nLab homotopy coproduct

Idea

Homotopy coproducts are a special case of homotopy colimits, when the indexing diagram is a discrete category.

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.

Examples

In simplicial sets with simplicial weak equivalences, topological spaces with weak homotopy equivalences, and chain complexes with quasi-isomorphisms, homotopy coproducts can be computed as ordinary coproducts because weak equivalences are closed under small coproducts.

In commutative differential graded algebras with quasi-isomorphisms homotopy coproducts can be computed by resolving each object by a cofibrant commutative differential graded algebra?, and then computing their ordinary tensor product.

Related notions

• coproduct

• homotopy colimit

• homotopy pushout

• homotopy coequalizer

• homotopy realization