nLab homotopy product

Idea

Homotopy products are Cartesian products in homotopy theory, hence are a special case of homotopy limits for the case that the 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) Cartesian product of the resulting family $\{\mathrm{R}A_i\}_{i\in I}$ of fibrant objects.

Examples

Example

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

Example

(homotopy products of topological space)
In topological spaces with weak homotopy equivalences, as in the classical model structure on topological spaces, every object is fibant, so that homotopy products can be computed as ordinary Cartesian products, which here means product spaces.

Example

The same applies to the relative category of chain complexes of abelian groups and quasi-isomorphisms (Grothendieck 1957, section 1.5),

One way to see this is that in the projective model structure on chain complexes (when it exists) again every object is a fibrant object.

Related notions

• Cartesian 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.