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5.1.1 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 5.1.1.1. 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 5.1.1.2. 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 1.3.1.1). 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 5.1.1.1.

Example 5.1.1.3. 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 5.1.1.1 can be identified with the constant simplicial set having the value $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ (see Example 5.1.5.4).

Example 5.1.1.4. 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 5.1.5.5.

Example 5.1.1.5. 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 4.3.3.21). 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 5.1.1.6. 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 5.1.1.7. 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 5.1.1.8. 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.

Proposition 5.1.1.8 is a special case of the following more general assertion:

Proposition 5.1.1.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $B$ be a simplicial set, let $A \subseteq B$ be a simplicial subset which contains every vertex of $B$, and let $f: A \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then the fiber product $\operatorname{Fun}(B,\operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A,\operatorname{\mathcal{C}}) } \{ f \} $ is a Kan complex.

Proof. Corollary 4.4.5.3 guarantees the restriction map $\theta : \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(A, \operatorname{\mathcal{C}})$ is an isofibration, so that the fiber $\operatorname{Fun}(B,\operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A,\operatorname{\mathcal{C}}) } \{ f \} $ is an $\infty $-category. To show that it is a Kan complex, it will suffice to show that every morphism $u$ in $\operatorname{Fun}(B,\operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A,\operatorname{\mathcal{C}}) } \{ f \} $ is an isomorphism (Proposition 4.4.2.1). By virtue of Corollary 4.4.3.17, this is equivalent to the assertion that the image of $u$ in the $\infty $-category $\operatorname{Fun}( B, \operatorname{\mathcal{C}})$ is an isomorphism. This follows from Theorem 4.4.4.4, since for every vertex $b \in B$, the evaluation functor

\[ \operatorname{ev}_ b: \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \{ b\} , \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}} \]

factors through $\operatorname{Fun}(A, \operatorname{\mathcal{C}})$ and therefore carries $u$ to the identity morphism $\operatorname{id}_{ f(b)}$. $\square$

Remark 5.1.1.10. 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 1.3.3.1) 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 1.3.3.7. 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 5.1.1.11 (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 3.1.6.10, this map is a Kan fibration whose domain is a contractible Kan complex. It follows that the long exact sequence of Theorem 3.2.5.1 yields isomorphisms $\pi _{n}( \operatorname{Hom}_{X}(x,x), \operatorname{id}_{x} ) \simeq \pi _{n+1}(X,x)$ for $n \geq 0$.

Example 5.1.1.12. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ denote its Duskin nerve (Construction 2.3.1.1), 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 2.3.5.14 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 2.3.5.1. 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 $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$.

It will sometimes be convenient to work with a relative version of Construction 5.1.1.1.

Construction 5.1.1.13. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, let $X$ and $Y$ be vertices of $\operatorname{\mathcal{C}}$, and let $e: q(X) \rightarrow q(Y)$ be an edge of the simplicial set $\operatorname{\mathcal{D}}$. We let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{e}$ denote the fiber product $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( q(X), q(Y) ) } \{ e\} $, which we regard as a simplicial subset of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$.

Example 5.1.1.14. In the situation of Construction 5.1.1.13, suppose that the simplicial $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( q(X), q(Y) )$ is isomorphic to $\Delta ^0$ (this condition is satisfied, for example, if $\operatorname{\mathcal{D}}$ is the nerve of a partially ordered set). Then the inclusion map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{e} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is an isomorphism.

Example 5.1.1.15. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets and let $X$ and $Y$ be vertices of $\operatorname{\mathcal{C}}$ having the same image $D = q(X) = q(Y)$ in $\operatorname{\mathcal{D}}$. Then we have a canonical isomorphism of simplicial sets

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{ \operatorname{id}_{D} } \simeq \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{D} }(X,Y), \]

where $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ denotes the fiber of $q$ over the vertex $D$.

Remark 5.1.1.16. Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix { \operatorname{\mathcal{C}}\ar [r]^{F} \ar [d]^{q} & \operatorname{\mathcal{C}}' \ar [d] \\ \operatorname{\mathcal{D}}\ar [r]^{ \overline{F} } & \operatorname{\mathcal{D}}'. } \]

Let $X$ and $Y$ be vertices of $\operatorname{\mathcal{C}}$, and let $e: q(X) \rightarrow q(Y)$ be an edge of the simplicial set $\operatorname{\mathcal{D}}$. Then composition with $F$ induces an isomorphism of simplicial sets

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{e} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}'}( F(X), F(Y) )_{ \overline{F}(e) }. \]

Remark 5.1.1.17. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, let $X$ and $Y$ be vertices of $\operatorname{\mathcal{C}}$, and let $e: q(X) \rightarrow q(Y)$ be an edge of $\operatorname{\mathcal{D}}$. Form a pullback diagram of simplicial sets

\[ \xymatrix { \operatorname{\mathcal{C}}' \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{q} \\ \Delta ^1 \ar [r]^{e} & \operatorname{\mathcal{D}}, } \]

so that $X$ lifts uniquely to a vertex $\widetilde{X} \in \operatorname{\mathcal{C}}'$ lying over the vertex $0 \in \Delta ^1$, and $Y$ lifts uniquely to a vertex $\widetilde{Y} \in \operatorname{\mathcal{C}}'$ lying over the vertex $1 \in \Delta ^1$. Remark 5.1.1.16 and Example 5.1.1.14 supply isomorphisms

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{e} \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}'}( \widetilde{X}, \widetilde{Y} )_{ \operatorname{id}_{\Delta ^1} } = \operatorname{Hom}_{\operatorname{\mathcal{C}}'}( \widetilde{X}, \widetilde{Y} ). \]

Proposition 5.1.1.18. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets, let $X$ and $Y$ be vertices of $\operatorname{\mathcal{C}}$, and let $e: q(X) \rightarrow q(Y)$ be an edge of $\operatorname{\mathcal{D}}$. Then the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{e}$ is a Kan complex.

Proof. Form a pullback diagram of simplicial sets

\[ \xymatrix { \operatorname{\mathcal{C}}' \ar [r] \ar [d]^{q'} & \operatorname{\mathcal{C}}\ar [d]^{q} \\ \Delta ^1 \ar [r]^{e} & \operatorname{\mathcal{D}}. } \]

Since $q$ is an inner fibration, the morphism $q'$ is also an inner fibration (Remark 4.1.1.5), so that $\operatorname{\mathcal{C}}'$ is an $\infty $-category (Remark 4.1.1.9). Remark 5.1.1.17 then supplies an isomorphism of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{e}$ with a simplicial set of the form $\operatorname{Hom}_{\operatorname{\mathcal{C}}'}( \widetilde{X}, \widetilde{Y} )$, which is a Kan complex by virtue of Proposition 5.1.1.8. $\square$

In the special case where $\operatorname{\mathcal{D}}$ is an $\infty $-category, we can prove a slightly stronger assertion:

Proposition 5.1.1.19. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, and let $e: q(X) \rightarrow q(Y) )$ be a morphism in $\operatorname{\mathcal{D}}$. Then the induced map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( q(X), q(Y) )$ is a Kan fibration of simplicial sets.

Remark 5.1.1.20. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, and let $e: q(X) \rightarrow q(Y) )$ be a morphism in $\operatorname{\mathcal{D}}$. By construction, we have a pullback diagram of simplicial sets

5.5
\begin{equation} \begin{gathered}\label{equation:homotopy-pullback-square-of-morphism-spaces} \xymatrix { \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{e} \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \ar [d] \\ \{ e\} \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( q(X), q(Y) ). } \end{gathered} \end{equation}

It follows from Proposition 5.1.1.19 that the vertical maps in this diagram are Kan fibrations, so that (5.5) is also a homotopy pullback square. Stated more informally, we have a homotopy fiber sequence

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{e} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( q(X), q(Y) ). \]

Exercise 5.1.1.21. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of simplicial sets, and let $X$ and $Y$ vertices of $\operatorname{\mathcal{C}}$. Show that the induced map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}(q(X), q(Y) )$ is a Kan fibration.

Proposition 5.1.1.19 is an immediate consequence of the following more general assertion:

Proposition 5.1.1.22. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, let $B$ be a simplicial set, let $A \subseteq B$ be a simplicial subset which contains every vertex of $B$, and let $f: A \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then the induced map

\[ \operatorname{Fun}( B, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{C}}) } \{ f\} \rightarrow \operatorname{Fun}( B, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \{ q \circ f \} \]

is a Kan fibration of simplicial sets.

Proof. It follows from Proposition 5.1.1.8 that the simplicial sets $\operatorname{Fun}( B, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{C}}) } \{ f\} $ and $\operatorname{Fun}( B, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \{ q \circ f \} $ are Kan complexes. It will therefore suffice to show that $\theta $ is an isofibration (Corollary 4.4.3.8). This follows from the observation that $\theta $ is a pullback of the restriction map

\[ \operatorname{Fun}( B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( B, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}( A, \operatorname{\mathcal{C}}), \]

which is an isofibration by virtue of Variant 4.4.5.8. $\square$

Proof of Proposition 5.1.1.19. Apply Proposition 5.1.1.22 in the special case $B = \Delta ^1$ and $A = \operatorname{\partial \Delta }^1$. $\square$