nLab homotopy weighted colimit

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 $V$, a $V$-enriched model category $C$, and a small $V$-enriched category $J$.

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

We have a left Quillen bifunctor

$$V^{J^{op}} \times C^J \to C$$

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.

Related concept

• homotopy coend

• fat geometric realization

References

See Section 9.2 in

• Emily Riehl, Categorical Homotopy Theory, New Mathematical Monographs 24, Cambridge University Press, 2014.

and for simplicially based theories,

• Jean-Marc Cordier and Timothy Porter, Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1-54, (pdf)

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

Other references:

• Lukáš Vokřínek, Homotopy weighted colimits, arXiv:1201.2970.

• Nicola Gambino, Weighted limits in simplicial homotopy theory, Journal of Pure and Applied Algebra 214 7 (2010) 1193–1199 [doi:10.1016/j.jpaa.2009.10.006]

• Michael Shulman, Homotopy limits and colimits and enriched homotopy theory, arXiv:math/0610194.

• Sergey Arkhipov, Sebastian Ørsted, Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories (arXiv:1807.03266).