In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen equivalent to one another and to Rezk's complete Segal space model structure on the category of simplicial spaces. Here, we show that these results still hold if we instead use groupoid or "invertible" cases. Namely, we show that model structures on the categories of simplicial groupoids, Segal pregroupoids, and invertible simplicial spaces are all Quillen equivalent to one another and to the standard model structure on the category of spaces. We prove this result using two different approaches to invertible complete Segal spaces and Segal groupoids.
Deep Dive into Adding inverses to diagrams II: Invertible homotopy theories are spaces.
In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen equivalent to one another and to Rezk’s complete Segal space model structure on the category of simplicial spaces. Here, we show that these results still hold if we instead use groupoid or “invertible” cases. Namely, we show that model structures on the categories of simplicial groupoids, Segal pregroupoids, and invertible simplicial spaces are all Quillen equivalent to one another and to the standard model structure on the category of spaces. We prove this result using two different approaches to invertible complete Segal spaces and Segal groupoids.
The notion of homotopy theories as mathematical objects is becoming a useful tool in topology as more mathematical structures are being viewed from a homotopical or higher-categorical viewpoint. Currently, there are four known models for homotopy theories: simplicial categories, Segal categories, complete Segal spaces, and quasi-categories. There are corresponding model category structures for each; the first three were shown to be Quillen equivalent to each other by the author [7], and the fourth was shown to be equivalent to the first by Joyal [19]; explicit equivalences between the fourth and the other two are also given by Joyal and Tierney [21]. Each of these models has proved to be useful in different contexts. Simplicial categories are naturally models for homotopy theories, in that they arise naturally from model categories, or more generally from categories with weak equivalences, via Dwyer and Kan's simplicial localization techniques [11], [13]. For this reason, one important motivation for studying any of these models is to understand specific homotopy theories and relationships between them. Quasi-categories, on the other hand, are more clearly a generalization of categories and more suited to constructions that look like those appearing in category theory. In fact, both Joyal [20] and Lurie [22] have written extensively on extending category theory to quasi-category theory. The objects in all four models are often called (∞, 1)-categories, to indicate that they can be regarded as categories with n-morphisms for any n ≥ 1, but for which these n-morphisms are all invertible whenever n > 1.
In this current paper, we would like to show that the first three models can be restricted to the groupoid case without much difficulty. Such structures could be called (∞, 1)-groupoids, but they are really (∞, 0)-categories, since even the 1morphisms are invertible in this case. In fact, we go on to prove that these model structures are Quillen equivalent to the standard model structure on the category of simplicial sets (and therefore to the standard model structure on the category of topological spaces). It has been proposed by a number of people, beginning with Grothendieck [16], that ∞-groupoids, or (∞, 0)-categories, should be models for homotopy types of spaces, so this result can be seen as further evidence for this “homotopy hypothesis.” Many authors have proved results in this area, including Tamsamani [29], Berger [3], Cisinski [10], Paoli [24], Biedermann [8], and Barwick [2], and a nice overview is given by Baez [1].
We should further note here that this comparison actually encompasses an invertible version of the fourth model, that of quasi-categories, since a quasi-category with inverses is just a Kan complex, and the fibrant objects in the standard model category structure on the category of simplicial sets are precisely the Kan complexes.
Organization of the Paper. In section 2, we give a new proof of the existence of a model structure on the category of simplicial groupoids. In sections 3 and 4, we define invertible versions of complete Segal spaces and Segal categories using the category I∆ op rather than ∆ op as a means of encoding inverses. We prove the existence of appropriate model category structures as well. In section 5, we prove that these simplicial groupoid, Segal groupoid, and invertible complete Segal space model structures are Quillen equivalent to one another and to the standard model category structure on the category of simplicial sets. In section 6, we give an alternate approach to invertible versions of Segal categories and complete Segal spaces by changing the projection maps in the category ∆ op , and we again show that we have a zig-zag of Quillen equivalences between the resulting model categories.
We refer the reader to the previous paper [4] for our notations and conventions regarding simplicial objects and model categories.
Acknowledgments. I would like to thank André Joyal and Simona Paoli for discussions on the material in this paper, as well as the referee for suggestions for its improvement.
A simplicial category is a category C enriched over simplicial sets, or a category such that, for objects x and y of C, there is a simplicial set of morphisms Map C (x, y) between them.
Recall that the category of components π 0 C of a simplicial category C is the category with the same objects as C and such that Hom π0C (x, y) = π 0 Map C (x, y).
We use the following notion of equivalence of simplicial categories. Definition 2.1. [11, 2.4] A functor f : C → D between two simplicial categories is a Dwyer-Kan equivalence if it satisfies the following two conditions:
• (W1) for any objects x and y of C, the induced map
is a weak equivalence of simplicial sets, and
Dwyer and Kan proved in [12, 2.5] that there is a model structure on the category SGpd of small simplicial groupoids with the Dwyer-Kan equivalences as weak equivalences. In fact, they went on to show that t
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