# '22

36th Summer Topology Conference

July 18-22, 2022

University of Vienna, Department of Mathematics
Oskar-Morgenstern-Platz 1, 1090 Vienna, AUSTRIA

### "Monads and their Eilenberg-Moore categories of the categories of enriched lattice-valued convergence spaces"

#### Ahsanullah, T M G

In 2012, Jäger first pointed out that the category of convergence approach spaces, CAP is simultaneously a reflective and coreflective subcategory of the category of enriched lattice-valued convergence spaces, $L$-CONV; this stimulates much interest to look into various aspects of these categories. Following the category of convergence approach transformation monoids, as introduced by Colebunders et al., we proved that the category of convergence approach transformation groups, CAPTGRP is isomorphic to a reflective subcategory of the category of enriched lattice-valued convergence transformation groups, $L$-CONVTGRP. In this talk our motivations are to construct various monads and their associated algebras on three types of subcategories stemmed from (a) the category of enriched lattice-valued convergence spaces, $L$-CONV [Jäger, 2012; Orpen-Jäger, 2012]; (b) the category of convergence approach spaces, CAP [Lowen, 1997]; (c) the category of probabilistic convergence spaces under $t$-norm, $PCONV_t$ [Herrlich-Zhang, 1998]. Specifically, constructing a monad $\mathbb{F}$ on $\mathfrak{C}:= L$- CONVGRP$\times L$-CONV, where $L$-CONVGRP denotes the category of enriched lattice-valued convergence groups, we show that the category $L$-CONVTGRP over $\mathfrak{C}$ is isomorphic to $\mathfrak{C}^{\mathbb{F}}$ - the category of Eilenberg-Moore algebras. In this way, one can achieve other monads related to (b) and (c), respectively. Finally, we are interested to find possible relationship between the monads, and their corresponding Eilenberg-Moore categories.

« back