Homotopy weighted colimits

TL;DR

This paper defines homotopy weighted colimits in a model category setting, describes their decomposition via homotopy tensors and colimits, and explores their behavior in different contexts like simplicial sets and dg-categories.

Contribution

It introduces a homotopy version of weighted colimits, providing a decomposition method and analyzing their properties in various model categories.

Findings

Homotopy weighted colimits can be decomposed into homotopy tensors and colimits.

In simplicial sets, weighted homotopy colimits are equivalent to conical homotopy colimits.

In dg-categories, desuspension cannot be obtained from conical homotopy colimits.

Abstract

Let V be a cofibrantly generated closed symmetric monoidal model category and M a model V-category. We say that a weighted colimit W*D of a diagram D weighted by W is a homotopy weighted colimit if the diagram D is pointwise cofibrant and the weight W is cofibrant in the projective model structure on [C^op,V]. We then proceed to describe such homotopy weighted colimits through homotopy tensors and ordinary (conical) homotopy colimits. This is a homotopy version of the well known isomorphism W*D=\int^C(W\tensor D). After proving this homotopy decomposition in general we study in some detail a few special cases. For simplicial sets tensors may be replaced up to weak equivalence by conical homotopy colimits and thus the weighted homotopy colimits have no added value. The situation is completely different for model dg-categories where the desuspension cannot be constructed from conical homotopy colimits. In the last section we characterize those V-functors inducing a Quillen equivalence on the enriched presheaf categories.