Top (category theory)

1. the category with topological spaces as objects and continuous maps as morphisms.
2. the category with topological manifolds as objects and continuous maps as morphisms. Compare Diff and PL. For a detailed study of Top in this sense (and its relations with the other categories of manifolds), see 

• Kirby, Robion C.; Siebenmann, Laurence C. (1977). Foundational Essays on Topological Manifolds. Smoothings, and Triangulations. Princeton, New Jersey: Princeton University Press. ISBN 0-691-08190-5.