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A category consists of two things: a collection of Objects and, for each pair of Objects, a Morphism (sometimes called an ``arrow'') from one to another. An Object is some mathematical structure (e.g., a Group, Vector Space, or Differentiable Manifold) and a Morphism is a Map between two Objects. The Morphisms are then required to satisfy some fairly natural conditions; for instance, the Identity Map between any object and itself is always a Morphism, and the composition of two Morphisms (if defined) is always a Morphism.
One usually requires the Morphisms to preserve the mathematical structure of the objects. So if the objects are all groups, a good choice for a Morphism would be a group Homomorphism. Similarly, for vector spaces, one would choose linear maps, and for differentiable manifolds, one would choose differentiable maps.
In the category of Topological Spaces, homomorphisms are usually continuous maps between topological spaces. However, there are also other category structures having Topological Spaces as objects, but they are not nearly as important as the ``standard'' category of Topological Spaces and continuous maps.
See also Abelian Category, Allegory, Eilenberg-Steenrod Axioms, Groupoid, Holonomy, Logos, Monodromy, Topos
References
Freyd, P. J. and Scedrov, A. Categories, Allegories. Amsterdam, Netherlands: North-Holland, 1990.