On lax limits in infinity categories

TL;DR

This paper introduces partially lax limits in infinity-categories, providing a unified formula that generalizes existing results for ordinary and lax limits, with applications to enriched categories and operads.

Contribution

It defines partially lax limits in infinity-categories and derives a general formula using the Grothendieck construction, extending prior work on limits.

Findings

Provides a formula for partially lax limits and colimits

Generalizes Lurie's formula for ordinary limits

Extends Gepner-Haugseng-Nikolaus's work for lax limits

Abstract

We introduce partially lax limits of infinity-categories, which interpolate between ordinary limits and lax limits. Most naturally occurring examples of lax limits are only partially lax; we give examples arising from enriched categories and operads. Our main result is a formula for partially lax limits and colimits in terms of the Grothendieck construction. This generalizes a formula of Lurie for ordinary limits and of Gepner-Haugseng-Nikolaus for fully lax limits.