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Let $\mathcal{A}$ and $\mathcal{B}$ be monoidal categories and let $R:\mathcal{A} \rightarrow \mathcal{B}$ be a lax monoidal functor. If $R$ has a left adjoint $L$, it is well-known that the two adjoints induce functors $\overline{R}={\sf Alg}(R):{\sf Alg}(\mathcal{A})\rightarrow {\sf Alg }(\mathcal{B})$ and $\underline{L}={\sf Coalg(L)}:{\sf Coalg}(\mathcal{B})\rightarrow {\sf Coalg}(\mathcal{A})$ respectively. The pair $(L,R)$ is called "liftable" if the functor $\overline{R}$ has a left adjoint and if the functor $\underline{L}$ has a right adjoint. A pleasing fact is that, when $\mathcal{A}$, $\mathcal{B}$ 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 $\mathcal{A}$ for $\overline{R}$ 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 $\overline{L}$, 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 $G$., The previous verision has been revised by means of weak coreflections and initial objects. The study of pre-rigid categories has been extrapolated and expanded to become an independent research line (arXiv:2201.03952)

The fundamental notion of separability for commutative algebras was interpreted in categorical setting where also the stronger notion of heavily separability was introduced. These notions were extended to (co)algebras in monoidal categories, in particular to cowreaths. In this paper, we consider the cowreath $$ \left( A\otimes H_{4}^{op}, H_{4}, \psi \right) $$ A ⊗ H 4 op , H 4 , ψ , where $$H_{4}$$ H 4 is the Sweedler 4-dimensional Hopf algebra over a field k and $$A=Cl(\alpha , \beta , \gamma )$$ A = C l ( α , β , γ ) is the Clifford algebra generated by two elements G, X with relations $$G^{2}=\alpha $$ G 2 = α , $$X^{2}=\beta $$ X 2 = β and $$XG+GX=\gamma $$ X G + G X = γ , $$ (\alpha , \beta , \gamma \in k $$ ( α , β , γ ∈ k ) which becomes naturally an $$H_{4}$$ H 4 -comodule algebra. We show that, when $$\textrm{char}\left( k \right) \ne 2, $$ char k ≠ 2 , this cowreath is always separable and h-separable as well.

It is known that the so-called monadic decomposition, applied to the adjunction connecting the category of bialgebras to the category of vector spaces via the tensor and the primitive functors, returns the usual adjunction between bialgebras and (restricted) Lie algebras. Moreover, in this framework, the notions of augmented monad and combinatorial rank play a central role. In order to set these results into a wider context, we are led to substitute the monadic decomposition by what we call the adjoint decomposition. This construction has the advantage of reducing the computational complexity when compared to the first one. We connect the two decompositions by means of an embedding and we investigate its properties by using a relative version of Grothendieck fibration. As an application, in this wider setting, by using the notion of augmented monad, we introduce a notion of combinatorial rank that, among other things, is expected to give some hints on the length of the monadic decomposition.

The purpose of this paper is to study Rota-Baxter operators for BiHom-associative algebras. Moreover, we introduce and discuss the properties of the notions of BiHom-(tri)dendriform algebra, BiHom-Zinbiel algebra and BiHom-quadri-algebra. We construct the free Rota-Baxter BiHom-associative algebra and present some observations about categories and functors related to Rota-Baxter structures.

Motivated by an example related to the tensor algebra, a stronger version of the notion of separable functor, called heavily separable (h-separable for short), was introduced and investigated in [1] . Here we study h-coseparable coalgebras in monoidal categories with special concern with the monoidal category T A ♯ of right transfer morphisms through an algebra A in a monoidal category. We characterize the h-separability of the forgetful functor from the category of entwined modules associated to a cowreath to the base category using suitable Casimir morphisms. Even if there are non trivial examples of h-coseparable coalgebras over a field [1, Theorem 4.4] , here we provide non trivial examples of h-coseparable coalgebras in the monoidal category T A ⊗ H o p # where H = H 4 is the Sweedler 4-dimensional Hopf algebra over a field k and A = C l ( α , β , γ ) the Clifford algebra.

We contribute to the study of Rota–Baxter operators on types of algebras other than associative and Lie algebras. If A is an algebra of a certain type and R is a Rota–Baxter operator on A, one can define a new multiplication on A by means of R and the previous multiplication and ask under what circumstances the new algebra is of the same type as A. Our first main result deals with such a situation in the case of BiHom-Lie algebras. Our second main result is a BiHom analogue of Aguiar’s theorem that shows how to obtain a pre-Lie algebra from a Rota–Baxter operator of weight zero on a Lie algebra. The BiHom analogue does not work for BiHom-Lie algebras, but for a new concept we introduce here, called left BiHom-Lie algebra, at which we arrived by defining first the BiHom version of Leibniz algebras.

We study BiHom-Novikov-Poisson algebras, which are twisted generalizations of Novikov-Poisson algebras and Hom-Novikov-Poisson algebras, and find that BiHom-Novikov-Poisson algebras are closed under tensor products and several kinds of perturbations. Necessary and sufficient conditions are given under which BiHom-Novikov-Poisson algebras give rise to BiHom-Poisson algebras., 21 pages

In this paper monoidal Hom-Lie algebras, Lie color algebras, Lie superalgebras and other type of generalized Lie algebras are recovered by means of an iterated construction, known as monadic decomposition of functors, which is based on Eilenberg–Moore categories. To this aim we introduce the notion of Milnor–Moore category as a monoidal category for which a Milnor–Moore type Theorem holds. We also show how to lift the property of being a Milnor–Moore category whenever a suitable monoidal functor is given and we apply this technique to provide examples.

We introduce the concept of a $\{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\}$ -Rota–Baxter operator, as a twisted version of a Rota–Baxter operator of weight zero. We show how to obtain a certain $\{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\}$ -Rota–Baxter operator from a solution of the associative (Bi)Hom-Yang–Baxter equation, and, in a compatible way, a Hom-pre-Lie algebra from an infinitesimal Hom-bialgebra.

We introduce and study infinitesimal BiHom-bialgebras, BiHom-Novikov algebras, BiHom-Novikov-Poisson algebras, and find some relations among these concepts. Our main result is to show how to obtain a left BiHom-pre-Lie algebra from an infinitesimal BiHom-bialgebra, Comment: 22 pages. v2, final version, we modified the definition of a BiHom-Novikov-Poisson algebra in section 3

Prompted by an example related to the tensor algebra, we introduce and investigate a stronger version of the notion of separable functor that we call heavily separable. We test this notion on several functors traditionally connected to the study of separability.

In our paper we heavily used the result that two constituent bialgebroids in a Hopf algebroid possess isomorphic comodule categories. This statement was based on [T. Brzezinski, A note on coring extensions, Ann. Univ. Ferrara Sez. VII Sci. Mat. LI (2005) 15–27. A corrected version is available at http://arxiv.org/abs/math/0410020v3 , Theorem 2.6], whose proof turned out to contain an unjustified step. Here we prove the main results in our paper without using [T. Brzezinski, A note on coring extensions, Ann. Univ. Ferrara Sez. VII Sci. Mat. LI (2005) 15–27. A corrected version is available at http://arxiv.org/abs/math/0410020v3 , Theorem 2.6] and the derived isomorphism of comodule categories.