nLab
Ho(Cat)

Definition of Ho(Cat)

Ho(Cat) is a name for the homotopy category of Cat. That is, $\mathrm{Ho}(\mathrm{Cat})$ 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.

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 $\mathrm{Ho}(\mathrm{Cat})$ as defined above is equivalent to the category obtained from $\mathrm{Cat}$ by forcing all equivalences of categories to be isomorphisms. This is for the same reason that the category $\mathrm{hTop}$ of topological spaces and homotopy classes of continuous maps is equivalent to the category obtained from $\mathrm{Top}$ by inverting the homotopy equivalences (namely, the existence of cylinder objects and/or path objects). In particular, the isomorphisms in $\mathrm{Ho}(\mathrm{Cat})$ are precisely the equivalences of categories.

Subcategories of Ho(Cat)

Some notable full subcategories of $\mathrm{Ho}(\mathrm{Cat})$ include

• $\mathrm{Ho}(\mathrm{Gpd})$, 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 class?es of group homomorphism?s. This can be identified with the full subcategory of $\mathrm{Ho}(\mathrm{Gpd})$ whose objects are the connected groupoids. This category sometimes arises in the study of gerbes.

Ho(Cat)-categories

Like the homotopy category of any model category, $\mathrm{Ho}(\mathrm{Cat})$ has products and coproducts, and is in particular a cartesian monoidal category. Therefore, we can talk about categories enriched over $\mathrm{Ho}(\mathrm{Cat})$. Such a ”$\mathrm{Ho}(\mathrm{Cat})$-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 ${\mathrm{id}}_x\in C(x,x)$
• 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 $\mathrm{Ho}(\mathrm{Cat})$-category and a bicategory is that in a $\mathrm{Ho}(\mathrm{Cat})$-category, no coherence axioms are required of the associator? and unitor? isomorphisms; they are merely required to exist. Thus a $\mathrm{Ho}(\mathrm{Cat})$-category can be thought of as an “incoherent bicategory.” In particular, any bicategory has an underlying $\mathrm{Ho}(\mathrm{Cat})$-category.

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

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

Other limits and colimits

Although $\mathrm{Ho}(\mathrm{Cat})$ 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 $\mathrm{Ho}(\mathrm{Gpd})$) fails to have pullbacks.

Consider the cospan

$$\begin{array}{ccc}& & \mathbb{Z}/3\\ & & {\downarrow }^j\\ \mathbb{Z}/2& \underset{i}{\to }& S_3\end{array}$$\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 $a=(\mathrm{1,2})$ and $b=(\mathrm{1,2,3})$, 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 $\mathrm{Ho}(\mathrm{Cat})$, and suppose that this cospan had a pullback

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

in $\mathrm{Ho}(\mathrm{Cat})$ or $\mathrm{Ho}(\mathrm{Gpd})$.

Note that for any category $C$, the set $\mathrm{Ho}(\mathrm{Cat})(1,C)$ is the set of isomorphism classes of objects in $C$ (where $1$ is the terminal category). Therefore, any pullbacks that exist in $\mathrm{Ho}(\mathrm{Cat})$ 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 $\sigma :jf\cong ig$, 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

$$\begin{array}{ccc}Q& \stackrel{h}{\to }& \mathbb{Z}/3\\ {}^k\downarrow & \cong & {\downarrow }^j\\ \mathbb{Z}/2& \underset{i}{\to }& S_3.\end{array}$$\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 $\mathrm{Ho}(\mathrm{Cat})$, there must be a functor $\ell :Q\to P$ 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 $(a\mathrm{,1}):1\to a$ and $b\in \mathbb{Z}/3$ is the image of $(1,b):b\to 1$. 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 $\mathrm{Ho}(\mathrm{Gpd})$ 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