Definition of Ho(Cat)

Ho(Cat) is a name for the homotopy category of Cat. That is, Ho(Cat)Ho(Cat) 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 nn-category. Sometimes this is called the 1-truncation and denoted τ 1\tau_1.

David Roberts: I would think that τ 1(C)\tau_1(C) 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 Ho(Cat)Ho(Cat) as defined above is equivalent to the category obtained from CatCat by forcing all equivalences of categories to be isomorphisms (by localizing). This is for the same reason that the category hTophTop of topological spaces and homotopy classes of continuous maps is equivalent to the category obtained from TopTop by inverting the homotopy equivalences (namely, the existence of cylinder objects and/or path objects). Indeed, a cylinder object for a category CC is the product category C×IC \times I where II is the category with two objects 0 and 1 and an isomorphism 010 \to 1. 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 CatCat.

Subcategories of Ho(Cat)

Some notable full subcategories of Ho(Cat)Ho(Cat) include

  • Ho(Gpd)Ho(Gpd), the homotopy category of the category Gpd of groupoids. Note that this is equivalent to the homotopy category of (unbased) homotopy 1-types.
  • The category whose objects are groups and whose morphisms are conjugacy classes of group homomorphisms. This can be identified with the full subcategory of Ho(Gpd)Ho(Gpd) whose objects are the connected groupoids. This category sometimes arises in the study of gerbes.


Like the homotopy category of any model category, Ho(Cat)Ho(Cat) has products and coproducts, and is in particular a cartesian monoidal category. Therefore, we can talk about categories enriched over Ho(Cat)Ho(Cat). Such a “Ho(Cat)Ho(Cat)-category” consists of

  • a collection of objects x,y,zx,y,z
  • for each pair of objects, a category C(x,y)C(x,y)
  • for each object xx, an objects id xC(x,x)id_x\in C(x,x)
  • for each triple of objects, a functor C(y,z)×C(x,y)C(x,z)C(y,z)\times C(x,y)\to C(x,z)

such that the usual associativity and unit diagrams for an enriched category commute up to isomorphism. The difference between a Ho(Cat)Ho(Cat)-category and a bicategory is that in a Ho(Cat)Ho(Cat)-category, no coherence axioms are required of the associator and unitor isomorphisms; they are merely required to exist. Thus a Ho(Cat)Ho(Cat)-category can be thought of as an “incoherent bicategory.” In particular, any bicategory has an underlying Ho(Cat)Ho(Cat)-category.

Although Ho(Cat)Ho(Cat)-categories are not very useful, there are some interesting things that can be said about them. For instance:

  • Any Ho(Cat)Ho(Cat)-category which is equivalent, as a Ho(Cat)Ho(Cat)-category, to a bicategory, is itself in fact a bicategory.
  • Any 2-functor between bicategories which induces an equivalence of underlying Ho(Cat)Ho(Cat)-categories is in fact itself an equivalence of bicategories (or “biequivalence”).

Other limits and colimits

Although Ho(Cat)Ho(Cat) has products and coproducts, like most homotopy categories it is not well-endowed with other limits. The following concrete example shows that it (and also Ho(Gpd)Ho(Gpd)) fails to have pullbacks.

Consider the cospan

/3 j /2 i S 3\array{&& \mathbb{Z}/3\\ && \downarrow^j\\ \mathbb{Z}/2 & \underset{i}{\to} & S_3}

where the two arrows are inclusions of subgroups. That is, we choose a 2-cycle and a 3-cycle in S 3S_3, say a=(1,2)a=(1,2) and b=(1,2,3)b=(1,2,3), and identify /2\mathbb{Z}/2 and /3\mathbb{Z}/3 with the subgroups generated by aa and bb respectively. Regard these groups as connected groupoids and thus as objects of Ho(Cat)Ho(Cat), and suppose that this cospan had a pullback

P f /3 g j /2 i S 3\array{P & \overset{f}{\to} & \mathbb{Z}/3\\ ^g \downarrow && \downarrow^j\\ \mathbb{Z}/2 & \underset{i}{\to} & S_3}

in Ho(Cat)Ho(Cat) or Ho(Gpd)Ho(Gpd).

Note that for any category CC, the set Ho(Cat)(1,C)Ho(Cat)(1,C) is the set of isomorphism classes of objects in CC (where 11 is the terminal category). Therefore, any pullbacks that exist in Ho(Cat)Ho(Cat) must induce pullbacks of sets of isomorphism classes of objects, and so PP 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 P/2P\to \mathbb{Z}/2 and P/3P\to \mathbb{Z}/3 representing ff and gg, respectively. We also choose a natural isomorphism σ:jfig\sigma\colon j f \cong i g, which consists of an element σS 3\sigma\in S_3 such that j(f(c))=σi(g(c))σ 1j(f(c)) = \sigma \cdot i(g(c)) \cdot \sigma^{-1} for all cPc\in P.

Now let QQ be the 2-pullback

Q h /3 k j /2 i S 3.\array{Q & \overset{h}{\to} & \mathbb{Z}/3\\ ^k \downarrow & \cong & \downarrow^j\\ \mathbb{Z}/2 & \underset{i}{\to} & S_3.}

Then the objects of QQ are the elements of S 3S_3, and the morphisms from xx to yy consist of pairs (u,v)/2×/3(u,v)\in \mathbb{Z}/2 \times \mathbb{Z}/3 such that i(u)x=yj(v)i(u)\cdot x = y \cdot j(v). Since the square defining QQ commutes in Ho(Cat)Ho(Cat), there must be a functor :QP\ell\colon Q\to P such that fhf\ell\cong h and gkg\ell\cong k.

Now every element of /2\mathbb{Z}/2 or /3\mathbb{Z}/3 is the image of some morphism of QQ under kk or hh, respectively. For instance, a/2a\in \mathbb{Z}/2 is the image of (a,1):1a(a,1)\colon 1 \to a and b/3b\in \mathbb{Z}/3 is the image of (1,b):b1(1,b)\colon b\to 1. Therefore, since hh and kk factor through ff and gg up to isomorphism, ff and gg must be surjective as monoid homomorphisms.

Let c 1c_1 be such that f(c 1)=bf(c_1)=b. If g(c 1)g(c_1) is not the identity, let c=c 1c=c_1. Otherwise g(c 1)=1g(c_1)=1 and there is some c 2c_2 with g(c 2)=ag(c_2)=a. If f(c 2)f(c_2) is not the identity, then let c=c 2c=c_2. Otherwise f(c 2)=1f(c_2)=1 and let c=c 1c 2c = c_1\cdot c_2. In either case, neither f(c)f(c) nor g(c)g(c) is the identity. Therefore, neither j(f(c))j(f(c)) nor i(g(c))i(g(c)) is the identity, and moreover j(f(c))j(f(c)) is a 3-cycle and i(g(c))i(g(c)) is a 2-cycle in S 3S_3. But the element σ\sigma conjugates i(g(c))i(g(c)) to j(f(c))j(f(c)), a contradiction.

Since all the categories involved were groupoids (except possibly PP), the same argument shows that Ho(Gpd)Ho(Gpd) 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 π 1(P)\pi_1(P) instead of PP itself).

See also

category: category

Revised on March 27, 2015 14:22:27 by Beppe Metere (