# Weighted limits in an $(\infty,1)$-category

## TL;DR

This paper generalizes the concept of weighted limits to quasi-categories, providing a new framework that unifies classical and enriched category theory approaches, with consistency results in specific cases.

## Contribution

It introduces a notion of weighted limit in quasi-categories, extending classical and enriched limits, and establishes connections with existing theories in homotopy coherent and enriched contexts.

## Key findings

- Weighted limits are defined in quasi-categories via weighted cones.
- The new definition agrees with homotopy weighted limits in enriched categories.
- Comparison with existing approaches suggests consistency in complete, tensored, and cotensored quasi-categories.

## Abstract

We introduce the notion of weighted limit in an arbitrary quasi-category, suitably generalizing ordinary limits in a quasi-category, and classical weighted limits in an ordinary category. This is accomplished by generalizing Joyal's approach: we identify a meaningful construction for the quasi-category of weighted cones over a diagram in a quasi-category, whose terminal object is the weighted limit of the considered diagram. When the quasi-category arises as the homotopy coherent nerve of a category enriched over Kan complexes, we use techniques by Riehl-Verity to show that the weighted limit agrees with the homotopy weighted limit in the sense of enriched category theory, for which explicit constructions are available. When the quasi-category is complete, tensored and cotensored over the quasi-category of spaces, we discuss a possible comparison of our definition of weighted limit with the approach by Gepner-Haugseng-Nikolaus.