Bipartite coalgebras and a reduction functor for coradical square complete coalgebras

Justyna Kosakowska; Daniel Simson

Displaying similar documents to “Bipartite coalgebras and a reduction functor for coradical square complete coalgebras”

Hall algebras of two equivalent extriangulated categories

Shiquan Ruan, Li Wang, Haicheng Zhang (2024)

Czechoslovak Mathematical Journal

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For any positive integer $n$ , let $A_{n}$ be a linearly oriented quiver of type $A$ with $n$ vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories $ℳ_{n + 1}$ and $ℱ_{n}$ , where $ℳ_{n + 1}$ and $ℱ_{n}$ are the two extriangulated categories corresponding to the representation category of $A_{n + 1}$ and the morphism category of projective representations of $A_{n}$ , respectively. As a...

Limits and colimits in certain categories of spaces of continuous functions

Marvin W. Grossman

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CONTENTSIntroduction................................................................................................................................................................................5§ 1. Notation and preliminaries.............................................................................................................................................6§ 2. Epimorphisms and monomorphisms.........................................................................................................................7§...

Cotorsion pairs in comma categories

Yuan Yuan, Jian He, Dejun Wu (2024)

Czechoslovak Mathematical Journal

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Let $𝒜$ and $ℬ$ be abelian categories with enough projective and injective objects, and $T : 𝒜 \to ℬ$ a left exact additive functor. Then one has a comma category $(ℬ ↓ T)$ . It is shown that if $T : 𝒜 \to ℬ$ is $𝒳$ -exact, then $(^{⊥} 𝒳, 𝒳)$ is a (hereditary) cotorsion pair in $𝒜$ and $(^{⊥} 𝒴, 𝒴)$ ) is a (hereditary) cotorsion pair in $ℬ$ if and only if $((\binom{^{⊥} 𝒳}{^{⊥} 𝒴})), 〈 𝐡 (𝒳, 𝒴) 〉)$ is a (hereditary) cotorsion pair in $(ℬ ↓ T)$ and $𝒳$ and $𝒴$ are closed under extensions. Furthermore, we characterize when special preenveloping classes in abelian categories $𝒜$ and $ℬ$ can induce special preenveloping...

Stable tubes in extriangulated categories

Li Wang, Jiaqun Wei, Haicheng Zhang (2022)

Czechoslovak Mathematical Journal

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Let $𝒳$ be a semibrick in an extriangulated category. If $𝒳$ is a $τ$ -semibrick, then the Auslander-Reiten quiver $Γ (ℱ (𝒳))$ of the filtration subcategory $ℱ (𝒳)$ generated by $𝒳$ is $ℤ 𝔸_{\infty}$ . If $𝒳 = {X_{i}}_{i = 1}^{t}$ is a $τ$ -cycle semibrick, then $Γ (ℱ (𝒳))$ is $ℤ 𝔸_{\infty} / τ_{𝔸}^{t}$ .

On category $𝒪$ for cyclotomic rational Cherednik algebras

Iain G. Gordon, Ivan Losev (2014)

Journal of the European Mathematical Society

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We study equivalences for category $𝒪_{p}$ of the rational Cherednik algebras $𝐇_{p}$ of type $G_{ℓ} (n) = {(μ_{ℓ})}^{n} ⋊ 𝔖_{n}$ : a highest weight equivalence between $𝒪_{p}$ and $𝒪_{σ (p)}$ for $σ \in 𝔖_{ℓ}$ and an action of $𝔖_{ℓ}$ on an explicit non-empty Zariski open set of parameters $p$ ; a derived equivalence between $𝒪_{p}$ and $𝒪_{p^{'}}$ whenever $p$ and $p^{'}$ have integral difference; a highest weight equivalence between $𝒪_{p}$ and a parabolic category $𝒪$ for the general linear group, under a non-rationality assumption on the parameter $p$ . As a consequence, we confirm special cases...

On the structure of triangulated categories with finitely many indecomposables

Claire Amiot (2007)

Bulletin de la Société Mathématique de France

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We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field $k$ . We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category $𝒯$ is of the form $ℤ Δ / G$ where $Δ$ is a disjoint union of simply-laced Dynkin diagrams and $G$ a weakly admissible group of automorphisms of $ℤ Δ$ . Then we prove that for ‘most’ groups $G$ , the category $𝒯$ is standard, ...

Generic representations of orthogonal groups: projective functors in the category $ℱ_{q u a d}$

Christine Vespa (2008)

Fundamenta Mathematicae

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We continue the study of the category of functors $ℱ_{q u a d}$ , associated to ₂-vector spaces equipped with a nondegenerate quadratic form, initiated in J. Pure Appl. Algebra 212 (2008) and Algebr. Geom. Topology 7 (2007). We define a filtration of the standard projective objects in $ℱ_{q u a d}$ ; this refines to give a decomposition into indecomposable factors of the first two standard projective objects in $ℱ_{q u a d}$ : $P_{H ₀}$ and $P_{H ₁}$ . As an application of these two decompositions, we give a complete description of the polynomial...

Higher-dimensional Auslander-Reiten sequences

Jiangsha Li, Jing He (2024)

Czechoslovak Mathematical Journal

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Zhou and Zhu have shown that if $𝒞$ is an $(n + 2)$ -angulated category and $𝒳$ is a cluster tilting subcategory of $𝒞$ , then the quotient category $𝒞 / 𝒳$ is an $n$ -abelian category. We show that if $𝒞$ has Auslander-Reiten $(n + 2)$ -angles, then $𝒞 / 𝒳$ has Auslander-Reiten $n$ -exact sequences.

One-sided Gorenstein subcategories

Weiling Song, Tiwei Zhao, Zhaoyong Huang (2020)

Czechoslovak Mathematical Journal

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We introduce the right (left) Gorenstein subcategory relative to an additive subcategory $𝒞$ of an abelian category $𝒜$ , and prove that the right Gorenstein subcategory $r 𝒢 (𝒞)$ is closed under extensions, kernels of epimorphisms, direct summands and finite direct sums. When $𝒞$ is self-orthogonal, we give a characterization for objects in $r 𝒢 (𝒞)$ , and prove that any object in $𝒜$ with finite $r 𝒢 (𝒞)$ -projective dimension is isomorphic to a kernel (or a cokernel) of a morphism from an object in $𝒜$ with finite $𝒞$ -projective...

On almost complex structures from classical linear connections

Jan Kurek, Włodzimierz M. Mikulski (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $ℳ f_{m}$ be the category of $m$ -dimensional manifolds and local diffeomorphisms and let $T$ be the tangent functor on $ℳ f_{m}$ . Let $𝒱$ be the category of real vector spaces and linear maps and let $𝒱_{m}$ be the category of $m$ -dimensional real vector spaces and linear isomorphisms. We characterize all regular covariant functors $F : 𝒱_{m} \to 𝒱$ admitting $ℳ f_{m}$ -natural operators $\tilde{J}$ transforming classical linear connections $\nabla$ on $m$ -dimensional manifolds $M$ into almost complex structures $\tilde{J} (\nabla)$ on $F (T) M = ⋃_{x \in M} F (T_{x} M)$ .

The Roquette category of finite $p$ -groups

Serge Bouc (2015)

Journal of the European Mathematical Society

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Let $p$ be a prime number. This paper introduces the Roquette category $ℛ_{p}$ of finite $p$ -groups, which is an additive tensor category containing all finite $p$ -groups among its objects. In $ℛ_{p}$ , every finite $p$ -group $P$ admits a canonical direct summand $\partial P$ , called the edge of $P$ . Moreover $P$ splits uniquely as a direct sum of edges of Roquette $p$ -groups, and the tensor structure of $ℛ_{p}$ can be described in terms of such edges. The main motivation for considering this category is that the additive functors...

Two results of $n$ -exangulated categories

Jian He, Jing He, Panyue Zhou (2024)

Czechoslovak Mathematical Journal

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M. Herschend, Y. Liu, H. Nakaoka introduced $n$ -exangulated categories, which are a simultaneous generalization of $n$ -exact categories and $(n + 2)$ -angulated categories. This paper consists of two results on $n$ -exangulated categories: (1) we give an equivalent characterization of axiom (EA2); (2) we provide a new way to construct a closed subfunctor of an $n$ -exangulated category.

The general rigidity result for bundles of $A$ -covelocities and $A$ -jets

Jiří M. Tomáš (2017)

Czechoslovak Mathematical Journal

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Let $M$ be an $m$ -dimensional manifold and $A = 𝔻_{k}^{r} / I = ℝ \oplus N_{A}$ a Weil algebra of height $r$ . We prove that any $A$ -covelocity $T_{x}^{A} f \in T_{x}^{A *} M$ , $x \in M$ is determined by its values over arbitrary $max {width A, m}$ regular and under the first jet projection linearly independent elements of $T_{x}^{A} M$ . Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result $T^{A *} M ≃ T^{r *} M$ without coordinate computations, which improves and generalizes the partial...

On the unit group of a semisimple group algebra $𝔽_{q} S L (2, ℤ_{5})$

Rajendra K. Sharma, Gaurav Mittal (2022)

Mathematica Bohemica

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We give the characterization of the unit group of $𝔽_{q} S L (2, ℤ_{5})$ , where $𝔽_{q}$ is a finite field with $q = p^{k}$ elements for prime $p > 5,$ and $S L (2, ℤ_{5})$ denotes the special linear group of $2 \times 2$ matrices having determinant $1$ over the cyclic group $ℤ_{5}$ .