94 research outputs found
Higher quasi-categories vs higher Rezk spaces
We introduce a notion of n-quasi-categories as fibrant objects of a model
category structure on presheaves on Joyal's n-cell category \Theta_n. Our
definition comes from an idea of Cisinski and Joyal. However, we show that this
idea has to be slightly modified to get a reasonable notion. We construct two
Quillen equivalences between the model category of n-quasi-categories and the
model category of Rezk \Theta_n-spaces showing that n-quasi-categories are a
model for (\infty, n)-categories. For n = 1, we recover the two Quillen
equivalences defined by Joyal and Tierney between quasi-categories and complete
Segal spaces.Comment: 44 pages, v2: terminology changed (see Remark 5.27), Corollary 7.5
added, appendix A added, references added, v3: reorganization of Sections 5
and 6, more informal comments, new section characterizing strict n-categories
whose nerve is an n-quasi-category, numbering has change
A Quillen's Theorem A for strict -categories I: the simplicial proof
The aim of this paper is to prove a generalization of the famous Theorem A of
Quillen for strict -categories. This result is central to the homotopy
theory of strict -categories developed by the authors. The proof
presented here is of a simplicial nature and uses Steiner's theory of augmented
directed complexes. In a subsequent paper, we will prove the same result by
purely -categorical methods.Comment: 51 pages, in French, v2: extended introduction, journal versio
Hidden Symmetry of the Differential Calculus on the Quantum Matrix Space
A standard bicovariant differential calculus on a quantum matrix space is considered. The principal result of this work is in observing
that the is in fact a
-module differential algebra.Comment: 5 page
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