Path categories and resolutions

The path category P (X) of a simplicial set X defines a functor X 7→ P (X) which is left adjoint to the nerve functor B : cat → sSet. The first section of this paper contains an elementary description of this functor, with some of its basic properties. This construction specializes to a path category functor for cubical sets which appears in the second section. The path category construction is not a standard homotopy invariant for simplicial sets: a map X → Y of simplicial sets which induces a homotopy equivalence |X| → |Y | of their topological realizations can fail to induce a homotopy equivalence |BP (X)| → |BP (Y )|. The path category functor is, on the other hand, a strong invariant for Joyal’s quasi-category model structure for simplicial sets [5], [6] and is, more generally, a theoretical building block for various approaches to higher category theory. But there is a question: given a simplicial set X with vertices x and y, how do you “compute” the morphism set P (X)(x, y)? Is it empty or not? When is it finite? What does it mean for this set to have more than one element? The purpose of this paper is to display techniques which give answers to these questions in specific cases of interest. The first answers that we get are counterintuitive from the point of view of ordinary homotopy theory, even for elementary examples. The path category P (X) only sees the 2-skeleton of a space X (as one might expect), and its morphism sets detect 2-dimensional holes in X. But they also detect missing 2-simplices in spaces which are contractible in the usual sense: it is shown here (in the first section) that the path categories of the outer horns Λ0,Λ 3 3 are not posets, while the path categories of all inner horns are posets. The non-triviality of the category P (Λ0) effectively means that the path category detects sources in finite oriented simplicial complexes, and the non-triviality of P (Λ3) means that the functor detects sinks. Joyal’s quasi-category model structure gives methods for identifying P (X) up to equivalence of categories, since every quasi-category weak equivalence X → Y induces an equivalence of categories P (X) → P (Y ). This is useful for dealing with global qualitative questions, but one wants some device for determining the morphism sets P (X)(x, y) explicitly.