5.2.1 Morphism Spaces

Let $\operatorname{\mathcal{C}}$ be a category. To every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, one can associate a set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ of morphisms from $X$ to $Y$. Our goal in this section is to explain a counterpart of this construction in the setting of $\infty $-categories.

Construction Let $\operatorname{\mathcal{C}}$ be a simplicial set containing a pair of vertices $X$ and $Y$. We let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ denote the simplicial set given by the fiber product

\[ \{ X\} \times _{ \operatorname{Fun}(\{ 0\} , \operatorname{\mathcal{C}}) } \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{C}}) } \{ Y\} . \]

We will typically be interested in this construction only in the case where $\operatorname{\mathcal{C}}$ is an $\infty $-category; if this condition is satisfied, we will refer to $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ as the space of morphisms from $X$ to $Y$.

Remark Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pair of objects $X$ and $Y$. Recall that a morphism from $X$ to $Y$ is an edge $e: \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ satisfying $e(0) = X$ and $e(1) = Y$ (Definition It follows that morphisms from $X$ to $Y$ can be identified with vertices of the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ of Construction

Example Let $\operatorname{\mathcal{C}}$ be an ordinary category containing objects $X$ and $Y$, which we will identify with objects of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Then the morphism space $\operatorname{Hom}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( X, Y)$ of Construction can be identified with the constant simplicial set having the value $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ (see Example

Example Let $X$ be a topological space containing a pair of points $x$ and $y$, which we regard as objects of the $\infty $-category $\operatorname{Sing}_{\bullet }(X)$. Then we have a canonical isomorphism of Kan complexes

\[ \operatorname{Hom}_{ \operatorname{Sing}_{\bullet }(X) }(x, y) \simeq \operatorname{Sing}_{\bullet }( P_{x,y} ), \]

where $P_{x,y}$ denotes the topological space of continuous paths $p: [0,1] \rightarrow X$ satisfying $p(0) = x$ and $p(1) = y$ (equipped with the compact-open topology). See Example

Example Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, so that the join $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ is also an $\infty $-category (Proposition Then the morphism spaces in $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ are described by the formula

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}}(X,Y) \simeq \begin{cases} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) & \textnormal{if } X,Y \in \operatorname{\mathcal{C}}\\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}(X,Y) & \textnormal{if } X,Y \in \operatorname{\mathcal{D}}\\ \Delta ^{0} & \textnormal{if } X \in \operatorname{\mathcal{C}}, Y \in \operatorname{\mathcal{D}}\\ \emptyset & \textnormal{if } X \in \operatorname{\mathcal{D}}, Y \in \operatorname{\mathcal{C}}. \end{cases} \]

Example Let $\operatorname{\mathcal{C}}$ be a simplicial set containing vertices $X$ and $Y$. Let $K$ be a simplicial set, and let $\underline{X}, \underline{Y}: K \rightarrow \operatorname{\mathcal{C}}$ be the constant maps taking the values $X$ and $Y$, respectively. Then there is a canonical isomorphism of simplicial sets

\[ \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) }( \underline{X}, \underline{Y} ) \simeq \operatorname{Fun}(K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ). \]

Remark Let $\operatorname{\mathcal{C}}$ be a simplicial set containing vertices $X$ and $Y$, which we also regard as vertices of the opposite simplicial set $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Then there is a canonical isomorphism of simplicial sets $\operatorname{Hom}_{\operatorname{\mathcal{C}}^{\operatorname{op}} }( X, Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)^{\operatorname{op}}$.

Proposition Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is a Kan complex.

Proof. By definition, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ can be identified with the fiber of the restriction map

\[ \theta : \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}}) \]

over the vertex $(X,Y) \in \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}})$. Corollary guarantees that $\theta $ is an isofibration, so that $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is an $\infty $-category. To show that it is a Kan complex, it will suffice to show that every morphism $u$ in $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is an isomorphism (Proposition By virtue of Corollary, it will suffice to show that the image of $u$ in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ is an isomorphism. This follows from Theorem, since evaluation functors

\[ \operatorname{ev}_0: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(\{ 0\} , \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}}\quad \quad \operatorname{ev}_1: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(\{ 1\} , \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}} \]

carry $u$ to the identity morphisms $\operatorname{id}_{X}$ and $\operatorname{id}_{Y}$, respectively. $\square$

Remark Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pair of morphisms $f,g: X \rightarrow Y$ having the same source and target. Then $f$ and $g$ are homotopic (Definition if and only if they belong to the same connected component of the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$: this follows from the characterization of Corollary Consequently, we obtain a bijection $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) \simeq \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) )$.

Example (Loop Spaces). Let $(X,x)$ be a pointed Kan complex. The Kan complex $\operatorname{Hom}_{X}(x,x)$ is often denoted by $\Omega (X)$ and referred to as the based loop space of $X$. Note that it can be identified with the fiber over $x$ of the evaluation map

\[ q: \{ x\} \times _{ \operatorname{Fun}(\{ 0\} , X) } \operatorname{Fun}( \Delta ^1, X) \rightarrow \operatorname{Fun}(\{ 1\} , X) = X. \]

By virtue of Example, this map is a Kan fibration whose domain is a contractible Kan complex. It follows that the long exact sequence of Theorem yields isomorphisms $\pi _{n}( \operatorname{Hom}_{X}(x,x), \operatorname{id}_{x} ) \simeq \pi _{n+1}(X,x)$ for $n \geq 0$.

Example Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ denote its Duskin nerve (Construction, and let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$ (which we identify with vertices of the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$). For any category $\operatorname{\mathcal{E}}$, Corollary supplies a bijection

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors $U: \operatorname{Tw}(\operatorname{\mathcal{E}})^{\operatorname{op}} \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$} \} \ar [d] \\ \{ \textnormal{Morphisms of simplicial sets $\operatorname{N}_{\bullet }(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}( X, Y)$} \} }, \]

where $\operatorname{Tw}(\operatorname{\mathcal{E}})$ denotes the twisted arrow category of Construction Specializing to categories of the form $[n] = \{ 0 < 1 < \cdots < n\} $, we see that the simplicial set $\operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}( X, Y)$ is given by the construction

\[ ( [n] \in \operatorname{{\bf \Delta }}) \mapsto \{ \textnormal{Functors from $\operatorname{Tw}([n])^{\operatorname{op}}$ to $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$} \} . \]

In other words, we can identify $n$-simplices of the simplicial set $\operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}( X, Y)$ with commutative diagrams

\[ \xymatrix@C =20pt{ & & & & f_{0,n} & & & & \\ & & & f_{0,n-1} \ar@ {=>}[ur] & & f_{1,n} \ar@ {=>}[ul] & & & \\ & & f_{0,n-2} \ar@ {=>}[ur] & & f_{1,n-1} \ar@ {=>}[ul] \ar@ {=>}[ur] & & f_{2,n} \ar@ {=>}[ul] & & \\ & \cdots \ar@ {=>}[ur] & & \cdots \ar@ {=>}[ur] \ar@ {=>}[ul] & & \cdots \ar@ {=>}[ul] \ar@ {=>}[ur] & & \cdots \ar@ {=>}[ul] & \\ f_{0,0} \ar@ {=>}[ur] & & f_{1,1} \ar@ {=>}[ul] \ar@ {=>}[ur] & & \cdots \ar@ {=>}[ul] \ar@ {=>}[ur] & & f_{n-1,n-1} \ar@ {=>}[ul] \ar@ {=>}[ur] & & f_{n,n} \ar@ {=>}[ul] } \]

in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$.