Model $\infty$-categories I: some pleasant properties of the $\infty$-category of simplicial spaces
Abstract
Both simplicial sets and simplicial spaces are used pervasively in homotopy theory as presentations of spaces, where in both cases we extract the "underlying space" by taking geometric realization. We have a good handle on the category of simplicial sets in this capacity; this is due to the existence of a suitable model structure thereon, which is particularly convenient to work with since it enjoys the technical properties of being *proper* and of being *cofibrantly generated*. This paper is devoted to showing that, if one is willing to work -categorically, then one can manipulate simplicial spaces exactly as one manipulates simplicial sets. Precisely, this takes the form of a proper, cofibrantly generated model structure on the *-category* of simplicial spaces, the definition of which we also introduce here.
Keywords
Cite
@article{arxiv.1412.8411,
title = {Model $\infty$-categories I: some pleasant properties of the $\infty$-category of simplicial spaces},
author = {Aaron Mazel-Gee},
journal= {arXiv preprint arXiv:1412.8411},
year = {2015}
}
Comments
Changes in v2: added appendix of notation (for entire sequence of papers); greatly expanded appendix of conventions, especially with regards to working model-independently (i.e. "in the $\infty$-category of $\infty$-categories"); clarified relationship with existing literature. (Note that some numberings have changed.)