homotopy weighted colimit


Limits and colimits

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



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.


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: