Smash Product - Adjoint Relationship

Adjoint Relationship

Adjoint functors make the analogy between the tensor product and the smash product more precise. In the category of R-modules over a commutative ring R, the tensor functor (– ⊗_R A) is left adjoint to the internal Hom functor Hom(A,–) so that:

In the category of pointed spaces, the smash product plays the role of the tensor product. In particular, if A is locally compact Hausdorff then we have an adjunction

where Hom(A,Y) is the space of based continuous maps together with the compact-open topology.

In particular, taking A to be the unit circle S1, we see that the suspension functor Σ is left adjoint to the loop space functor Ω.