Rigidification of quasi-categories
Quasi-categories are simplicial sets where all inner horns admit a filler (whereas Kan complexes admit fillers for all horns). They provide a model for \((\infty, 1)\)-categories where one thinks of the vertices as objects and 1-simplices as morphisms. The inner horn fillers allows one to compose morphisms, but this is defined only up to homotopy. Lurie defines in [\textit{J. Lurie}, Higher topos theory. Annals of Mathematics Studies 170. Princeton, NJ: Princeton University Press. (2009; Zbl 1175.18001)] a rigidification functor taking a quasi-category \(K\) to \(\mathfrak C(K)\), a category enriched over simplicial sets (a ``simplicial category''). The morphisms between each pair of objects form a simplicial set and composition is well-defined and associative. This functor is in fact left adjoint to the coherent nerve and part of a Quillen equivalence between the category of simplicial sets equipped with the Joyal model structure, where fibrant objects are quasi-categories, and the category of simplicial categories equipped with Bergner's model structure. The simplicial category \(\mathfrak C(\Delta^n)\) can be described or defined by nerves of posets and, since \(\mathfrak C\) is a left adjoint, \(\mathfrak C(S)\) is the colimit, taken over all simplices of the simplicial set \(S\), of simplicial categories \(\mathfrak C(\Delta^n)\). This formula does not give much concrete information because the colimit takes place in \(s\mathcal C at\)\dots The aim of the article is to provide such information by finding new models for the mapping spaces. The first observation is that the rigidification has a nerve description for necklaces, ordered wedges of \(\Delta^{n_i}\)'s where the final vertex of \(\Delta^{n_i}\) is glued to the initial one of \(\Delta^{n_{i+1}}\). The colimit over all simplices of \(S\) can thus be replaced by a colimit taken over certain morphisms from necklaces. A careful analysis of this colimit allows the authors to give a very explicit description of the mapping space \(\mathfrak C(K) (a, b)\) for vertices \(a, b\). This description is difficult to use to identify the homotopy type of these mapping spaces, but here comes the twist! The above colimit has the same homotopy type as the homotopy colimit taken over the necklaces. Even better, this homotopy colimit is identified with the nerve of a category of necklaces, more precisely, \(\mathfrak C(S)(a, b)\) is weakly equivalent to the nerve of the category of necklaces \(T\) together with a map to \(S\) sending the initial vertex of the first ``bead'' \(\Delta^{n_1}\) to \(a\) and the final vertex of the last bead to~\(b\). The choice of necklaces is flexible and, by choosing instead products of necklaces, a short proof of the equivalence of simplicial categories \(\mathfrak C(X \times Y) \rightarrow \mathfrak C(X) \times \mathfrak C(Y)\) is obtained. In the sequel [Algebr. Geom. Topol. 11, No. 1, 263--325 (2011; Zbl 1214.55013)] to this paper, these techniques are used to give a new proof of the Quillen equivalence between quasi-categories and simplicial categories.
scientific article