nLab
homotopy product

Idea

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

Computation

In any model category, the homotopy product of a family of objects $\{A_i\}_{i\in I}$ can be computed by fibrantly replacing each $A_i$ and computing the (ordinary) product of the resulting family $\{RA_i\}_{i\in I}$ of fibrant replacements.

Examples

In simplicial sets with simplicial weak equivalences, finite homotopy products can be computed by taking their (ordinary) product, because finite products preserve weak equivalences. More generally, homotopy products can be computed by applying Kan’s Ex^∞ functor to each simplicial set and taking their (ordinary) product.

In topological spaces with weak homotopy equivalences, homotopy products can be computed as ordinary products. The same applies to the relative category of chain complexes of abelian groups and quasi-isomorphisms (Grothendieck 1957, section 1.5).

Related notions

• product

• homotopy limit

• homotopy pullback

• homotopy equalizer

• homotopy totalization

References

• Alexander Grothendieck, Sur quelques points d'algèbre homologique, Tôhoku Math. J. vol 9, n.2, 3, 1957.