Related papers: D\'ecalage and Kan's simplicial loop group functor
Given a simplicial group G, there are two known classifying simplicial set constructions, the Kan classifying simplicial set Wbar G and Diag N G, where N denotes the dimensionwise nerve. They are known to be weakly homotopy equivalent. We…
We study the simplicial coalgebra of chains on a simplicial set with respect to three notions of weak equivalence. To this end, we construct three model structures on the category of reduced simplicial sets for any commutative ring R. The…
The simplicial extension of any functor from Sets to Sets which commutes with directed colimits takes weak equivalences to weak equivalences. The goal of the present paper is construct a framework which can be used to proof results of this…
We extend Goodwillie's classification of finitary linear functors to arbitrary small functors. That is we show that every small linear simplicial functor from spectra to simplicial sets is weakly equivalent to a filtered colimit of…
The diagonal lemma asserts that if a map of bisimplicial sets is a levelwise weak equivalence in the Kan-Quillen model structure, then it induces a weak equivalence of the diagonal simplicial sets. In this short note, we observe that the…
By careful analysis of the comparison map from a simplicial set to its image under Kan's ex-infinity functor we obtain a new and combinatorial proof that it is a weak homotopy equivalence. Moreover, we obtain a presentation of it as a…
We prove that the classification diagram functor from the category of marked simplicial sets to the category of bisimplicial sets carries cartesian equivalences to Rezk equivalences. As a corollary, we obtain Mazel-Gee's theorem on…
We establish an explicit comparison between two constructions in homotopy theory: the left adjoint of the homotopy coherent nerve functor, also known as the rigidification functor, and the Kan loop groupoid functor. This is achieved by…
In this note we show that a semisimplicial set with the weak Kan condition admits a simplicial structure, provided any object allows an idempotent self-equivalence. Moreover, any two choices of simplicial structures give rise to equivalent…
We develop an alternative to the May-Thomason construction used to compare operad based infinite loop machines to that of Segal, which relies on weak products. Our construction has the advantage that it can be carried out in $Cat$, whereas…
If all objects of a simplicial combinatorial model category \cat A are cofibrant, then there exists the homotopy model structure on the category of small functors $\sS^{\cat A}$, where the fibrant objects are homotopy functors, i.e.,…
We will construct an algebraic weak factorisation system on the category of 01 substitution sets such that the R-algebras are precisely the Kan fibrations together with a choice of Kan filling operation. The proof is based on Garner's small…
A simplicial set is said to be non-singular if the representing map of each non-degenerate simplex is degreewise injective. The inclusion into the category of simplicial sets, of the full subcategory whose objects are the non-singular…
We consider the d\'ecalage construction $\operatorname{Dec}$ and its right adjoint $T$. These functors are induced on the category of simplicial objects valued in any bicomplete category $\mathcal{C}$ by the ordinal sum. We identify $T…
We prove the following generalization of a classical result of Adams: for any pointed and connected topological space $(X,b)$, that is not necessarily simply connected, the cobar construction of the differential graded (dg) coalgebra of…
Generalizing a result of Dwyer and Kan for simplicial categories, we characterize the morphisms of multi-sorted simplicial algebraic theories and simplicial coloured operads which induce a Quillen equivalence between the corresponding…
Associated to each small category $C$, there is a category of $C$-shaped diagrams of simplicial sets and an $\infty$-category of $NC$-shaped homotopy coherent diagrams of spaces. We present a functor which exhibits the latter as the…
When one studies the structure (e.g. graded ideals, graded subspaces, radicals, ...) or graded polynomial identities of graded algebras, the grading group itself does not play an important role, but can be replaced by any other group that…
We discuss Lurie's (derived) bar and cobar constructions, the classical ones for simplicial groups and sets (due to Eilenberg-MacLane and Kan), and the classical ones for differential graded (co)algebras (due to Eilenberg-MacLane and Adams)…
For any 1-reduced simplicial set $K$ we define a canonical, coassociative coproduct on $\Om C(K)$, the cobar construction applied to the normalized, integral chains on $K$, such that any canonical quasi-isomorphism of chain algebras from…