Academic literature on the topic 'Skew monoidal category'

AbstractIncidence 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.

For defining a K-semialgebra A, we use Katsov's tensor product which makes the category K-Smod monoidal. Further, if A is a K-semialgebra then AΔ is a KΔ-algebra and A embeds in AΔ. The subtractive and strong partial actions of a group are defined on A. A subtractive partial action α of a group G on A can be extended to a partial action of G on AΔ which helps in globalization of α. A strong partial action on A has a unique subtractive globalization. We also discuss the associativity of the skew group semiring A ×α G.

Abstract Let $R$ be the skew group algebra of a finite group acting on the path algebra of a quiver. This article develops both theoretical and practical methods to do computations in the Morita-reduced algebra associated to $R$. Reiten and Riedtmann proved that there exists an idempotent $e$ of $R$ such that the algebra $eRe$ is both Morita equivalent to $R$ and isomorphic to the path algebra of some quiver, which was described by Demonet. This article gives explicit formulas for the decomposition of any element of $eRe$ as a linear combination of paths in the quiver described by Demonet. This is done by expressing appropriate compositions and pairings in a suitable monoidal category, which takes into account the representation theory of the finite group.

AbstractWe exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids—or in a straightforward generalisation, the category of modules over a commutative rig k. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad Q. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad Q involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal k-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category—thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.

AbstractWe study Serre structures on categories enriched in pivotal monoidal categories, and apply this to study Serre structures on two types of graded k-linear categories: categories with group actions and categories with graded hom spaces. We check that Serre structures are preserved by taking orbit categories and skew group categories, and describe the relationship with graded Frobenius algebras. Using a formal version of Auslander-Reiten translations, we show that the derived category of a d-representation finite algebra is fractionally Calabi-Yau if and only if its preprojective algebra has a graded Nakayama automorphism of finite order. This connects various results in the literature and gives new examples of fractional Calabi-Yau algebras.

AbstractLet $$\mathcal {A}$$ A and $$\mathcal {B}$$ B be monoidal categories and let $$R:\mathcal {A}\rightarrow \mathcal {B}$$ R : A → B be a lax monoidal functor. If R has a left adjoint L, it is well-known that the two adjoints induce functors $${\overline{R}}={\textsf {Alg}}(R):\textsf {Alg}({\mathcal {A}})\rightarrow {\textsf {Alg}}({\mathcal {B}})$$ R ¯ = Alg ( R ) : Alg ( A ) → Alg ( B ) and $${\underline{L}}={\textsf {Coalg}}(L):{ \textsf {Coalg}}({\mathcal {B}})\rightarrow \textsf {Coalg}({\mathcal {A}})$$ L ̲ = Coalg ( L ) : Coalg ( B ) → Coalg ( A ) respectively. The pair (L, R) is called liftable if the functor $${\overline{R}}$$ R ¯ has a left adjoint and if the functor $${\underline{L}}$$ L ̲ has a right adjoint. A pleasing fact is that, when $$\mathcal {A}$$ A , $$\mathcal {B}$$ B and R are moreover braided, a liftable pair of functors as above gives rise to an adjunction at the level of bialgebras. In this note, sufficient conditions on the category $$\mathcal {A}$$ A for $${\overline{R}}$$ R ¯ to possess a left adjoint, are given. Natively these conditions involve the existence of suitable colimits that we interpret as objects which are simultaneously initial in four distinguished categories (among which the category of epi-induced objects), allowing for an explicit construction of $${\overline{L}}$$ L ¯ , under the appropriate hypotheses. This is achieved by introducing a relative version of the notion of weakly coreflective subcategory, which turns out to be a useful tool to compare the initial objects in the involved categories. We apply our results to obtain an analogue of Sweedler’s finite dual for the category of vector spaces graded by an abelian group G endowed with a bicharacter. When the bicharacter on G is skew-symmetric, a lifted adjunction as mentioned above is explicitly described, inducing an auto-adjunction on the category of bialgebras “colored” by G.