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11.4 Obsolete Constructions

The tags in this section were removed from the main text because they make use of a definition or convention which was modified.

Definition 11.4.0.1. Let $(X,x)$ and $(Y,y)$ be pointed simplicial sets, and suppose we are given a pair of pointed maps $f_0, f_1: X \rightarrow Y$. A pointed homotopy from $f_0$ to $f_1$ is a morphism $h: \Delta ^1 \times X_{} \rightarrow Y_{}$ for which $f_0 = h|_{ \{ 0\} \times X_{}}$, $f_1 = h|_{ \{ 1\} \times X_{} }$, and $h|_{ \Delta ^1 \times \{ x\} }$ is the degenerate edge associated to the vertex $y \in Y$.

Remark 11.4.0.2. The formation of Eilenberg-MacLane spaces $A \mapsto \mathrm{K}(A,n)$ is defined for every integer $n$. However, it is only interesting for $n \geq 0$: if $n$ is negative, then the simplicial abelian group $\mathrm{K}(A,n)$ is trivial (that is, it is isomorphic to $\Delta ^0$ as a simplicial set).

Example 11.4.0.3. The horn $\Lambda ^{0}_{0}$ is the empty simplicial set (and therefore coincides with the boundary $\operatorname{\partial \Delta }^{0}$).

Remark 11.4.0.5. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be a morphism of simplicial sets and let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of simplicial sets. Then the projection map $\pi : \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ restricts to a projection map $\pi ^{\operatorname{CCart}}: \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$. Moreover, for each vertex $C \in \operatorname{\mathcal{C}}$, the isomorphism $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \simeq \operatorname{Fun}_{ / \operatorname{\mathcal{D}}_ C }( \operatorname{\mathcal{D}}_ C, \operatorname{\mathcal{E}}_ C )$ of Remark 4.5.9.8 restricts to an isomorphism of full subcategories $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}) \simeq \operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{D}}_ C }( \operatorname{\mathcal{D}}_ C, \operatorname{\mathcal{E}}_ C )$.

Proposition 11.4.0.6. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $U$ is exponentiable (Definition 4.5.9.10).

$(2)$

Let $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ be a simplicial subset for which the restriction $U|_{\operatorname{\mathcal{D}}_0}$ is exponentiable, let $T: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ be an isofibration in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}}$, and set $\operatorname{\mathcal{E}}_0 = \operatorname{\mathcal{D}}_0 \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}'_0 = \operatorname{\mathcal{D}}_0 \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}'$. Then the induced map

\[ \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Res}_{ \operatorname{\mathcal{D}}_0/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}_0 ) \times _{ \operatorname{Res}_{\operatorname{\mathcal{D}}_0/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}'_0) } \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}' ) \]

is also an isofibration.

$(3)$

For every isofibration $T: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}}$, the induced map

\[ \operatorname{Res}_{ \operatorname{\mathcal{D}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Res}_{\operatorname{\mathcal{D}}/ \operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}') \]

is also an isofibration.

$(4)$

For every isofibration of $\infty $-categories $T_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}'_0$, the induced map

\[ \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{E}}_0 ) \rightarrow \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{E}}'_0 ) \]

is also an isofibration.

Construction 11.4.0.7 (Direct Images). Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be morphisms of simplicial sets. For every integer $n \geq 0$, we let $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})_{n}$ denote the collection of pairs $(\sigma , f)$, where $\sigma $ is an $n$-simplex of $\operatorname{\mathcal{C}}$ and $f: \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is a morphism for which the composition

\[ \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\xrightarrow {f} \operatorname{\mathcal{E}}\xrightarrow {V} \operatorname{\mathcal{D}} \]

coincides with projection onto the second factor. Note that every nondecreasing function $\alpha : [m] \rightarrow [n]$ induces a map

\[ \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})_{n} \rightarrow \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})_{m} \quad \quad (\sigma ,f) \mapsto (\alpha ^{\ast }(\sigma ), f' ), \]

where $f'$ denotes the composite map

\[ \Delta ^{m} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\xrightarrow { \alpha \times \operatorname{id}} \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\xrightarrow {f} \operatorname{\mathcal{E}}. \]

This construction is compatible with composition, and therefore endows $\{ \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}})_{n} \} _{n \geq 0}$ with the structure of a simplicial set $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) = \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})_{\bullet }$ which we will refer to as the direct image of $\operatorname{\mathcal{E}}$ along $U$.

Note that the construction $(\sigma ,f) \mapsto \sigma $ determines a morphism of simplicial sets $\pi : \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$. Moreover, there is a tautological evaluation map $\operatorname{ev}: \operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}$, which carries an $n$-simplex $( \widetilde{\sigma }, (\sigma ,f) )$ of the fiber product $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ to the $n$-simplex of $\operatorname{\mathcal{E}}$ given by the composite map $\Delta ^{n} \xrightarrow { \operatorname{id}\times \widetilde{\sigma } } \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\xrightarrow {f} \operatorname{\mathcal{E}}$.

Remark 11.4.0.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will often abuse terminology by referring to a functor $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ if there exists a natural transformation $\alpha : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{C}}) } \rightarrow \mathscr {H}|_{\operatorname{Tw}(\operatorname{\mathcal{C}})}$ which satisfies condition $(\ast )$ of Definition 8.3.5.1. In this case, we will say that $\alpha $ exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$.

Notation 11.4.0.9 (The Universal Mapping Simplex). Let $\operatorname{Set_{\Delta }}$ denote the category of simplicial sets and let $\operatorname{N}_{\bullet }( \operatorname{Set_{\Delta }})$ denote its nerve. For each $n \geq 0$, we can identify $\operatorname{N}_{n}(\operatorname{Set_{\Delta }})$ with the set of functors from the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \} $ to the category $\operatorname{Set_{\Delta }}$. In what follows, we will typically denote such a functor by $\overrightarrow {X}$ and will denote its value on an integer $0 \leq i \leq n$ by $X(i)$. Let $\operatorname{N}^{+}_{n}(\operatorname{Set_{\Delta }})$ denote the set of pairs $( \overrightarrow {X}, \sigma )$, where $\overrightarrow {X}: [n] \rightarrow \operatorname{Set_{\Delta }}$ is a functor and $\sigma $ is an $n$-simplex of the simplicial set $X(0)$. Every nondecreasing function $\alpha : [m] \rightarrow [n]$ determines a map $\alpha ^{\ast }: \operatorname{N}^{+}_{n}(\operatorname{Set_{\Delta }}) \rightarrow \operatorname{N}^{+}_{m}(\operatorname{Set_{\Delta }})$, which carries $( \overrightarrow {X}, \sigma )$ to the pair $( \overrightarrow {X} \circ \alpha , \sigma ' )$ where $\sigma '$ denotes the composite map

\[ \Delta ^{m} \xrightarrow {\alpha } \Delta ^{n} \xrightarrow {\sigma } X(0)x \rightarrow X( \alpha (0) ). \]

The construction $[n] \mapsto \operatorname{N}^{+}_{n}( \operatorname{Set_{\Delta }})$ determines a simplicial set $\operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }})$, which we will refer to as the universal mapping simplex.

Remark 11.4.0.10. The universal mapping simplex $\operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }})$ is equipped with a morphism of simplicial sets $\pi : \operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})$, given on $n$-simplices by the formula $\pi ( \overrightarrow {X}, \sigma ) = \overrightarrow {X}$. We will refer to $\pi $ as the projection map. Unwinding the definitions, we see that for every simplicial set $K$, there is a canonical isomorphism of simplicial sets

\[ K \simeq \{ K\} \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }}) } \operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }}). \]

Definition 11.4.0.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. We say that a left fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is corepresentable by $X$ if there exists an initial object $\widetilde{X} \in \operatorname{\mathcal{E}}$ satisfying $U( \widetilde{X} ) = X$. In this case, we say that the object $\widetilde{X}$ exhibits $U$ as a left fibration corepresented by $X$. We say that the left fibration $U$ is corepresentable if it is corepresentable by $X$ for some object $X \in \operatorname{\mathcal{C}}$.

We say that a right fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is representable by $X$ if there exists a final object $\widetilde{X} \in \operatorname{\mathcal{E}}$ satisfying $U( \widetilde{X} ) = X$. In this case, we say that the object $\widetilde{X}$ exhibits $U$ as a right fibration represented by $X$. We say that the fibration $U$ is representable if it is representable by $X$ for some object $X \in \operatorname{\mathcal{C}}$.

Warning 11.4.0.14. Throughout most of this book, we will often employ the following conventions:

  • If $\operatorname{\mathcal{C}}$ is an $\infty $-category and $W$ is a collection of morphisms of $\operatorname{\mathcal{C}}$, then we let $\operatorname{\mathcal{C}}[W^{-1}]$ denote a localization of $\operatorname{\mathcal{C}}$ with respect to $W$ (Remark 6.3.2.2).

  • If $\operatorname{\mathcal{C}}$ is an ordinary category, we abuse terminology by identifying $\operatorname{\mathcal{C}}$ with the associated $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.

Beware that these conventions are in conflict with one another. If $W$ is a collection of morphisms in an ordinary category $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}[W^{-1}]$ denotes the strict localization of Remark 6.3.0.2, then the $\infty $-categorical localization $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})[W^{-1}]$ is generally not equivalent to the nerve $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}[W^{-1}] )$. To avoid confusion, we will henceforth use the notation $\operatorname{\mathcal{C}}[W^{-1}]$ only for the $\infty $-categorical notion of localization, unless otherwise specified.

Notation 11.4.0.18. This tag became obsolete because the convention for $2$-categories of paths was changed.

Remark 11.4.0.19. This tag referred to the use of the term “equivalence” in higher category theory, which has now been replaced by “isomorphism”.

Remark 11.4.0.23. This tag was removed because the definition of $2$-category was changed.

Let $\operatorname{\mathcal{C}}$ be a strictly unitary $2$-category. Then the triangle identity $(T)$ of Definition 2.2.1.1 can be stated more simply as the assertion that for every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$, the associativity constraint $\alpha _{g, \operatorname{id}_ Y, f}$ coincides with the identity map from $g \circ (\operatorname{id}_{Y} \circ f) = g \circ f = (g \circ \operatorname{id}_{Y} ) \circ f$ to itself.

Remark 11.4.0.24. By swapping the roles of the monomorphisms $i: A_{} \hookrightarrow B_{}$ and $i': A'_{} \hookrightarrow B'_{}$ in the proof of Proposition 11.6.0.135, we obtain a proof of Theorem 3.1.3.1 (which is essentially the same as the proof given in §3.1.3).

Construction 11.4.0.25 (The $\infty $-Category of Pointed Spaces). Let $(\operatorname{Set_{\Delta }})_{\ast }$ denote the category whose objects are pointed simplicial sets $(X,x)$ and pointed morphisms between them. We regard $(\operatorname{Set_{\Delta }})_{\ast }$ as a simplicial category, where the simplicial set of morphisms from $(X,x)$ to $(Y,y)$ is given by the fiber product $\operatorname{Fun}(X,Y) \times _{ \operatorname{Fun}( \{ x\} , Y) } \{ y\} $. Let $(\operatorname{Set}_{\Delta }^{\circ })_{\ast }$ denote the full subcategory of $(\operatorname{Set_{\Delta }})_{\ast }$ spanned by the pointed Kan complexes, so that $(\operatorname{Set}_{\Delta }^{\circ })_{\ast }$ inherits the structure of a simplicial category. We let $\operatorname{\mathcal{S}}_{\ast }$ denote the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( (\operatorname{Set}_{\Delta }^{\circ })_{\ast } )$. Corollary 3.1.3.3 (and Remark 3.1.1.9) guarantee that the simplicial category $(\operatorname{Set}_{\Delta }^{\circ })_{\ast }$ is locally Kan, so the simplicial set $\operatorname{\mathcal{S}}_{\ast } = \operatorname{N}_{\bullet }^{\operatorname{hc}}( (\operatorname{Set}_{\Delta }^{\circ })_{\ast } )$ is an $\infty $-category (Theorem 2.4.5.1). We will refer to $\operatorname{\mathcal{S}}_{\ast }$ as the $\infty $-category of pointed spaces.

Remark 11.4.0.26. The definition of the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ can be viewed as a special case of Construction 2.4.6.1, applied to the simplicial category $(\operatorname{Set}_{\Delta }^{\circ })_{\ast }$ of Construction 11.4.0.25. Invoking Proposition 2.4.6.9, we see that the category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ of Construction 3.2.1.12 can be identified with the homotopy category of the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$ (as suggested by the notation).

Remark 11.4.0.27. In §, we will give a different description of the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$ (at least up to equivalence): it can be realized as the $\infty $-category of pointed objects of the $\infty $-category $\operatorname{\mathcal{S}}$ (that is, the full subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{S}})$ spanned by those diagrams $f: X \rightarrow Y$ where Kan complex $X$ is contractible; see Example ). Beware that the analogous statement does not hold at the level of homotopy categories: there is no formal mechanism to extract the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ from the unpointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ (see Warning 3.2.1.11).