Quasi-2-Segal sets

Segal spaces are a model of ``categories up to homotopy,'' defined as simplicial spaces satisfying a condition that encodes unique (up to homotopy) composition of morphisms. In the last decade, the study of simplicial spaces which satisfy a particular weaker algebraic condition was independently begun by both Dyckerhoff-Kapranov, who call them 2-Segal spaces, and Galvez-Kock-Tonks, who call them decomposition spaces. The idea of the 2-Segal condition is that composition is no longer necessarily unique or defined at all, but there is still have a kind of associativity. A key example of 2-Segal spaces is the output of Waldhausen's S-construction from algebraic K-theory, and there are many other examples from diverse areas of math, from geometry and homological algebra to probability and combinatorics.

Meanwhile, quasi-categories are an alternative model of up-to-homotopy categories whose theory has been massively developed in the past two decades, thanks largely due to Joyal and Lurie, and they have become vital tools in many areas of algebraic topology, algebraic geometry, and beyond. The ultimate goal of this thesis is to define a 2-Segal analogue of quasi-categories, which we call quasi-2-Segal sets, with the hope that it will allow one to harness the technology of quasi-categories in 2-Segal theory.

One of the fundamental results from quasi-categories is that there is an associated model structure on the category of simplicial sets, due to Joyal. We therefore set out to construct an analogous model structure for quasi-2-Segal sets. However, finding a model structure with a more general class of fibrant objects than a given model structure is a nontrivial and open-ended task. We devote a significant portion of this work to investigating how Cisinski's machinery allows us to construct model structures on the category of simplicial sets whose fibrant objects generalize quasi-categories.

Chapter 2 of this thesis is devoted to the general study of Cisinski model structures on the category of simplicial sets. We identify a lifting condition which captures the homotopical behavior of quasi-categories without the algebraic aspects and show that there is a model structure whose fibrant objects are precisely those which satisfy this condition. We also identify a localization of this model structure whose fibrant objects satisfy a ``special horn lifting'' property similar to the one satisfied by quasi-categories.

In Chapter 3, we study the minimal model structure on simplicial sets, which is the model structure with the smallest class of weak equivalences coming from Cisinski's theory. We show that the fibrant objects in the minimal model structure are characterized by a lifting condition with respect to maps which resemble the horn inclusions that define Kan complexes.

In Chapter 4, we define and study quasi-2-Segal sets, applying the work in the preceding chapters. We show that quasi-2-Segal sets enjoy a similar relationship with 2-Segal spaces as the one quasi-categories have with Segal spaces. In particular, we construct a model structure on the category of simplicial sets whose fibrant objects are the quasi-2-Segal sets which is Quillen equivalent to a model structure for complete 2-Segal spaces (where our notion of completeness comes from one of the equivalent characterizations of completeness for Segal spaces). We also prove a path space criterion, which says that a simplicial set is a quasi-2-Segal set if and only if its path spaces (also called d\'ecalage) are quasi-categories, as well as an edgewise subdivision criterion.