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A category is given by two pieces of data: a class of objects and, for any two objects X and Y, a set of morphisms from X to Y. Morphisms are often depicted as arrows between those objects. In the case of a concrete category, X and Y are sets of some kind and a morphism f is a function from X to Y satisfying some condition; this example supplies the notation f: X -> Y. But not every category is concrete, so these aren't the only types of morphisms.

Some examples of morphisms are homomorphisms from the categories studied in universal algebra (such as those of groups, rings, and so on), continuous functions between topological spaces, elements of a group when the group is thought of a special kind of category, paths in a single topological space (which form a groupoid), functors between categories, and many more.

Variants and subclasses of morphism:

An epimorphism with a one-sided inverse is called a split epimorphism. An monomorphism with a one-sided inverse is called a split monomorphism.