An action of $A$ on $X$ is a map $F colon A imes X o X$ such that $Fert_X = mathrm{id} colon X o X$. The restriction $Fert_A colon A o X$ of an action is called a cyclic map. Special cases of these notions include group actions and the Gottlieb groups of a space, each of which has been studied extensively. We prove some general results about actions and their Eckmann-Hilton duals. For instance, we classify the actions on an $H$-space that are compatible with the $H$-structure. As a corollary, we prove that if any two actions $F$ and $F'$ of $A$ on $X$ have cyclic maps $f$ and $f'$ with $Omega f = Omega f'$, then $Omega F$ and $Omega F'$ give the same action of $Omega A$ on $Omega X$. We introduce a new notion of the category of a map $g$ and prove that $g$ is cocyclic if and only if the category is less than or equal to $1$. From this we conclude that if $g$ is cocyclic, then the Berstein-Ganea category of $g$ is $le 1$. We also briefly discuss the relationship between a map being cyclic and its cocategory being $le 1$.
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机译:$ A $对$ X $的操作是映射$ F 冒号A 倍X 到X $,使得$ F vert_X = mathrm {id} 冒号X 到X $。一个动作对X $的限制$ F vert_A 冒号A 被称为循环映射。这些概念的特殊情况包括群体行为和空间的Gottlieb群体,每一个都经过了广泛的研究。我们证明了有关动作及其Eckmann-Hilton对偶的一些一般结果。例如,我们对与$ H $结构兼容的$ H $空间上的动作进行分类。作为推论,我们证明如果$ X $上$ A $的两个动作$ F $和$ F'$具有带有$ Omega f = Omega f'$的循环映射$ f $和$ f'$,然后$ Omega F $和$ Omega F'$对$ Omega X $给出$ Omega A $相同的作用。我们引入了映射$ g $的类别的新概念,并证明并且仅当类别小于或等于$ 1 $时,$ g $是同环的。由此得出的结论是,如果$ g $是同环的,则$ g $的Berstein-Ganea类别为$ le 1 $。我们还简要地讨论了循环图和其共分类是$ le 1 $之间的关系。
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