arXiv subjects

Adnan H. Abdulwahid

Publications and source records attributed to Adnan H. Abdulwahid.

3 recordsLinked to original sources

arXiv2016

Cofree Objects in The Centralizer and The Center Categories

We study cocompleteness, co-wellpoweredness and generators in the centralizer category of an object or morphism in a monoidal category, and the center or the weak center of a monoidal category. We explicitly give some answers for when colimits, cocompleteness, co-wellpoweredness and generators in these monoidal categories can be inherited from their base monidal categories. Most importantly, we investigate cofree objects of comonoids in these monoidal categories.

math.CT↗

arXiv2016

$n$-representations of Quivers

Let $n \geq 2$. We introduce the notion of $n$-representations of quivers, and we explicitly provide concrete examples of $2$-representations of quivers. We establish the categories of $n$-representations and investigate kernels and cokernels in the categories of $n$-representations of quivers. Further, we construct them in terms of kernels and cokernels of morphisms in the usual categories of quiver representations. We show that every morphism in the categories of $n$-representations has a canonical decomposition. Most importantly, we prove that the categories of $n$-representations of quivers are $k$-linear abelian categories.

math.RT↗

arXiv2016

$n$-quivers and A Universal Investigation of $n$-representations of Quivers

We have two parallel goals of this paper. First, we investigate and construct cofree coalgebras over $n$-representations of quivers, limits and colimits of $n$-representations of quivers, and limits and colimits of coalgebras in the monoidal categories of $n$-representations of quivers. Second, we introduce a generalization for quivers and show that this generalization can be seen as essentially the same as $n$-representations of quivers. This is significantly important because it allows us to build a new quiver $\mathscr{Q}_{\!_{(\Qq_1, \Qq_2,..., \Qq_n)}}$, called $n$-quiver, from any given quivers $\Qq_1,\Qq_2,...,\Qq_n$, and enable us to identify each category $Rep_k(\Qq_j)$ of representations of a quiver $\Qq_j$ as a full subcategory of the category $Rep_k(\mathscr{Q}_{\!_{(\Qq_1, \Qq_2,..., \Qq_n)}})$ of representations of $\mathscr{Q}_{\!_{(\Qq_1, \Qq_2,..., \Qq_n)}}$ for every $j \in \{1,\,\,2,\,\,..., \,\,n\}$.

math.RT↗