Weak cartesian properties of simplicial sets

TL;DR

This paper introduces weaker completeness conditions for simplicial sets, extending their characterization by relaxing the pullback property to weak pullbacks, with applications to quasicategories, compositories, gleaves, and bar constructions.

Contribution

It defines and analyzes weaker completeness conditions for simplicial sets, providing criteria and examples, and connects these to database theory and algebraic structures.

Findings

Some completeness conditions enable lifting against certain simplices.

Reduced criteria for checking properties via pushout squares in Δ.

Examples include quasicategories, compositories, gleaves, and bar constructions.

Abstract

Many special classes of simplicial sets, such as the nerves of categories or groupoids, the 2-Segal sets of Dyckerhoff and Kapranov, and the (discrete) decomposition spaces of G\'{a}lvez, Kock, and Tonks, are characterized by the property of sending certain commuting squares in the simplex category $\Delta$ to pullback squares of sets. We introduce weaker analogues of these properties called completeness conditions, which require squares in $\Delta$ to be sent to weak pullbacks of sets, defined similarly to pullback squares but without the uniqueness property of induced maps. We show that some of these completeness conditions provide a simplicial set with lifts against certain subsets of simplices first introduced in the theory of database design. We also provide reduced criteria for checking these properties using factorization results for pushouts squares in $\Delta$, which we characterize completely, along with several other classes of squares in $\Delta$. Examples of simplicial sets with completeness conditions include quasicategories, many of the compositories and gleaves of Flori and Fritz, and bar constructions for algebras of certain classes of monads. The latter is our motivating example.