continuous functions between fixed spaces). One can likewise define a homotopy of loops keeping the base point fixed. The relation of being homotopic is an equivalence relation on paths in a topological space. The equivalence class of a path under this relation is called the homotopy class of often denoted Path composition One can compose paths in a topological space in the following manner. Suppose is a path from to and is a path from to . The path is defined as the path obtained by first traversing and then traversing : Clearly path composition is only defined when the terminal point of coincides with the initial point of If one considers all loops based at a point then path composition is a binary operation. Path composition, whenever defined, is not associative due to the difference in parametrization. However it associative up to path-homotopy. That is, Path composition defines a group structure on the set of homotopy classes of loops based at a point in The resultant group is called the fundamental group of based at usually denoted In situations calling for associativity of path composition "on the nose," a path in may instead be defined as a continuous map from an interval to for any real (Such a path is called a Moore path.) A path of this kind has a length defined as Path composition is then defined as before with the following modification: Whereas with the previous definition, , and all have length (the length of the domain of the map), this definition makes What made associativity fail for the previous definition is that although and have the same length, namely the midpoint of occurred between and whereas the midpoint of occurred between and . With this modified definition and have the same length, namely and the same midpoint, found at in both and ; more generally they have the same parametrization throughout. Fundamental groupoid There is a categorical picture of paths which is sometimes useful. Any topological space gives rise to a category where the objects are the points of and the morphisms are the homotopy classes of paths. Since any morphism in this category is an isomorphism this category is a groupoid, called the fundamental groupoid of Loops in this category are the endomorphisms (all of which are actually automorphisms). The automorphism group of a point in is just the fundamental group based at .