Associativity Data in an (∞, 1)-category

A popular slogan is that (∞, 1)-categories (also called quasi-categories or ∞categories) sit somewhere between categories and spaces, combining some of the features of both. The analogy with spaces is fairly clear, at least to someone who is happy to regard spaces as Kan complexes, which are simplicial sets in which every horn can be filled. The analogy with categories is somewhat more subtle. We choose to model (∞, 1)-categories as quasi-categories, which are a particular type of simplicial set. When we regard the 0-simplices of an quasi-category as its objects and the 1-simplices as its morphisms, we can define a weak composition law that is well-defined, unital, and associative only up to homotopy. This means exactly that when we replace the 1-simplices by homotopy classes of 1-simplices we obtain an ordinary category, called the homotopy category of our quasi-category. However, a lot of data is lost when we replace an quasi-category by its homotopy category. In particular, there exists a 3-simplex that witnesses the fact that a particular composite (hg)f of 1-simplices f , g, and h (where some composite hg of g and h has already been chosen) is also a composite of gf (a chosen composite of f and g) and h. This is well-known by those who are familiar with the construction of the homotopy category mentioned above, but a question remains: what sort of associativity data is provided by the n-simplices, for n > 3? Each n-simplex of an quasi-category exhibits some composite of the n morphisms which make up its spine. We will argue that these simplices can be regarded as “unbiased associahedra” in the sense that they witness the “commutativity” of their boundaries, which are n − 1-simplices of the same form. In fact, one may choose all but one of these n − 1-simplices to be any witnesses that you want, subject to some obvious constraints. We will describe in detail the combinatorial analogies between these constraints and Stasheff’s associahedra in what follows, at least in low (n ≤ 6) dimensions. Before turning to these concrete details, we should acknowledge a more conceptual explanation for the experts in this field, due to Jacob Lurie, for the phenomena we’ll describe below. The theory of quasi-categories is equivalent to the theory of simplicial (or topological) categories. One feature of this equivalence is that the mapping spaces in any quasi-category (however one chooses to extract them) are homotopy equivalent to mapping spaces in a simplicial category, where there is a strictly associative multiplication. Transporting this structure along the homotopy equivalences will give an A∞ structure on the mapping spaces in the original quasicategory. By definition, A∞-spaces are algebras for the non-Σ operad {Kn} with objects given by Stasheff’s associahedra, accounting for their appearance in what follows. We begin by explaining the construction of the homotopy category of an quasicategory, emphasizing the role played by the 2and 3-simplices. The reader who