推出范畴的上核

讨论推出与推出之间的关系,定义了推出态射,在此基础上建立了推出范畴影l^＊．影l是以范畴留中的推出为对象,推出态射为态射构成的范畴．并进一步证明了推出范畴中上核存在的条件．     

s-幂等态射与幂等态射

给出了s-幂等态射的定义,在Abel范畴中讨论了s－幂等态射的定义,在Ａbel范围中讨论了s-幂等态射与幂等态射的关系,这些结果应用到p-除环上的矩阵范畴,得到p-除环上矩阵的相应结论。     

具有广义分解的态射幂的广义逆

研究了具有广义分解的态射幂的Moore-Penrose逆的存在条件及其表达式，给出群逆的表达式，并得到了态射幂的Moore-Penrose与群逆之间的关系，推广了具有泛分解的广义逆的相应的结果。     

Facets of Descent, II

Methods of internal-category theory are applied to show that the split epimorphisms in a category C are exactly the morphisms which are effective for descent with respect to any fibration over C (or to any C-indexed category). In the same context, composition-cancellation rules for effective descent morphisms are established and being applied to (suitably defined) locally-split epimorphisms.     

Prequantale态射及其性质

给出了Prequantale态射的定义,证明了Prequantale中的特殊元在Prequantale态射的右伴随下不变的性质.找到了Prequantale中态射与蕴涵运算的关系,得到了Prequantale上的一个闭映射是态射的充要条件.     

Abel范畴中的态射方程

讨论了以下问题：１）方程αχβ＝γ在其有解时的性质；２）方程组｛αχβ＝γ，σχτ＝ρ有解的充要条件及通解表达式；３）方程αψβ＋σψτ＝γ有解的充要条件及通解表达式。     

Liberal理论态射的一些性质

该文构作了一个ｌｉｂｅｒａｌ理论范Ｔｈ讨论了ｌｉｂｅｒａｌ理论态射的一些性质，得到了Ｐｅｒｓｉｓｔｅｎｔ函子了的一个刻划定理，同时也得到了Ｆ－ｆｒｅｅ，Ｆ－ｇｅｎｅｒａｔｅｄ，Ｆ－ｐｒｉｍｅ模型的一些结果。     

Components of the Fundamental Category II

In this article we carry on the study of the fundamental category (Goubault and Raussen, Dihomotopy as a tool in state space analysis. In: Rajsbaum, S. (ed.) LATIN 2002: Theoretical Informatics. Lecture Notes in Computer Science, vol. 2286, Cancun, Mexico, pp. 16–37, Springer, Berlin Heidelberg New York, 2002; Goubault, Homology, Homotopy Appl., 5(2): 95–136, 2003) of a partially ordered topological space (Nachbin, Topology and Order, Van Nostrand, Princeton, 1965; Johnstone, Stone Spaces, Cambridge University Press, Cambridge, MA, 1982), as arising in e.g. concurrency theory (Fajstrup et al., Theor. Comp. Sci. 357: 241–278, 2006), initiated in (Fajstrup et al., APCS, 12(1): 81–108, 2004). The “algebra” of dipaths modulo dihomotopy (the fundamental category) of such a po-space is essentially finite in a number of situations. We give new definitions of the component category that are more tractable than the one of Fajstrup et al. (APCS, 12(1): 81–108, 2004), as well as give definitions of future and past component categories, related to the past and future models of Grandis (Theory Appl. Categ., 15(4): 95–146, 2005). The component category is defined as a category of fractions, but it can be shown to be equivalent to a quotient category, much easier to portray. A van Kampen theorem is known to be available on fundamental categories (Grandis, Cahiers Topologie Géom. Différentielle Catég., 44: 281–316, 2003; Goubault, Homology, Homotopy Appl., 5(2): 95–136, 2003), we show in this paper a similar theorem for component categories (conjectured in Fajstrup et al. (APCS, 12(1): 81–108, 2004). This proves useful for inductively computing the component category in some circumstances, for instance, in the case of simple PV mutual exclusion models (Goubault and Haucourt, A practical application of geometric semantics to static analysis of concurrent programs. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005 – Concurrency Theory: 16th International Conference, San Francisco, USA, August 23–26. Lecture Notes in Computer Science, vol. 3653, pp. 503–517, Springer, Berlin Heidelberg New York, 2005), corresponding to partially ordered subspaces of R ⁿ minus isothetic hyperrectangles. In this last case again, we conjecture (and give some hints) that component categories enjoy some nice adjunction relations directly with the fundamental category.