S-系的覆盖、包络与相对同调代数

设S是幺半群, SF,WPF和CP分别代表强平坦，弱拉回平坦以及满足条件(P)的S-系的类。2008年，Mahmoudi和Renshaw开始了幺半群上S-系的平坦覆盖的研究，他们是通过余本质的满同态来定义覆盖的概念。到了2012年，Baily和 Renshaw开始将Enochs在模范畴中所研究的覆盖思想运用于幺半群上的S-系范畴中。基于Baily 和 Renshaw所定义的新的覆盖的概念，本项目将考虑关于同构和有向极限封闭的S-系的分类问题；给出每一个循环右S-系具有SF(WPF,CP)-覆盖的充要条件；研究所有右S-系具有SF(WPF,CP)-覆盖的幺半群的刻画。假设X表示任意S-系的类, 我们将定义和研究S-系的X-包络，尽量将Enochs的工作与幺半群上的S-系范畴紧密结合，重点考虑内射，弱内射和P-内射性。我们的工作将继续回答S-系理论中的新的公开问题。

Let S be a monoid，SF(WPF,CP) the classes of S-acts which has strongly flat(weakly pullback flat, condition (P)) property respectively. In 2008 Mahmoudi and Renshaw initiate a study of flat S-covers of S-acts over monoids, and their definition of cover concern coessential epimorphisms. In 2012 Baily and Renshaw attempt to initiate the study of Enochs' notion of cover on module category to the category of acts over monoids. Based on Baily and Renshaw's new notion of covers of S-acts, this program will classify those classes of S-acts which are closed under isomorphisms and directed colimts, give the sufficient and necessary conditions that every cyclic right S-act has a SF(WPF,CP)-cover, and investiagte characterizations of monoids over which every right S-act has a SF(WPF,CP )-cover. If X denotes any class of S-acts, we will introduce and investigate X-envelopes of S-acts and attempt to emulate some of Enochs' work for the category of acts over monoids and concentrate on injective, weakly injective and P-injective properties. Our work will continute to answer some new open problems on covers and envelopes of S-acts.