Homotopy coproducts can be defined in any relative category, just like homotopy colimits, but practical computations are typically carried out in presence of additional structures such as model structures.

Computation

In any model category, the homotopy coproduct of a family of objects $\{A_i\}_{i\in I}$ can be computed by cofibrantly replacing? each $A_i$ and computing the (ordinary) coproduct of the resulting family $\{QA_i\}_{i\in I}$ of cofibrant replacements.

If weak equivalences are closed under small coproducts, then homotopy coproducts can be computed as ordinary coproducts, because the map $$\coprod_{i\in I}QA_i\to \coprod_{i\in I}A_i$$ is a weak equivalence.