nLab
homotopy weighted colimit

Contents

Context

Limits and colimits

(,1)(\infty,1)-Category theory

Contents

Idea

Homotopy weighted colimits (alias weighted homotopy colimits) are the analog of weighted colimits in homotopy theory.

In relative categories

... ...

In model categories

For the special case of model categories, we can define homotopy weighted colimits as follows.

Fix a monoidal model category VV, a VV-enriched model category CC, and a small VV-enriched category JJ.

For simplicity, assume all enriched hom objects of JJ are cofibrant. If this is not the case, we can first cofibrantly replace JJ in the Dwyer-Kan model structure on enriched categories.

We have a left Quillen bifunctor

V J op×C JCV^{J^{op}} \times C^J \to C

given by the ordinary weighted colimit functor.

The homotopy weighted colimit can then be defined as the left derived Quillen bifunctor of the weighted colimit functor.

References

See Section 9.2 in

and for simplicially based theories,

(That article uses the older terminology of ‘indexed colimits’ rather than the `weighted' one.)

Other references:

Last revised on September 18, 2021 at 05:15:33. See the history of this page for a list of all contributions to it.