Fibrewise convergence and exponential laws

Abstract

We show that the category $\mathbf{Conv_{B}}$ of convergence spaces over $B$ is a convenient category for any $B \in \mathbf{Conv}$ . It is shown that without any condition on spaces the category $\mathbf{Conv_{B}}$ and the category $\mathbf{Conv_{B}^{B}}$ of sectioned convergence spaces over $B$ hold various exponential laws in a natural way. In $\mathbf{Conv_{B}}$, we can construct exponential object in terms of function spaces. Our fibrewise mapping space structure generalizes the fibrewise compact-open topology in some case.