Ho(Cat)

**Ho(Cat)** is a name for the homotopy category of Cat. That is, $Ho(Cat)$ is the category

- whose objects are (small) categories, and
- whose morphisms are natural isomorphism classes of functors.

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 $n$-category. Sometimes this is called the 1-truncation and denoted $\tau_1$.

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

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

- $Ho(Gpd)$, 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 $Ho(Gpd)$ whose objects are the connected groupoids. This category sometimes arises in the study of gerbes.

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

- a collection of objects $x,y,z$
- for each pair of objects, a category $C(x,y)$
- for each object $x$, an objects $id_x\in C(x,x)$
- for each triple of objects, a functor $C(y,z)\times C(x,y)\to C(x,z)$

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

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

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

Although $Ho(Cat)$ 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 $Ho(Gpd)$) fails to have pullbacks.

Consider the cospan

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

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

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

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

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

Now let $Q$ be the 2-pullback

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

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

Now every element of $\mathbb{Z}/2$ or $\mathbb{Z}/3$ is the image of some morphism of $Q$ under $k$ or $h$, respectively. For instance, $a\in \mathbb{Z}/2$ is the image of $(a,1)\colon 1 \to a$ and $b\in \mathbb{Z}/3$ is the image of $(1,b)\colon b\to 1$. Therefore, since $h$ and $k$ factor through $f$ and $g$ up to isomorphism, $f$ and $g$ must be surjective as monoid homomorphisms.

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

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

category: category

Revised on March 27, 2015 14:22:27
by Beppe Metere
(147.163.15.64)