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5.5.6 $\infty $-Categories with a Distinguished Object

In this section, we study pairs $(\operatorname{\mathcal{C}}, C)$, where $\operatorname{\mathcal{C}}$ is a (small) $\infty $-category and $C \in \operatorname{\mathcal{C}}$ is a distinguished object. Our goal is to organize the collection of such pairs into an $\infty $-category. We consider several variants of this construction which are related by inclusion maps

\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}_{\ast }) \hookrightarrow \operatorname{\mathcal{QC}}_{\ast } \hookrightarrow \operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}} \hookrightarrow \operatorname{ \pmb {\mathcal{QC}} }_{\ast }^{\operatorname{lax}}; \]

their interrelationships can be described informally as follows:

  • Morphisms from $(\operatorname{\mathcal{C}}, C)$ to $(\operatorname{\mathcal{D}}, D)$ in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}_{\ast })$ are given by functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which satisfy $F(C) = D$ (that is, $F$ is strictly compatible with the choice of distinguished objects).

  • Morphisms from $(\operatorname{\mathcal{C}}, C)$ to $(\operatorname{\mathcal{D}}, D)$ in the $\infty $-category $\operatorname{\mathcal{QC}}_{\ast }$ are given by pairs $(F,\alpha )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor and $\alpha : F(C) \rightarrow D$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$ (that is, $F$ is compatible with the choice of distinguished objects up to isomorphism). The inclusion $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}_{\ast }) \hookrightarrow \operatorname{\mathcal{QC}}_{\ast }$ is an equivalence of $\infty $-categories (Proposition 5.5.6.6).

  • Morphisms from $(\operatorname{\mathcal{C}}, C)$ to $(\operatorname{\mathcal{D}}, D)$ in the $\infty $-category $\operatorname{\mathcal{QC}}^{\operatorname{lax}}_{\ast }$ are given by pairs $(F,\alpha )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor and $\alpha : F(C) \rightarrow D$ is an morphism in the $\infty $-category $\operatorname{\mathcal{D}}$ which is not required to be an isomorphism; this $\infty $-category contains $\operatorname{\mathcal{QC}}_{\ast }$ as a (non-full) subcategory (Remark 5.5.6.13).

  • The simplicial set $\operatorname{ \pmb {\mathcal{QC}} }_{\ast }^{\operatorname{lax}}$ is an $(\infty ,2)$-category having the same objects and morphisms as $\operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}}$, but which also contains information about non-invertible natural transformations between functors (see Example 5.5.6.14).

Construction 5.5.6.1. Let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of $\infty $-categories (Construction 5.5.4.1), and regard the Kan complex $\Delta ^{0}$ as an object of $\operatorname{\mathcal{QC}}$. We let $\operatorname{\mathcal{QC}}_{\ast }$ denote the coslice simplicial set $\operatorname{\mathcal{QC}}_{\Delta ^{0} / }$.

Proposition 5.5.6.2. The simplicial set $\operatorname{\mathcal{QC}}_{\ast }$ is an $\infty $-category, and the projection map $\operatorname{\mathcal{QC}}_{\ast } \rightarrow \operatorname{\mathcal{QC}}$ is a left fibration of $\infty $-categories.

Proof. By virtue of Proposition 5.5.4.3, the simplicial set $\operatorname{\mathcal{QC}}$ is an $\infty $-category. It follows that for every object $\operatorname{\mathcal{C}}\in \operatorname{\mathcal{QC}}$, the projection map $\operatorname{\mathcal{QC}}_{\operatorname{\mathcal{C}}/} \rightarrow \operatorname{\mathcal{QC}}$ is a left fibration (Corollary 4.3.6.8). Taking $\operatorname{\mathcal{C}}= \Delta ^{0}$, we conclude that the projection map $\operatorname{\mathcal{QC}}_{\ast } \rightarrow \operatorname{\mathcal{QC}}$ is a left fibration, so that $\operatorname{\mathcal{QC}}_{\ast }$ is an $\infty $-category (Remark 4.2.1.5). $\square$

Example 5.5.6.3 (Objects and Morphisms of $\operatorname{\mathcal{QC}}_{\ast }$). The low-dimensional simplices of the $\infty $-category $\operatorname{\mathcal{QC}}_{\ast }$ are easy to describe:

  • The objects of $\operatorname{\mathcal{QC}}_{\ast }$ can be identified with pairs $(\operatorname{\mathcal{C}}, C)$, where $\operatorname{\mathcal{C}}$ is a (small) $\infty $-category and $C \in \operatorname{\mathcal{C}}$ is an object (which we identify with the morphism $\Delta ^{0} \rightarrow \operatorname{\mathcal{C}}$ taking the value $C$).

  • Let $(\operatorname{\mathcal{C}},C)$ and $(\operatorname{\mathcal{D}},D)$ be objects of $\operatorname{\mathcal{QC}}_{\ast }$. A morphism from $(\operatorname{\mathcal{C}}, C)$ to $(\operatorname{\mathcal{D}}, D)$ in the $\infty $-category $\operatorname{\mathcal{QC}}_{\ast }$ can be identified with a pair $(F, \alpha )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories and $\alpha : F(C) \rightarrow D$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$.

Warning 5.5.6.4. By analogy with Definition 3.2.1.1, it would be natural to refer to the objects $(\operatorname{\mathcal{C}}, C)$ of $\operatorname{\mathcal{QC}}_{\ast }$ as pointed $\infty $-categories. We will avoid using this terminology, since it conflicts with another (related but distinct) notion of pointed $\infty $-category that we will consider later (Definition ).

Remark 5.5.6.5 (Comparison with Pointed Spaces). Let us regard the $\infty $-category of spaces $\operatorname{\mathcal{S}}$ as a full subcategory of the $\infty $-category $\operatorname{\mathcal{QC}}$ (Remark 5.5.4.8). The inclusion $\operatorname{\mathcal{S}}\hookrightarrow \operatorname{\mathcal{QC}}$ determines a functor of coslice $\infty $-category $\operatorname{\mathcal{S}}_{\ast } \rightarrow \operatorname{\mathcal{QC}}_{\ast }$, This functor restricts to an isomorphism from $\operatorname{\mathcal{S}}_{\ast }$ with the full subcategory of $\operatorname{\mathcal{QC}}_{\ast }$ spanned by those pairs $(\operatorname{\mathcal{C}}, C)$, where $\operatorname{\mathcal{C}}$ is a Kan complex.

Let $\operatorname{QCat}$ denote the ordinary category whose objects are (small) $\infty $-categories and whose morphisms are functors, and let $\operatorname{QCat}_{\ast }$ denote the coslice category $\operatorname{QCat}_{\Delta ^{0}/ }$. The simplicial enrichment of $\operatorname{QCat}$ (described in Construction 5.5.4.1) determines a simplicial enrichment of the coslice category $\operatorname{QCat}_{\ast }$ (see Variant 5.5.2.3), and Construction 5.5.2.17 yields a comparison map coslice comparison functor

\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}_{\ast } ) = \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}_{ \Delta ^0 / } ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{QCat})_{ \Delta ^{0} / } = \operatorname{\mathcal{QC}}_{\ast }. \]

Proposition 5.5.6.6. The coslice comparison functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}_{\ast } ) \rightarrow \operatorname{\mathcal{QC}}_{\ast }$ is an equivalence of $\infty $-categories.

Proof. By virtue of Theorem 5.5.2.21, it will suffice to show that for every pair of objects $(\operatorname{\mathcal{C}},C), (\operatorname{\mathcal{D}}, D) \in \operatorname{\mathcal{QC}}_{\ast }$, the restriction map

\[ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } = \operatorname{Hom}_{\operatorname{QCat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{QCat}}( \{ C\} , \operatorname{\mathcal{D}}) = \operatorname{Fun}( \{ C\} , \operatorname{\mathcal{D}})^{\simeq } \]

is a Kan fibration. This follows from Proposition 4.4.3.6, since the restriction functor $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \{ C\} , \operatorname{\mathcal{D}})$ is an isofibration of $\infty $-categories (Corollary 4.4.5.3). $\square$

Warning 5.5.6.7. The coslice comparison functor $U: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}_{\ast } ) \rightarrow \operatorname{\mathcal{QC}}_{\ast }$ of Proposition 5.5.6.6 is bijective on vertices: objects of either $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}_{\ast } )$ and $\operatorname{\mathcal{QC}}_{\ast }$ can be identified with pairs $(\operatorname{\mathcal{C}}, C)$, where $\operatorname{\mathcal{C}}$ is an $\infty $-category and $C$ is an object of $\operatorname{\mathcal{C}}$. However, it is not bijective on edges (and is therefore not an isomorphism of simplicial sets). If $(\operatorname{\mathcal{C}},C)$ and $(\operatorname{\mathcal{D}},D)$ are objects of $\operatorname{\mathcal{QC}}_{\ast }$, then a morphism from $(\operatorname{\mathcal{C}}, C)$ to $(\operatorname{\mathcal{D}}, D)$ in the $\infty $-category $\operatorname{\mathcal{QC}}_{\ast }$ can be identified with a pair $(F, \alpha )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories and $\alpha : F(C) \rightarrow D$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$. The pair $(F, \alpha )$ belongs to the image of $U$ if and only if the isomorphism $\alpha $ is a degenerate edge of $\operatorname{\mathcal{D}}$ (which guarantees in particular that $F(C) = D$).

We now introduce an enlargement of the $\infty $-category $\operatorname{\mathcal{QC}}_{\ast }$.

Construction 5.5.6.8. Let $\operatorname{ \pmb {\mathcal{QC}} }$ denote the $(\infty ,2)$-category of $\infty $-categories (Construction 5.5.5.1), and regard the Kan complex $\Delta ^{0}$ as an object of $\operatorname{ \pmb {\mathcal{QC}} }$. We let $\operatorname{ \pmb {\mathcal{QC}} }^{\operatorname{lax}}_{\ast }$ denote the coslice simplicial set $\operatorname{ \pmb {\mathcal{QC}} }_{\Delta ^{0} / }$.

Proposition 5.5.6.9. The simplicial set $\operatorname{ \pmb {\mathcal{QC}} }_{\ast }^{\operatorname{lax}}$ is an $(\infty ,2)$-category. Moreover, the projection map $\operatorname{ \pmb {\mathcal{QC}} }_{\ast }^{\operatorname{lax}} \rightarrow \operatorname{ \pmb {\mathcal{QC}} }$ is an interior fibration of $(\infty ,2)$-categories.

Proof. It follows from Proposition 5.5.5.2 that $\operatorname{ \pmb {\mathcal{QC}} }$ is an $(\infty ,2)$-category. The desired conclusion now follows from Corollary 5.4.3.4 and Proposition 5.4.3.1. $\square$

Definition 5.5.6.10. We let $\operatorname{\mathcal{QC}}^{\operatorname{lax}}_{\ast }$ denote the pith of the $(\infty ,2)$-category $\operatorname{ \pmb {\mathcal{QC}} }^{\operatorname{lax}}_{\ast }$ (see Construction 5.4.5.1).

Proposition 5.5.6.11.

$(1)$

The simplicial set $\operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}}$ is an $\infty $-category.

$(2)$

The projection map $U: \operatorname{ \pmb {\mathcal{QC}} }_{\ast } = \operatorname{ \pmb {\mathcal{QC}} }_{\Delta ^{0} / } \rightarrow \operatorname{ \pmb {\mathcal{QC}} }$ restricts to a functor

\[ U_0: \operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}} = \operatorname{Pith}( \operatorname{ \pmb {\mathcal{QC}} }_{\ast }^{\operatorname{lax}} ) \rightarrow \operatorname{Pith}(\operatorname{ \pmb {\mathcal{QC}} }) = \operatorname{\mathcal{QC}}. \]
$(3)$

The diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}} \ar [r] \ar [d]^{U_0} & \operatorname{ \pmb {\mathcal{QC}} }_{\ast }^{\operatorname{lax}} \ar [d]^{U} \\ \operatorname{\mathcal{QC}}\ar [r] & \operatorname{ \pmb {\mathcal{QC}} }} \]

is a pullback square of simplicial sets.

$(4)$

The functor $U_0$ is a cocartesian fibration of $\infty $-categories.

Proof. Assertion $(1)$ follows from Proposition 5.4.5.6. Since $U$ is an interior fibration (Proposition 5.5.6.9), assertions $(2)$ and $(3)$ follow from Proposition 5.4.7.9. Assertion $(4)$ is a special case of Corollary 5.4.7.10. $\square$

Example 5.5.6.12 (Objects and Morphisms of $\operatorname{\mathcal{QC}}^{\operatorname{lax}}_{\ast }$). The inclusion of simplicial sets $\operatorname{\mathcal{QC}}\hookrightarrow \operatorname{ \pmb {\mathcal{QC}} }$ induces a functor of $\infty $-categories $\iota : \operatorname{\mathcal{QC}}_{\ast } \hookrightarrow \operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}}$. The functor $\iota $ is bijective on vertices. In particular, we can identify the objects of $\operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}}$ with pairs $(\operatorname{\mathcal{C}}, C)$, where $\operatorname{\mathcal{C}}$ is a (small) $\infty $-category and $C \in \operatorname{\mathcal{C}}$ is an object. However, it is not bijective on objects. Unwinding the definitions, we see that a morphism $\widetilde{F}$ from $(\operatorname{\mathcal{C}}, C)$ to $(\operatorname{\mathcal{D}}, D)$ in the $\infty $-category $\operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}}$ can be identified with a pair $(F,\alpha )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories and $\alpha : F(C) \rightarrow D$ is a morphism in the $\infty $-category $\operatorname{\mathcal{D}}$. For every such pair $(F,\alpha )$, the following conditions are equivalent:

  • The morphism $\widetilde{F} = (F,\alpha )$ belongs to the image of the inclusion map $\iota : \operatorname{\mathcal{QC}}_{\ast } \hookrightarrow \operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}}$.

  • The morphism $\alpha : F(C) \rightarrow D$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$.

  • The morphism $\widetilde{F}$ is $U_0$-cocartesian, where $U_0: \operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}} \rightarrow \operatorname{\mathcal{QC}}$ is the cocartesian fibration of Proposition 5.5.6.11.

Remark 5.5.6.13. The inclusion map $\iota : \operatorname{\mathcal{QC}}_{\ast } \hookrightarrow \operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}}$ is an isomorphism from $\operatorname{\mathcal{QC}}_{\ast }$ to the (non-full) subcategory of $\operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}}$ spanned by those morphisms which satisfy the conditions of Example 5.5.6.12. In other words, the projection map $\operatorname{\mathcal{QC}}_{\ast } \rightarrow \operatorname{\mathcal{QC}}$ is the underlying left fibration of the cocartesian fibration $\operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}} \rightarrow \operatorname{\mathcal{QC}}$ (see Corollary 5.4.7.11).

Note that the inclusion map $\operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}} = \operatorname{Pith}( \operatorname{ \pmb {\mathcal{QC}} }_{\ast }^{\operatorname{lax}} ) \hookrightarrow \operatorname{ \pmb {\mathcal{QC}} }_{\ast }^{\operatorname{lax}}$ is bijective on simplices of dimension $\leq 1$ (Remark 5.4.5.2). However, it is not degenerate at the level of $2$-simplices.

Example 5.5.6.14 ($2$-Simplices of $\operatorname{ \pmb {\mathcal{QC}} }_{\ast }^{\operatorname{lax}}$). By virtue of Example 5.5.6.12, a morphism of simplicial sets $\sigma _0: \operatorname{\partial \Delta }^2 \rightarrow \operatorname{ \pmb {\mathcal{QC}} }_{\ast }^{\operatorname{lax}}$ can be identified with the following data:

  • A collection of $\infty $-categories $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ equipped with distinguished objects $C \in \operatorname{\mathcal{C}}$, $D \in \operatorname{\mathcal{D}}$, and $E \in \operatorname{\mathcal{E}}$.

  • A collection of functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, and $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$.

  • A collection of morphisms $\alpha : F(C) \rightarrow D$, $\beta : G(D) \rightarrow E$, and $\gamma : H(C) \rightarrow E$ in the $\infty $-categories $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$.

Unwinding the definitions, we see that extending $\sigma _0$ to a $2$-simplex $\sigma $ of $\operatorname{ \pmb {\mathcal{QC}} }_{\ast }^{\operatorname{lax}}$ is equivalent to choosing a natural transformation of functors $\mu : (G \circ F) \rightarrow H$ and a morphism of simplicial sets $\theta : \operatorname{\raise {0.1ex}{\square }}^{2} \rightarrow \operatorname{\mathcal{E}}$ whose restriction to the boundary $\operatorname{\partial \raise {0.1ex}{\square }}^2$ is indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ (G \circ F)(C) \ar [r]^-{ \mu (C) } \ar [d]^-{G(\alpha )} & H(C) \ar [d]^{\gamma } \\ G(D) \ar [r]^-{ \beta } & E. } \]

Moreover:

  • The $2$-simplex $\sigma $ belongs to the image of $\operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}} \hookrightarrow \operatorname{ \pmb {\mathcal{QC}} }_{\ast }^{\operatorname{lax}}$ if and only if $\mu : G \circ F \rightarrow H$ is an isomorphism in the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.

  • The $2$-simplex $\sigma $ belongs to the image of $\operatorname{\mathcal{QC}}_{\ast } \hookrightarrow \operatorname{ \pmb {\mathcal{QC}} }_{\ast }^{\operatorname{lax}}$ if and only $\mu $, $\alpha $, $\beta $, and $\gamma $ are all isomorphisms.

  • The $2$-simplex $\sigma $ belongs to the image of $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}_{\ast } ) \hookrightarrow \operatorname{ \pmb {\mathcal{QC}} }_{\ast }^{\operatorname{lax}}$ if and only if $\mu $, $\alpha $, $\beta $, and $\gamma $ are identity morphisms (so that $H = G \circ F$, $D = F(C)$, and $E = G(D)$) and the morphism $\theta : \operatorname{\raise {0.1ex}{\square }}^{2} \rightarrow \operatorname{\mathcal{E}}$ is constant.