相关论文: Realizability of Univalence: Modest Kan complexes

For a Kan complex with a vertex, we have the notion of its simplicial homotopy groups. In this paper, for a weak complicial set in the sense of Verity with a vertex, we construct monoids which are a generalization of simplicial homotopy…

It is well known that the underlying simplicial set of any simplicial group is a Kan complex. Roughly speaking, Kan complex is an infinite-dimensional analogue of groupoid, and the relation between groupoids and categories resembles that…

We show that the fibrant objects in the minimal model structure on the category of simplicial sets are characterized by a lifting condition with respect to maps which resemble the horn inclusions that define Kan complexes.

This article shows several new methods for proofs on Kan complexes while using them to give a compact introduction to the homotopy groups of these complexes. Then more advanced objects are studied starting with homology and the Hurewicz…

Ehrenborg, Govindaiah, Park, and Readdy recently introduced the van der Waerden complex, a pure simplicial complex whose facets correspond to arithmetic progressions. Using techniques from combinatorial commutative algebra, we classify when…

We present an accessible account of Voevodsky's construction of a univalent universe of Kan fibrations.

We present Voevodsky's construction of a model of univalent type theory in the category of simplicial sets. To this end, we first give a general technique for constructing categorical models of dependent type theory, using universes to…

We introduce the notion of an effective Kan fibration, a new mathematical structure that can be used to study simplicial homotopy theory. Our main motivation is to make simplicial homotopy theory suitable for homotopy type theory. Effective…

In Quillen's paper on rational homotopy theory, the category of 1-reduced simplicial sets is endowed with a family of model structures, the most prominent of which is the one in which the weak equivalences are the rational homotopy…

We construct a model structure on the category of ordered simplicial complexes, Quillen equivalent to the standard model structure on simplicial sets. This shows that simplicial complexes, which are fully combinatorial in nature, provide a…

We define and study the notion of a minimal Cohen-Macaulay simplicial complex. We prove that any Cohen-Macaulay complex is shelled over a minimal one in our sense, and we give sufficient conditions for a complex to be minimal…

An analogous construction of simplicial homotopy group for Kan complex can be applied to saturated complicial sets to give monoids. In this paper, we investigate how the construction of loop spaces of Kan complexes lifts to the complicial…

The category of dendroidal sets is an extension of that of simplicial sets, suitable for defining nerves of operads rather than just of categories. In this paper, we prove some basic properties of inner Kan complexes in the category of…

Univalent categories constitute a well-behaved and useful notion of category in univalent foundations. The notion of univalence has subsequently been generalized to bicategories and other structures in (higher) category theory. Here, we…

We explore an alternative definition of unit in a monoidal category originally due to Saavedra: a Saavedra unit is a cancellative idempotent (in a 1-categorical sense). This notion is more economical than the usual notion in terms of…

For a small involutive quantaloid $\mathcal{Q}$, it is shown that the category of separated complete $\mathcal{Q}$-categories and left adjoint $\mathcal{Q}$-functors is strictly monadic over the category of symmetric…

We study $d$-dimensional simplicial complexes that are PL embeddable in $\mathbb{R}^{d+1}$. It is shown that such a complex must satisfy a certain homological condition. The existence of this obstruction allows us to provide a systematic…

The purpose of this note is to point out that simplicial methods and the well-known Dold-Kan construction in simplicial homotopy theory can be fruitfully applied to convert link homology theories into homotopy theories. Dold and Kan prove…

Let X be a finite set of cardinality n. The Kalmanson complex K_n is the simplicial complex whose vertices are non-trivial X-splits, and whose facets are maximal circular split systems over X. In this paper we examine K_n from three…

The canonical map from the Kan subdivision of a product of finite simplicial sets to the product of the Kan subdivisions is a simple map, in the sense that its geometric realization has contractible point inverses.