Limits of $(\infty, 1)$-categories with structure and their lax morphisms

TL;DR

This paper extends the understanding of limits in various models of $( olinebreak\infty, 1)$-categories with structure, showing that certain limits of lax morphisms exist under minimal conditions, generalizing prior results.

Contribution

It generalizes Riehl and Verity's limit completeness results to multiple models of $(\infty, 1)$-categories with structure, including limits of lax morphisms like inserters and equifiers.

Findings

Limits of lax morphisms exist under minimal structure-preserving conditions.

Generalization to various models of $(\infty, 1)$-categories.

Existence of $\ abla$-limits such as inserters and equifiers.

Abstract

Riehl and Verity have established that for a quasi-category $A$ that admits limits, and a homotopy coherent monad on $A$ which does not preserve limits, the Eilenberg-Moore object still admits limits; this can be interpreted as a completeness result involving lax morphisms. We generalise their result to different models for $(\infty, 1)$-categories, with an abundant variety of structures. For instance, $(\infty, 1)$-categories with limits, Cartesian fibrations between $(\infty, 1)$-categories, and adjunctions between $(\infty, 1)$-categories. In addition, we show that these $(\infty, 1)$-categories with structure in fact possess an important class of limits of lax morphisms, including $\infty$-categorical versions of inserters and equifiers, when only one morphism in the diagram is required to be structure-preserving. Our approach provides a minimal requirement and a transparent explanation for several kinds of limits of $(\infty, 1)$-categories and their lax morphisms to exist.