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5.2 Morphism Spaces

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 $f$ of $\operatorname{\mathcal{C}}$ satisfying $d_1(f) = X$ and $d_0(f) = Y$ (Definition 1.3.1.1). Morphisms from $X$ to $Y$ can be identified with vertices of a simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$, given by the iterated fiber product

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

In §5.2.1, we show that the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is a Kan complex (Proposition 5.2.1.8), which we refer to as the space of morphisms from $X$ to $Y$ (Construction 5.2.1.1).

Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. We say that $F$ is fully faithful if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ is a homotopy equivalence of Kan complexes (Definition 5.2.2.1). We say that $F$ is essentially surjective if it induces a surjection $\pi _0( \operatorname{\mathcal{C}}^{\simeq } ) \rightarrow \pi _0( \operatorname{\mathcal{D}}^{\simeq } )$ on isomorphism classes of objects. In §5.2.2, we show that $F$ is an equivalence of $\infty $-categories if and only if it is both fully faithful and essentially surjective (Theorem 5.2.2.15). This is essentially a reformulation of the criterion of Theorem 4.5.4.1. Nevertheless, it can be quite useful: the mapping spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ are often more amenable to calculation than the Kan complex $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq }$.

In practice, it is often useful to work with a variant of Construction 5.2.1.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pair of objects $X$ and $Y$. We define simplicial sets $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ by the formulae

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X, Y) = \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Y\} \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}( X,Y) = \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y}. \]

We will refer to $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X, Y)$ as the left-pinched space of morphisms from $X$ to $Y$, and to $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}( X,Y)$ as the right-pinched space of morphisms from $X$ to $Y$. These simplicial sets are also Kan complexes, which can often be described very explicitly:

  • Let $\operatorname{\mathcal{C}}$ be a $(2,1)$-category containing objects $X$ and $Y$, and let $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ denote the Duskin nerve of $\operatorname{\mathcal{C}}$ (Construction 2.3.1.1). Then there are canonical isomorphisms of simplicial sets

    \[ \operatorname{Hom}_{\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}) }^{\mathrm{L}}(X, Y) \simeq \operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) \simeq \operatorname{Hom}_{\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}) }^{\mathrm{R}}(X, Y)^{\operatorname{op}}; \]

    see Example 5.2.5.11.

  • Let $\operatorname{\mathcal{C}}$ be a differential graded category containing objects $X$ and $Y$, and let $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ denote the differential graded nerve of $\operatorname{\mathcal{C}}$ (Definition 2.5.3.7). Then there is a canonical isomorphism of simplicial sets

    \[ \operatorname{Hom}^{\mathrm{R}}_{\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}( X, Y) \simeq \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } ), \]

    where $\mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } )$ denotes the Eilenberg-MacLane space associated to the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$ (Example 5.2.5.13).

  • Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category containing a pair of objects $X$ and $Y$, and let $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ denote the homotopy coherent nerve of $\operatorname{\mathcal{C}}$ (Definition 2.4.3.5). Then there are canonical homotopy equivalences

    \[ \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) }^{\mathrm{L}}( X,Y)^{\operatorname{op}} \leftarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) }^{\mathrm{R}}( X,Y); \]

    see Theorem 5.2.6.4. This is a special case of a more general result (where the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is assumed to be an $\infty $-category rather than a Kan complex), which we prove in §5.2.6.

In §5.2.5, we construct comparison maps

\[ \iota ^{\mathrm{L}}_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X, Y) \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \quad \quad \iota ^{\mathrm{R}}_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X, Y) \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), \]

which we refer to as the pinch inclusion maps, and show that they are homotopy equivalences of Kan complexes (Proposition 5.2.5.9). This follows from a more general statement about the relationship between (co)slice $\infty $-categories and comma objects (Theorem 5.2.4.14), which we formulate and prove in §5.2.4). Our proof will make use of a general detection principle for natural isomorphisms of diagrams (Theorem 5.2.3.8), which we explain in §5.2.3.

Structure

  • Subsection 5.2.1: Morphism Spaces
  • Subsection 5.2.2: Fully Faithful and Essentially Surjective Functors
  • Subsection 5.2.3: Digression: Categorical Mapping Cylinders
  • Subsection 5.2.4: The Comma Construction
  • Subsection 5.2.5: Pinched Morphism Spaces
  • Subsection 5.2.6: Morphism Spaces in the Homotopy Coherent Nerve