# nLab homotopy weighted colimit

Contents

### Context

#### Limits and colimits

limits and colimits

## (∞,1)-Categorical

### Model-categorical

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

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

# 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 $V$, 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.

## 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 09:15:33. See the history of this page for a list of all contributions to it.