On kernels of cellular covers

On cellular covers with free kernels

Recall that a homomorphism of $R$-modules $π: G\to H$ is called a {\it cellular cover} over $H$ if $π$ induces an isomorphism $π_*: \Hom_R(G,G)\cong \Hom_R(G,H),$ where $π_*(φ)= πφ$ for each $φ\in \Hom_R(G,G)$ (where maps are acting on the left). In this paper we show that every cotorsion-free module $K$ of finite rank can be realized as the kernel of a cellular cover of some cotorsion-free module of rank 2. In particular, every free abelian group of any finite rank appears then as the kernel of a cellular cover of a cotorsion-free abelian group of rank 2. This situation is best possible in th…

Cellular covers of local groups

We prove that, in the category of groups, the composition of a cellularization and a localization functor need not be idempotent. This provides a negative answer to a question of Emmanuel Dror Farjoun.

Cellular covers of cotorsion-free modules

In this paper we improve recent results dealing with cellular covers of $R$-modules. Cellular covers (sometimes called co-localizations) come up in the context of homotopical localization of topological spaces. They are related to idempotent cotriples, idempotent comonads or coreflectors in category theory. Recall that a homomorphism of $R$-modules $π: G\to H$ is called a {\it cellular cover} over $H$ if $π$ induces an isomorphism $π_*: \Hom_R(G,G)\cong \Hom_R(G,H),$ where $π_*(ϕ)= πϕ$ for each $ϕ\in \Hom_R(G,G)$ (where maps are acting on the left). On the one hand, we show that every cotorsio…

论细胞自动机

On Cellular Automata

本研究将细胞自动机推广至群宇宙，提出φ-细胞自动机，建立广义Curtis-Hedlund定理与线性理论，并定义覆盖映射，在循环图上推导商覆盖结果，深化代数与拓扑理解。

Linear cellular automata, duality and sofic groups

We produce for arbitrary non-amenable group $G$ and field $K$ a non-pre-injective, surjective linear cellular automaton. This answers positively Open Problem (OP-14) in Ceccherini-Silberstein and Coornaert's monograph "Cellular Automata and Groups". We also reprove in a direct manner, for linear cellular automata, the result by Capobianco, Kari and Taati that cellular automata over sofic groups are injective if and only if they are post-surjective. These results come from considerations related to matrices over group rings: we prove that a matrix's kernel and the image of its adjoint are mutua…