The Category MAP

In this paper we introduce a new category MAP of continuous maps and their morphisms as an extension of both the category TOPY (of continuous maps into a fixed space Y and their morphisms) and TOP (of topological spaces and continuous maps as morphisms). Several operations in MAP are introduced such as products, fibrewise products, inverse limits, sums, fibrewise sums and direct limits. Finally we take a look at compact (perfect) maps as an object in the category MAP.