathcal {P}])\) and \(\mathcal {C}/[{\Omega }{\mathscr{M}}]\cong (\textup {mod-}({\mathscr{M}}/[\mathcal {P}])^{\textup {op}})^{\textup {op}}\) provided that \({\mathscr{M}}\) is a cluster-tilting subcategory.

Higher Coverings of Racks and Quandles—Part II

This article is the second part of a series of three articles, in which we develop a higher covering theory of racks and quandles. This project is rooted in M. Eisermann’s work on quandle coverings, and the categorical perspective brought to the subject by V. Even, who characterizes coverings as those surjections which are categorically central, relatively to trivial quandles. We extend this work by applying the techniques from higher categorical Galois theory, in the sense of G. Janelidze, and in particular we identify meaningful higher-dimensional centrality conditions defining our higher coverings of racks and quandles. In this second article (Part II), we show that categorical Galois theory applies to the inclusion of the category of coverings into the category of surjective morphisms of racks or of quandles. We characterise the induced Galois-theoretic concepts of trivial covering, normal covering and covering in this two-dimensional context. The latter is described via our definition and study of double coverings, also called algebraically central double extensions of racks (and quandles). We define a suitable and well-behaved commutator which captures the zero-, one- and two-dimensional concepts of centralization in the category of quandles. We keep track of the links with the corresponding concepts in the category of groups.

τ-Cluster morphism categories and picture groups

τ-cluster morphism categories, introduced by Buan and Marsh, are a generalization of cluster morphism categories (defined by Igusa and Todorov). We show the classifying space of such a category is a cube complex, generalizing results of Igusa and Todorov and Igusa. Furthermore, the fundamental group of this space is the picture group of the algebra, first defined by Igusa, Todorov, and Weyman. Finally, we show that for Nakayama algebras, this space is a The key step is a combinatorial proof that, for Nakayama algebras, 2-simple minded collections are characterized by pairwise compatibility conditions, a fact not true in general.

Brauer-Thrall type theorems for submodule categories

Let Λ be an artin algebra and be a quasi-resolving subcategory of which is of finite type. Let be the full subcategory of the morphism category consisting of all monomorphisms in such that also lies in . In this paper, we state and prove Brauer-Thrall type theorems for . As applications, we provide necessary and sufficient conditions for the submodule category to be of finite type, whenever Λ is of finite representation type, as well as, for the lower 2 × 2 triangular matrix algebra to be of finite CM-type, whenever Λ is of finite CM-type.

Probabilistic morphisms and Bayesian nonparametrics

In this paper we develop a functorial language of probabilistic morphisms and apply it to some basic problems in Bayesian nonparametrics. First we extend and unify the Kleisli category of probabilistic morphisms proposed by Lawvere and Giry with the category of statistical models proposed by Chentsov and Morse-Sacksteder. Then we introduce the notion of a Bayesian statistical model that formalizes the notion of a parameter space with a given prior distribution in Bayesian statistics. {We revisit the existence of a posterior distribution, using probabilistic morphisms}. In particular, we give an explicit formula for posterior distributions of the Bayesian statistical model, assuming that the underlying parameter space is a Souslin space and the sample space is a subset in a complete connected finite dimensional Riemannian manifold. Then we give a new proof of the existence of Dirichlet measures over any measurable space using a functorial property of the Dirichlet map constructed by Sethuraman.

From subcategories to the entire module categories

Abstract In this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander–Reiten quiver of the subcategories is obtained only by the Auslander–Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, when Λ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the 2 × 2 {2\times 2} lower triangular matrix algebra over Λ and the stable Auslander algebra of Λ can be estimated by the category of modules over the stable Cohen–Macaulay Auslander algebra of Λ.

Pairwise compatibility for 2-simple minded collections

In $\tau$-tilting theory, it is often difficult to determine when a set of bricks forms a 2-simple minded collection. The aim of this paper is to determine when a set of bricks is contained in a 2-simple minded collection for a $\tau$-tilting finite algebra. We begin by extending the definition of mutation from 2-simple minded collections to more general sets of bricks (which we call semibrick pairs). This gives us an algorithm to check if a semibrick pair is contained in a 2-simple minded collection. We then use this algorithm to show that the 2-simple minded collections of a $\tau$-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) are given by pairwise compatibility conditions if and only if every vertex in the corresponding quiver has degree at most 2. As an application, we show that the classifying space of the $\tau$-cluster morphism category of a $\tau$-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) is an Eilenberg- MacLane space if every vertex in the corresponding quiver has degree at most 2.