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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 $d^{1}_1(f) = X$ and $d^{1}_0(f) = Y$ (Definition 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, which we refer to as the space of morphisms from $X$ to $Y$ (Construction

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 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 This is essentially a reformulation of the criterion of Theorem 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 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 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

  • 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 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

  • 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 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 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 §4.6.8.

In §4.6.5, 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 This follows from a more general statement about the relationship between (co)slice $\infty $-categories and oriented fiber products (Theorem, which we formulate and prove in §4.6.4). Our proof will make use of a general detection principle for natural isomorphisms of diagrams (Theorem, which we explain in §4.6.3.

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. In §4.6.9, 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 We show that this composition law is unital and associative up to homotopy (Propositions and, 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 and Remark


  • Subsection 4.6.1: Morphism Spaces
  • Subsection 4.6.2: Fully Faithful and Essentially Surjective Functors
  • Subsection 4.6.3: Digression: Categorical Mapping Cylinders
  • Subsection 4.6.4: Oriented Fiber Products
  • Subsection 4.6.5: Pinched Morphism Spaces
  • Subsection 4.6.6: Digression: Diagrams in Slice $\infty $-Categories
  • Subsection 4.6.7: Initial and Final Objects
  • Subsection 4.6.8: Morphism Spaces in the Homotopy Coherent Nerve
  • Subsection 4.6.9: Composition of Morphisms