Ho(Cat) is a name for the homotopy category of Cat. That is, is the category This is an instance of a general construction which, given a 2-category, or more generally an n-category, produces a 1-category with the same objects and whose morphisms are equivalence classes of 1-morphisms in the original -category. Sometimes this is called the 1-truncation and denoted . It can also be viewed as an instance of the homotopy category of a model category (or more generally a category with weak equivalences). The category as defined above is equivalent to the category obtained from by forcing all equivalences of categories to be isomorphisms (by localizing). This is for the same reason that the category of topological spaces and homotopy classes of continuous maps is equivalent to the category obtained from by inverting the homotopy equivalences (namely, the existence of cylinder objects and/or path objects). Indeed, a cylinder object for a category is the product category where is the category with two objects 0 and 1 and an isomorphism . It is not difficult to see that an isomorphism of functors is the same as a homotopy of functors with the respect to the canonical model structure on . Some notable full subcategories of include Like the homotopy category of any model category, has products and coproducts, and is in particular a cartesian monoidal category. Therefore, we can talk about categories enriched over . Such a “-category” consists of such that the usual associativity and unit diagrams for an enriched category commute up to isomorphism. The difference between a -category and a bicategory is that in a -category, no coherence axioms are required of the associator and unitor isomorphisms; they are merely required to exist. Thus a -category can be thought of as an “incoherent bicategory.” In particular, any bicategory has an underlying -category. Although -categories are not very useful, there are some interesting things that can be said about them. For instance: An...