2.2.8 The Homotopy Category of a $2$-Category
Every ordinary category can be regarded as a $2$-category having only identity $2$-morphisms (Remark 2.2.1.6). Conversely, to every $2$-category $\operatorname{\mathcal{C}}$ one can associate ordinary category $\operatorname{hPith}(\operatorname{\mathcal{C}})$ having the same objects, in which morphisms are given by isomorphism classes of $1$-morphisms in $\operatorname{\mathcal{C}}$. We will refer to $\operatorname{hPith}(\operatorname{\mathcal{C}})$ as the homotopy category of the $2$-category $\operatorname{\mathcal{C}}$ (Construction 2.2.8.12). It will be convenient to view this construction as a composition of two different operations:
To every $2$-category $\operatorname{\mathcal{C}}$, one can associate a subcategory $\operatorname{Pith}(\operatorname{\mathcal{C}}) \subseteq \operatorname{\mathcal{C}}$ by removing the non-invertible $2$-morphisms of $\operatorname{\mathcal{C}}$; we will refer to $\operatorname{Pith}(\operatorname{\mathcal{C}})$ as the pith of $\operatorname{\mathcal{C}}$ (Construction 2.2.8.9).
To every $2$-category $\operatorname{\mathcal{C}}$, one can associate an ordinary category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ by “collapsing” all $2$-morphisms of $\operatorname{\mathcal{C}}$ to identity $2$-morphisms (Construction 2.2.8.2). We will refer to $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as the coarse homotopy category of the $2$-category $\operatorname{\mathcal{C}}$.
We begin by formulating the latter construction more precisely.
Definition 2.2.8.1. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $\operatorname{\mathcal{H}}$ be an ordinary category, viewed as a $2$-category having only identity $2$-morphisms. We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{H}}$ exhibits $\operatorname{\mathcal{H}}$ as a coarse homotopy category of $\operatorname{\mathcal{C}}$ if, for every ordinary category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors of ordinary categories from $\operatorname{\mathcal{H}}$ to $\operatorname{\mathcal{E}}$} \} \ar [d] \\ \{ \textnormal{Functors of $2$-categories from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{E}}$} \} . } \]
It follows immediately from the definitions that if a $2$-category $\operatorname{\mathcal{C}}$ admits a coarse homotopy category $\operatorname{\mathcal{H}}$, then $\operatorname{\mathcal{H}}$ is uniquely determined up to isomorphism. We will prove existence by an explicit construction.
Construction 2.2.8.2 (The Coarse Homotopy Category of a $2$-Category). Let $\operatorname{\mathcal{C}}$ be a $2$-category. We define a category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as follows:
The objects of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ are the objects of $\operatorname{\mathcal{C}}$.
If $X$ and $Y$ are objects of $\operatorname{\mathcal{C}}$, then $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y)$ is the set of connected components of the simplicial set $\operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) )$.
For objects $X$, $Y$, and $Z$ of $\operatorname{\mathcal{C}}$, the composition of morphisms in $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is given by the map
\begin{eqnarray*} \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(Y,Z) \times \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) & = & \pi _0( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) ) \times \pi _0( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) \\ & \simeq & \pi _0( \operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) ) \\ & \xrightarrow {\circ } & \pi _0( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) ) \\ & = & \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Z). \end{eqnarray*}
We will refer to $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as the coarse homotopy category of $\operatorname{\mathcal{C}}$.
The terminology of Construction 2.2.8.2 is consistent with that of Definition 2.2.8.1, by virtue of the following:
Proposition 2.2.8.3. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ be the ordinary category of Construction 2.2.8.2, regarded as a $2$-category having only identity $2$-morphisms. Then there is a unique functor of $2$-categories $F: \operatorname{\mathcal{C}}\rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ with the following properties:
The functor $F$ carries each object of $\operatorname{\mathcal{C}}$ to itself (regarded as an object of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$).
The functor $F$ carries each $1$-morphism $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ to the connected component of $u$, regarded as a vertex of the nerve $\operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) )$.
Moreover, the functor $F$ exhibits $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as a coarse homotopy category of $\operatorname{\mathcal{C}}$, in the sense of Definition 2.2.8.1.
Proof.
The existence of $F$ follows from Example 2.2.4.14. Let $\operatorname{\mathcal{E}}$ be an ordinary category, and suppose we are given a functor of $2$-categories $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$. We wish to show that there is a unique functor of ordinary categories $\overline{G}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{E}}$ satisfying $G = \overline{G} \circ F$. The uniqueness is clear (since the functor $F$ is surjective on objects and on $1$-morphisms). To prove existence, we define $\overline{G}$ on objects by the formula $\overline{G}(X) = G(X)$ and on morphism by using the map of simplicial sets
\[ \operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{E}}}( G(X), G(Y) ) \]
and passing to connected components.
$\square$
Corollary 2.2.8.4. Let $\operatorname{Cat}$ denote the category of (small) categories and let $\operatorname{2Cat}$ denote the category of (small) $2$-categories (Definition 2.2.5.5. Then the inclusion $\operatorname{Cat}\hookrightarrow \operatorname{2Cat}$ admits a left adjoint, given on objects by the construction $\operatorname{\mathcal{C}}\mapsto \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.
In general, passage from a $2$-category $\operatorname{\mathcal{C}}$ to its coarse homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is a very destructive procedure: if $u,v: X \rightarrow Y$ are $1$-morphisms of $\operatorname{\mathcal{C}}$ having the same source and target, then the existence of any $2$-morphism $\gamma : u \Rightarrow v$ in $\operatorname{\mathcal{C}}$ guarantees that $u$ and $v$ have the same image in $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. For many purposes, it is more appropriate to work with a variant of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ which identifies only isomorphic $1$-morphisms of $\operatorname{\mathcal{C}}$ (Construction 2.2.8.12). First, let us introduce some terminology.
Definition 2.2.8.5. A $(2,1)$-category is a $2$-category $\operatorname{\mathcal{C}}$ with the property that every $2$-morphism in $\operatorname{\mathcal{C}}$ is invertible.
Example 2.2.8.7. Let $\operatorname{\mathcal{C}}$ be an ordinary category, viewed as a $2$-category having only identity $2$-morphisms (Remark 2.2.1.6). Then $\operatorname{\mathcal{C}}$ is a $(2,1)$-category.
Construction 2.2.8.9 (The Pith of a $2$-Category). Let $\operatorname{\mathcal{C}}$ be a $2$-category. We define a new $2$-category $\operatorname{Pith}(\operatorname{\mathcal{C}})$ as follows:
The objects of $\operatorname{Pith}(\operatorname{\mathcal{C}})$ are the objects of $\operatorname{\mathcal{C}}$.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the category $\underline{\operatorname{Hom}}_{\operatorname{Pith}(\operatorname{\mathcal{C}})}( X, Y)$ is the core $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)^{\simeq }$ of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ (see Construction 1.3.5.4).
The composition law, associativity constraints, and unit constraints of $\operatorname{Pith}(\operatorname{\mathcal{C}})$ are given by restricting the composition law, associativity constraints, and unit constraints of $\operatorname{\mathcal{C}}$.
Then $\operatorname{Pith}(\operatorname{\mathcal{C}})$ is a $(2,1)$-category which we will refer to as the pith of $\operatorname{\mathcal{C}}$.
More informally: for any $2$-category $\operatorname{\mathcal{C}}$, the $(2,1)$-category $\operatorname{Pith}(\operatorname{\mathcal{C}})$ is obtained by discarding the non-invertible $2$-morphisms of $\operatorname{\mathcal{C}}$.
Warning 2.2.8.11. In the situation of Remark 2.2.8.10, it is not true that a lax functor $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ factors through the pith $\operatorname{Pith}(\operatorname{\mathcal{C}})$ (even when $\operatorname{\mathcal{D}}$ is a $(2,1)$-category): any lax functor which admits such a factorization is automatically a functor, by virtue of Remark 2.2.8.8.
Construction 2.2.8.12 (The Homotopy Category of a $2$-Category). Let $\operatorname{\mathcal{C}}$ be a $2$-category. We define a category $\operatorname{hPith}(\operatorname{\mathcal{C}})$ as follows:
The objects of $\operatorname{hPith}(\operatorname{\mathcal{C}})$ are the objects of $\operatorname{\mathcal{C}}$.
If $X$ and $Y$ are objects of $\operatorname{\mathcal{C}}$, then $\operatorname{Hom}_{\operatorname{hPith}(\operatorname{\mathcal{C}})}(X,Y)$ is the set of isomorphism classes of objects in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$. If $f: X \rightarrow Y$ is a $1$-morphism from $X$ to $Y$, we typically denote its isomorphism class by $[f] \in \operatorname{Hom}_{ \operatorname{hPith}(\operatorname{\mathcal{C}}) }(X,Y)$.
The composition law on $\operatorname{hPith}(\operatorname{\mathcal{C}})$ is determined by the requirement that $[g] \circ [f] = [g \circ f]$ for every pair of composable $1$-morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ (this composition law is associative by virtue of the existence of the associativity contraints of the $2$-category $\operatorname{\mathcal{C}}$).
For every object $Y \in \operatorname{\mathcal{C}}$, the identity morphism from $Y$ to itself in $\operatorname{hPith}(\operatorname{\mathcal{C}})$ is the isomorphism class of the identity morphism $\operatorname{id}_{Y}$ in $\operatorname{\mathcal{C}}$. For $1$-morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$, the identities
\[ [\operatorname{id}_ Y] \circ [f] = [f] \quad \quad [g] \circ [\operatorname{id}_ Y] = [g] \]
follow from the existence of left and right unit constraints (see Construction 2.2.1.11).
We will refer to $\operatorname{hPith}(\operatorname{\mathcal{C}})$ as the homotopy category of $\operatorname{\mathcal{C}}$.
Warning 2.2.8.14. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $\operatorname{hPith}(\operatorname{\mathcal{C}})$ be the homotopy category of $\operatorname{\mathcal{C}}$, which we regard as a $2$-category having only identity $2$-morphisms. In general, there is no functor which directly relates $\operatorname{\mathcal{C}}$ to the homotopy category $\operatorname{hPith}(\operatorname{\mathcal{C}})$. Instead, there is a commutative diagram of $2$-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Pith}(\operatorname{\mathcal{C}}) \ar@ {^{(}->}[r] \ar [d] & \operatorname{\mathcal{C}}\ar [d] \\ \operatorname{hPith}(\operatorname{\mathcal{C}}) \ar [r] & \mathrm{h} \mathit{\operatorname{\mathcal{C}}}. } \]
Here the functor $\operatorname{hPith}(\operatorname{\mathcal{C}}) \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is bijective on objects and full: that is, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map
\[ \operatorname{Hom}_{ \operatorname{hPith}(\operatorname{\mathcal{C}})}(X,Y) = \pi _0( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)^{\simeq } ) \rightarrow \pi _0( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) = \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}} }(X,Y) \]
is surjective.
Example 2.2.8.15. Let $\operatorname{\mathcal{C}}$ be a $(2,1)$-category, so that $\operatorname{Pith}(\operatorname{\mathcal{C}}) = \operatorname{\mathcal{C}}$. In particular, the inclusion $\operatorname{Pith}(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{\mathcal{C}}$ induces an isomorphism of categories $\operatorname{hPith}(\operatorname{\mathcal{C}}) \simeq \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. In this situation, we will generally abuse notation by identifying $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ with $\operatorname{hPith}(\operatorname{\mathcal{C}})$ and referring to it as the homotopy category of $\operatorname{\mathcal{C}}$.
Beware that the analogous assertion does not hold if $U$ is only assumed to be a lax functor of $2$-categories.
Definition 2.2.8.17. Let $\operatorname{\mathcal{C}}$ be a $2$-category. We say that a $1$-morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ is an isomorphism if the homotopy class $[f]$ is an isomorphism in the homotopy category $\operatorname{hPith}(\operatorname{\mathcal{C}})$. Equivalently, $f$ is an isomorphism if there exists another $1$-morphism $g: Y \rightarrow X$ such that $g \circ f$ and $f \circ g$ are isomorphic to $\operatorname{id}_{X}$ and $\operatorname{id}_{Y}$ as objects of the categories $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$ and $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Y)$, respectively. In this case, $g$ is also an isomorphism in $\operatorname{\mathcal{C}}$, which we will refer to as a homotopy inverse to $f$.
Example 2.2.8.18. Let $\operatorname{\mathcal{C}}$ be an ordinary category, regarded as a $2$-category having only identity $2$-morphisms (Remark 2.2.1.6). Then a morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ is an isomorphism in the sense of Definition 2.2.8.17 if and only if it is an isomorphism in the usual sense: that is, if and only if there exists a morphism $g: Y \rightarrow X$ satisfying $g \circ f = \operatorname{id}_{X}$ and $f \circ g = \operatorname{id}_ Y$.
Warning 2.2.8.19. Let $\operatorname{\mathcal{C}}$ be a strict $2$-category. We can then consider two different notions of isomorphism in $\operatorname{\mathcal{C}}$:
We say that a morphism $f: X \rightarrow Y$ is a strict isomorphism if it is an isomorphism in the underlying category of $\operatorname{\mathcal{C}}$: that is, if there exists a $1$-morphism $g: Y \rightarrow X$ satisfying $g \circ f = \operatorname{id}_ X$ and $f \circ g = \operatorname{id}_ Y$.
We say that a morphism $f: X \rightarrow Y$ is an isomorphism if the homotopy class $[f]$ is an isomorphism in the homotopy category $\operatorname{hPith}(\operatorname{\mathcal{C}})$: that is, if there exists a $1$-morphism $g: Y \rightarrow X$ such that $g \circ f$ and $f \circ g$ are isomorphic to $\operatorname{id}_{X}$ and $\operatorname{id}_{Y}$ as objects of the categories $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$ and $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Y)$, respectively.
Every strict isomorphism in $\operatorname{\mathcal{C}}$ is an isomorphism. However, the converse is false in general (see Example 2.2.8.20).
Example 2.2.8.20. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between (small) categories. Then $F$ is an equivalence of categories if and only if it is an isomorphism when regarded as a $1$-morphism in the $2$-category $\mathbf{Cat}$ of Example 2.2.0.4.
We close this section by discussing a strengthening of Definition 2.2.8.5.
Definition 2.2.8.24. Let $\operatorname{\mathcal{C}}$ be a $2$-category. We say that $\operatorname{\mathcal{C}}$ is a $2$-groupoid if every $1$-morphism in $\operatorname{\mathcal{C}}$ is an isomorphism and every $2$-morphism of $\operatorname{\mathcal{C}}$ is an isomorphism.
Example 2.2.8.26. Let $\operatorname{\mathcal{C}}$ be an ordinary category. Then $\operatorname{\mathcal{C}}$ is a groupoid if and only if it is a $2$-groupoid (when viewed as a $2$-category having only identity $2$-morphisms).
Construction 2.2.8.27 (The Core of a $2$-Category). Let $\operatorname{\mathcal{C}}$ be a $2$-category. We define a new $2$-category $\operatorname{\mathcal{C}}^{\simeq }$ as follows:
The objects of $\operatorname{\mathcal{C}}^{\simeq }$ are the objects of $\operatorname{\mathcal{C}}$.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}^{\simeq } }( X, Y)$ is the full subcategory of $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)^{\simeq }$ spanned by the isomorphisms $f: X \rightarrow Y$.
The composition law, associativity constraints, and unit constraints of $\operatorname{\mathcal{C}}^{\simeq }$ are obtained by restricting the composition law, associativity constraints, and unit constraints of $\operatorname{\mathcal{C}}$ (which is well-defined by virtue of Remark 2.2.8.22).
We will refer to $\operatorname{\mathcal{C}}^{\simeq }$ as the core of the $2$-category $\operatorname{\mathcal{C}}$.
Example 2.2.8.28. Let $\operatorname{\mathcal{C}}$ be a category. Then the core $\operatorname{\mathcal{C}}^{\simeq } \subseteq \operatorname{\mathcal{C}}$ of Construction 1.3.5.4 coincides with the core $\operatorname{\mathcal{C}}^{\simeq } \subseteq \operatorname{\mathcal{C}}$ of Construction 2.2.8.27, where we regard $\operatorname{\mathcal{C}}$ as a $2$-category having only identity $2$-morphisms.