nLab Ho(Cat)

Definition of 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 $\tau_1$.

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 $Cat$.

Subcategories of Ho(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.

Ho(Cat)-categories

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”).

An example of a $Ho(Cat)$-category that does not come from any bicategory is sketched in this MathOverflow answer.

Other limits and colimits

Although $Ho(Cat)$ has products and coproducts, like most homotopy categories it is not well-endowed with other limits.

This section historically used the cospan

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

to demonstrate that $Ho(Cat)$ fails to have pullbacks. However, the pullback of this particular diagram does, in fact, exist in $Ho(Cat)$, as shown in this MathOverflow answer.

Related concepts

• canonical model structure

• Ho(CombModCat)

$(n+1,r+1)$-categories of (n,r)-categories

• Pos

• Set

  • Rel

• Grpd, ∞Grpd

• Cat

  • Ho(Cat)

  • AccCat

  • PrCat

  • LexCat

  • MonCat

  • VCat

  • CatAdj

  • Prof

  • Operad

• 2Cat

• ModCat, CombModCat

• (∞,1)Cat

  • Pr(∞,1)Cat

  • (∞,1)Operad

• (∞,n)Cat

category: category