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4.6 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 §4.6.1, we show that the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is a Kan complex (Proposition 4.6.1.8), which we refer to as the space of morphisms from $X$ to $Y$ (Construction 4.6.1.1).

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 4.6.2.1). 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 §4.6.2, we show that $F$ is an equivalence of $\infty $-categories if and only if it is both fully faithful and essentially surjective (Theorem 4.6.2.17). 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 $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq }$.

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. In §4.6.3, we associate to every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$ a morphism of Kan complexes

\[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z), \]

which is well-defined up to homotopy (Construction 4.6.3.9). We show that this composition law is unital and associative up to homotopy (Propositions 4.6.3.11 and 4.6.3.12), and therefore determines an enrichment of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ over the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$ (Construction 4.6.3.13 and Remark 4.6.3.14).

In practice, it is often useful to work with a variant of Construction 4.6.1.1. 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 4.6.6.12.

  • 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

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

    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 4.6.6.14).

  • 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

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

    see Theorem 4.6.7.5. 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 §4.6.7.

In §4.6.6, 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 4.6.6.9). This follows from a more general statement about the relationship between (co)slice $\infty $-categories and comma objects (Theorem 4.6.5.14), which we formulate and prove in §4.6.5). Our proof will make use of a general detection principle for natural isomorphisms of diagrams (Theorem 4.6.4.8), which we explain in §4.6.4.

Structure

  • Subsection 4.6.1: Morphism Spaces
  • Subsection 4.6.2: Fully Faithful and Essentially Surjective Functors
  • Subsection 4.6.3: Composition of Morphisms
  • Subsection 4.6.4: Digression: Categorical Mapping Cylinders
  • Subsection 4.6.5: The Comma Construction
  • Subsection 4.6.6: Pinched Morphism Spaces
  • Subsection 4.6.7: Morphism Spaces in the Homotopy Coherent Nerve