Category of topological spaces

The category Top has topological spaces as objects and continuous maps as morphisms. This is a category because the composition of two continuous maps is again continuous.

(Some authors use the name Top for the category with topological manifolds as objects and continuous maps as morphisms; Wikipedia follows the convention given above.)

The monomorphisms in Top are the injective continuous maps, the epimorphisms are the surjective continuous maps, and the isomorphisms are the homeomorphisms.

The empty set (considered as a topological space) is the initial object of Top; any singleton topological space is a terminal object. There are thus no zero objects in Top. Indeed, there aren't even zero morphisms in Top, and in particular the category is not preadditive.

The product in Top is given by the product topology on the cartesian product. Using the subspace topology for subsets of those products, one can then show that Top is a complete category. The coproduct is given by the disjoint union of topological spaces. By using the quotient topology, one can then show that Top is also cocomplete.

Top is not cartesian closed (and therefore also not a topos) since it does not have exponential objects.

We have a "forgetful" functor Top → Set which assigns to each topological space the underlying set, and to each continuous map the underlying function. This functor is faithful, and therefore Top is a concrete category. The forgetful functor has a left adjoint (which equips a given set with the discrete topology) and a right adjoint (which equips a given set with the trivial topology).