Topology and category for singular product spaces

For $\kappa$ a regular uncountable cardinal, the higher Baire and Cantor spaces ${}^{\kappa }\kappa$ and ${}^{\kappa }2$ (endowed with the $<\kappa$-box topology) have been relatively well-studied, but less is known about the case where $\kappa$ is singular. We will consider several spaces of functions and box topologies that could serve as higher Baire and Cantor spaces for singular cardinals. The ultimate focus of the article lies in studying cardinal characteristics of the ideal of $\kappa$-meagre subsets of these spaces.