an object $x$ of a category $mathbf{c}$ with finite limits is called exponentiable if the functor $-times x:mathbf{c}rightarrow mathbf{c}$ has a right adjoint. there are many characterizations of the exponentiable  spaces in the category $mathbf{top}$ of topological spaces. here, we study the exponentiable objects in the category $mathbf{stop}$ of soft topological spaces which is a generalizati...

We develop foundations for the category theory of $\infty$-categories parametrized by a base $\infty$-category. Our main contribution is indexed homotopy limits and colimits, which specializes to $G$-colimits $G$ finite group when chosen be orbit $G$. apply this show that $G$-$\infty$-category $G$-spaces freely generated under contractible $G$-space, thereby affirming conjecture Mike Hill.

The present paper has been an attempt to investigate the general fuzzy automata on the basis of complete residuated lattice-valued ($L$-GFAs). The study has been chiefly inspired from the work by Mockor cite{15, 16, 17}. Regarding this, the categorical issue of $L$-GFAs has been studied in more details. The main issues addressed in this research include: (1) investigating the relationship betwe...