# Lax limits of model categories

## TL;DR

This paper demonstrates that the lax limit of a diagram of simplicial combinatorial model categories, with the projective model structure, accurately models the lax limit of their underlying ∞-categories, extending to homotopy limits.

## Contribution

It establishes that the lax limit with the projective model structure presents the lax limit of underlying ∞-categories, even when the indexing category is simplicial.

## Key findings

- Lax limits of model categories model ∞-category lax limits.
- Results extend to homotopy limits and other intermediate limits.
- Applicable to diagrams indexed by simplicial categories.

## Abstract

For a diagram of simplicial combinatorial model categories, we show that the associated lax limit, endowed with the projective model structure, is a presentation of the lax limit of the underlying $\infty$-categories. Our approach can also allow for the indexing category to be simplicial, as long as the diagram factors through its homotopy category. Analogous results for the associated homotopy limit (and other intermediate limits) directly follow.