In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipline.

Besides algebraic topology, the theory has also been used in other areas of mathematics such as:

In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated weak Hausdorff or a CW complex.

In the same vein as above, a "map" is a continuous function, possibly with some extra constraints.

Often, one works with a pointed space—that is, a space with a "distinguished point", called a basepoint. A pointed map is then a map which preserves basepoints; that is, it sends the basepoint of the domain to that of the codomain. In contrast, a free map is one which needn't preserve basepoints.

The Cartesian product of two pointed spaces ${\displaystyle X,Y}$ are not naturally pointed. A substitute is the smash product ${\displaystyle X\wedge Y}$ which is characterized by the adjoint relation

that is, a smash product is an analog of a tensor product in abstract algebra (see tensor-hom adjunction). Explicitly, ${\displaystyle X\wedge Y}$ is the quotient of ${\displaystyle X\times Y}$ by the wedge sum ${\displaystyle X\vee Y}$.

Let I denote the unit interval ${\displaystyle [0,1]}$. A map