On a connection between fuzzy subgroups and $F$-inverse covers of inverse monoids

We define two categories, the category $\mathfrak{F}\mathfrak{G}$ of fuzzy subgroups, and $\mathfrak{F}\mathfrak{C}$ $F$-inverse covers inverse monoids, prove that fully embeds into $\mathfrak{F}\mathfrak{C}$. This shows that, at least from a categorical viewpoint, subgroups belong to standard mathematics as much they do one.