of the basepoint in under the quotient map. One can form the product of two pointed spaces as the topological product with serving as the basepoint. The coproduct in the category of pointed spaces is the , which can be thought of as the 'one-point union' of spaces. The smash product of two pointed spaces is essentially the quotient of the direct product and the wedge sum. We would like to say that the smash product turns the category of pointed spaces into a symmetric monoidal category with the pointed 0-sphere as the unit object, but this is false for general spaces: the associativity condition might fail. But it is true for some more restricted categories of spaces, such as compactly generated weak Hausdorff ones. The reduced suspension of a pointed space is (up to a homeomorphism) the smash product of and the pointed circle The reduced suspension is a functor from the category of pointed spaces to itself. This functor is left adjoint to the functor taking a pointed space to its loop space . See also References mathoverflow discussion on several base points and groupoids Topology Homotopy theory Categories in category theory Topological spaces