Related papers: On truncated quasi-categories

For a cofibrantly generated Quillen model category, we show that the cofibrant replacement functor constructed using the small object argument admits a cotriple structure. If all acyclic cofibrations are monomorphisms, the fibrant…

This expository article sets forth a self-contained and purely algebraic proof of a deep result of Quillen stating that the category of simplicial commutative algebras over a commutative ring is a model category. This is accomplished by…

We show that complete Segal spaces and Segal categories are Quillen equivalent to quasi-categories.

We develop a constructive model of homotopy type theory in a Quillen model category that classically presents the usual homotopy theory of spaces. Our model is based on presheaves over the cartesian cube category, a well-behaved…

This paper develops the foundations of a simplicial theory of weak omega-categories, which builds upon the insights originally expounded by Ross Street in his 1987 paper on oriented simplices. The resulting theory of weak complicial sets…

We prove that the category of dg-modules and dg-algebras in a Grothendieck quasi-abelian category are endowed with a Quillen model structure. This allows some flexibility in setting up a theory of derived algebraic geometry in the infinite…

We consider notions of metrized categories, and then approximate categorical structures defined by a function of three variables generalizing the notion of $2$-metric space. We prove an embedding theorem giving sufficient conditions for an…

In [Joyal] where the category $\Theta$ is first defined it is noted that the dimensional shift on $\Theta$ suggests an elegant presentation of the unreduced suspension on cellular sets. In this note we prove that the reduced suspension…

We show that every combinatorial model category can be obtained, up to Quillen equivalence, by localizing a model category of diagrams of simplicial sets. This says that any combinatorial model category can be built up from a category of…

It is proved that the category of simplicial complete bornological spaces over $\mathbb R$ carries a combinatorial monoidal model structure satisfying the monoid axiom. For any commutative monoid in this category the category of modules is…

We show that the category of simplicial sets is a co-reflective subcategory of the category of cubical sets with connections, with the inclusion given by a version of the straightening functor. We show that using the co-reflector, one can…

Diagrammatic sets admit a notion of internal equivalence in the sense of coinductive weak invertibility, with similar properties to its analogue in strict $\omega$-categories. We construct a model structure whose fibrant objects are…

We apply the Acyclicity Theorem of Hess, Kerdziorek, Riehl, and Shipley (recently corrected by Garner, Kedziorek, and Riehl) to establishing the existence of model category structure on categories of coalgebras over comonads arising from…

Much research has been done on structures equivalent to topological or simplicial groups. In this paper, we consider instead simplicial monoids. In particular, we show that the usual model category structure on the category of simplicial…

Adhesive and quasiadhesive categories provide a general framework for the study of algebraic graph rewriting systems. In a quasiadhesive category any two regular subobjects have a join which is again a regular subobject. Vice versa, if…

A quasi-schemoid is a small category whose morphisms are colored with appropriate combinatorial data. In this note, Mitchell's embedding theorem for a tame schemoid is established. The result allows us to give a cofibrantly generated model…

A category is called {\em split} if for every morphism $s\colon X\to Y$ there exists a morphism $t\colon Y\to X$ such that $s\circ t\circ s = s$. Let $C$ be a finite split category, let $k$ be a field of characteristic 0 and let $\alpha$ be…

We introduce and compare two approaches to equivariant homotopy theory in a topological or ordinary Quillen model category. For the topological model category of spaces, we generalize Piacenza's result that the categories of topological…

We prove that the quasicategories arising from models of Martin-L\"of type theory via simplicial localization are locally cartesian closed.

The study of topological quantum field theories increasingly relies upon concepts from higher-dimensional algebra such as n-categories and n-vector spaces. We review progress towards a definition of n-category suited for this purpose, and…