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 .
David Roberts: I would think that for a strict 2-category is the underlying 1-category. What is described here could be called the Poincaré category (I think that Benabou’s monograph on bicategories has this term). Maybe terminology as developed in the meantime, though.
Mike Shulman: Well, the uses of “truncation” I’ve seen always involves quotienting by equivalences, rather than discarding them. Discarding them only even makes sense in the strict situation (a bicategory has no underlying 1-category) and is an evil (and not often very useful) thing to do, so it doesn’t seem to me worth giving an important name to. “Poincare category” may also be a name for the same thing, but I prefer “truncation” as more evocative.
Beppe Metere: If I remember well, Benabou introduces two different constructions related to this discussion: the Poincarè category of a bicategory, where the arrows are connected components of 1-cells, and the classifying category, where the arrows are iso classes of 1-cells. Of course, these two categories coincide when the bicategory is locally groupoidal.
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:
Consider the cospan
where the two arrows are inclusions of subgroups. That is, we choose a 2-cycle and a 3-cycle in , say and , and identify and with the subgroups generated by and respectively. Regard these groups as connected groupoids and thus as objects of , and suppose that this cospan had a pullback
in or .
Note that for any category , the set is the set of isomorphism classes of objects in (where is the terminal category). Therefore, any pullbacks that exist in must induce pullbacks of sets of isomorphism classes of objects, and so must also have only one isomorphism class of objects; i.e. it must be a monoid, regarded as a one-object category. We choose monoid homomorphisms and representing and , respectively. We also choose a natural isomorphism , which consists of an element such that for all .
Now let be the 2-pullback
Then the objects of are the elements of , and the morphisms from to consist of pairs such that . Since the square defining commutes in , there must be a functor such that and .
Now every element of or is the image of some morphism of under or , respectively. For instance, is the image of and is the image of . Therefore, since and factor through and up to isomorphism, and must be surjective as monoid homomorphisms.
Let be such that . If is not the identity, let . Otherwise and there is some with . If is not the identity, then let . Otherwise and let . In either case, neither nor is the identity. Therefore, neither nor is the identity, and moreover is a 3-cycle and is a 2-cycle in . But the element conjugates to , a contradiction.
Since all the categories involved were groupoids (except possibly ), the same argument shows that doesn’t have pullbacks. Moreover, basically the same argument, regarding groupoids as connected 1-types, shows that the homotopy category of topological spaces doesn’t have pullbacks either (in this case the final contradiction is derived from instead of itself).