arXiv subjects
Publications and source records attributed to Vadim Leshkov.
We consider Lambek's invariants $\text{Ker}$ and $\text{Im}$ for commutative squares in Grandis homological categories. We prove that Nomura's null sequences exist in such categories and find sufficienct conditions for their exactness. We also prove the coincidence of left and right homology in Grandis homological categories.
We consider the Lambek invariants (introduced by Joachim Lambek in 1964) in the context of semiexact and homological categories in the sense of Grandis. We generalize the Lambek isomorphism theorem to semiexact and homological categories. Throughout the text, for completeness we give proofs of some lemmas which are known in the additive case for the case of semiexact and homological categories.
In the present work we describe the category $\mathsf{WC}_2$ of weighted 2-complexes and its subcategory $\mathsf{WC}_1$ of weighted graphs. Since a Coxeter group is defined by its Coxeter graph, the construction of Coxeter groups defines a functor from $\mathsf{WC}_1$ to the category of groups. We generalize the notion of a Coxeter group by extending the domain of the functor to the category $\mathsf{WC}_2$. It appears that the resulting functor generalizes the construction of Coxeter groups, Gauss pure braid groups $GVP_n$ (introduced by V. Bardakov, P. Bellingeri, and C. Damiani in 2015), $k$-free braid groups on $n$ strands $G_n^k$ (introduced by V. Manturov in 2015), and other quotients of Coxeter groups.