Smash product of spectra ($D\wedge E$)

The derived smash product of spectra is the symmetric monoidal product on the stable homotopy category. It has unit sphere spectrum, preserves cofiber sequences of spectra and homotopy colimits of spectra in each variable, and satisfies ${\Sigma }^{\infty }X\wedge {\Sigma }^{\infty }Y\simeq {\Sigma }^{\infty }(X\wedge Y)$.

The internal function spectrum is characterized by $[C,F(A,E){]}_{\mathrm{st}}=[C\wedge A,E{]}_{\mathrm{st}}$. For a based space $K$, notation $F(K,E)$ abbreviates $F({\Sigma }^{\infty }K,E)$.

If $E$ is a ring spectrum, then $F(X_{+},E)$ is a ring spectrum. Its multiplication is adjoint to evaluating the two functions on the two copies supplied by the diagonal of $X$, then multiplying in $E$. Its unit is the constant unit function. Coassociativity of the diagonal and associativity in $E$ prove the associative identity; the counit and unit identities prove both unit laws.