Attended and Delivered a talk at the 33rd Summer Conference on Topology and its Applications, Western Kentucky University, USA from July 17-20, 2018
Some Categorical aspects of probabilistic convergence transformation groups
In [3], we introduced a notion of probabilistic convergence transformation group with the help of the notion of probabilistic convergence group [2], generalizing topological transformation groups as well as Park's convergence transformation groups [5], and obtained various natural examples. Since PCONV, the category of probabilistic convergence spaces [4], and PUCS, the category of probabilistic uniform convergence spaces [1] are Cartesian closed, each of these possesses nice function space structure. In this talk, as a continuation of our investigations into the category PCONVTGRP of probabilistic convergence transformation groups, we look at the actions of probabilistic convergence groups on the function spaces: C_c(S, T) and C_u(Y, T); where S is a probabilistic convergence space, T is a probabilistic convergence group, and Y is a probabilistic uniform space [1], given the fact that every probabilistic convergence group and probabilistic uniform space give rise to probabilistic uniform convergence space. Furthermore, we present among other results, a construction of a monad [6] in the category PCONVGRP×PCONV such that the corresponding category of algebras is isomorphic to the category PCONVTGRP, where PCONVGRP stands for the category of probabilistic convergence groups [2].

[1] T. M. G. Ahsanullah and G. Jäger, Probabilistic uniform convergence spaces redefined, Acta Math. Hugarica 146(2)(2015), 376-390.
[2] T. M. G. Ahsanullah and G. Jäger, Probabilistic uniformization and probabilistic metrization of probabilistic convergence groups, Math. Slovaca 67(3)(2017), 985-1000.
[3] T. M. G. Ahsanullah and G. Jäger, Probabilistic convergence transformation groups, Math. Slovaca, to appear.
[4] G. Jäger, A convergence theory for probabilistic metric spaces, Quaest. Math. 38(2015), 587-599.
[5] W. R. Park, Convergence structures on homeomorphism groups, Math. Ann. 199(1972), 45-54.
[6] S. Mac Lane, Categories for the Working Mathematicians, Springer, 1971.