Subsection 7.6.5 (03HG): Sequential Limits and Colimits

7.6.5 Sequential Limits and Colimits

Throughout this section, we let $\operatorname{\mathbf{Z}}_{\geq 0}$ denote the set of nonnegative integers, endowed with its usual ordering.

Definition 7.6.5.1 (Towers). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A tower in $\operatorname{\mathcal{C}}$ is a functor $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$. We say that $\operatorname{\mathcal{C}}$ admits sequential limits if every tower in $\operatorname{\mathcal{C}}$ has a limit, and that $\operatorname{\mathcal{C}}$ admits sequential colimits if every diagram $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ) \rightarrow \operatorname{\mathcal{C}}$ has a colimit. We say that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves sequential limits if it preserves limits indexed by the simplicial set $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} )$, and that it preserves sequential colimits if it preserves colimits indexed by the simplicial set $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )$.

Notation 7.6.5.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will generally abuse notation by identifying a functor $X: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ) \rightarrow \operatorname{\mathcal{C}}$ with the collection of objects $\{ X(n) \} _{n \geq 0}$ and morphisms $f_{n}: X(n+1) \rightarrow X(n)$ obtained by the evaluating $X$ on the edges of $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )$ corresponding to ordered pairs of the form $(n,n+1)$; we will depict the pair $( \{ X(n) \} _{n \geq 0}, \{ f_ n \} _{n \geq 0} )$ as a diagram

\[ X(0) \xrightarrow { f_0 } X(1) \xrightarrow { f_1} X(2) \xrightarrow { f_2} X(3) \xrightarrow { f_3} X(4) \rightarrow \cdots \]

Similarly, we abuse notation by identifying towers $X: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ with diagrams

\[ \cdots \rightarrow X(4) \xrightarrow { f_3 } X(3) \xrightarrow { f_2 } X(2) \xrightarrow { f_1} X(1) \xrightarrow { f_0 } X(0). \]

Beware that the convention of Notation 7.6.5.2 is slightly abusive: the simplicial set $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )$ has nondegenerate simplices in every dimension, so a functor $X: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ) \rightarrow \operatorname{\mathcal{C}}$ is not literally determined by its underlying diagram

\[ X(0) \xrightarrow { f_0 } X(1) \xrightarrow { f_1} X(2) \xrightarrow { f_2} X(3) \xrightarrow { f_3} X(4) \rightarrow \cdots \]

However, the abuse is essentially harmless:

Remark 7.6.5.3. Let $\operatorname{Spine}[ \operatorname{\mathbf{Z}}_{\geq 0} ]$ denote the simplicial subset of $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )$ whose $k$-simplices are sequences of nonnegative integers $(n_0, n_1, \cdots , n_ k )$ satisfying $n_0 \leq n_1 \leq \cdots \leq n_ k \leq n_0 + 1$. Then $\operatorname{Spine}[ \operatorname{\mathbf{Z}}_{\geq 0} ]$ is a $1$-dimensional simplicial set, which corresponds (under the equivalence of Proposition 1.1.6.9) to the directed graph $G$ indicated in the diagram

\[ 0 \rightarrow 1 \rightarrow 2 \rightarrow 3 \rightarrow \cdots . \]

Moreover, the linearly ordered set $(\operatorname{\mathbf{Z}}_{\geq 0}, \leq )$ can be identified with the path category $\operatorname{Path}[G]$ of Construction 1.3.7.1. It follows that the inclusion map $\operatorname{Spine}[ \operatorname{\mathbf{Z}}_{\geq 0} ] \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )$ is inner anodyne (Proposition 1.5.7.3).

In particular, for any $\infty $-category $\operatorname{\mathcal{C}}$, the restriction map

\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{Spine}[ \operatorname{\mathbf{Z}}_{\geq 0} ], \operatorname{\mathcal{C}}) \]

is a trivial Kan fibration (Theorem 1.5.7.1). Stated more informally, every sequence of composable morphisms

\[ X(0) \xrightarrow { f_0 } X(1) \xrightarrow { f_1} X(2) \xrightarrow { f_2} X(3) \xrightarrow { f_3} X(4) \rightarrow \cdots \]

admits an essentially unique extension to a functor $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ) \rightarrow \operatorname{\mathcal{C}}$.

Example 7.6.5.4. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category, and suppose we are given a collection of objects $\{ X(n) \} _{n \geq 0}$ and morphisms $f_ n: X(n) \rightarrow X(n+1)$ in $\operatorname{\mathcal{C}}$. It follows from Remark 7.6.5.3 that the diagram

\[ X(0) \xrightarrow { f_0 } X(1) \xrightarrow { f_1} X(2) \xrightarrow { f_2} X(3) \xrightarrow { f_3} X(4) \rightarrow \cdots \]

can be extended to a functor $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$. In fact, there is a preferred choice of such an extension, which is uniquely determined by the requirement that it factors through the inclusion map $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$.

Remark 7.6.5.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and suppose we are given a tower $X: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$, which we depict as a diagram

\[ \cdots \rightarrow X(4) \xrightarrow { f_3 } X(3) \xrightarrow { f_2 } X(2) \xrightarrow { f_1} X(1) \xrightarrow { f_0 } X(0), \]

having a limit $\varprojlim (X)$. Then, for every object $Y \in \operatorname{\mathcal{C}}$, the map of sets

\[ \theta : \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, \varprojlim (X) ) \rightarrow \varprojlim ( \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, X(n) ) ) \]

is surjective. To prove this, suppose we are given a collection of morphisms $g_ n: Y \rightarrow X(n)$ satisfying $[f_{n}] \circ [g_{n+1} ] = [ g_ n ]$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. Then, for each $n \geq 0$, we can choose a $2$-simplex $\sigma _ n$ in $\operatorname{\mathcal{C}}$ as indicated in the diagram

\[ \xymatrix@C =50pt@R=50pt{ & X(n+1) \ar [dr]^-{ f_ n } & \\ Y \ar [ur]^{ g_{n+1} } \ar [rr]^-{ g_ n } & & X(n). } \]

Let $X_0$ denote the restriction of $X$ to the spine $\operatorname{Spine}[ \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} ] \subset \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} )$. Then the collection of $2$-simplices $\{ \sigma _ n \} _{n \geq 0}$ determines an extension of $X_0$ to a diagram $\overline{X}_0: \operatorname{Spine}[ \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} ]^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ carrying the cone point to the object $Y$. The isomorphism class of this extension can be identified with a morphism $[g]: Y \rightarrow \varprojlim (X_0) \simeq \varprojlim (X)$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, which is a preimage of the sequence $\{ [g_ n] \} _{n \geq 0}$ under the function $\theta $.

Warning 7.6.5.6. In the situation of Remark 7.6.5.5, the map

need not be injective. That is, the forgetful functor $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ generally does not preserve sequential limits (or colimits).

Example 7.6.5.7. Fix a prime number $p$. For every integer $n \geq 0$, let $p^{n} \operatorname{\mathbf{Z}}$ denote the cyclic subgroup of $\operatorname{\mathbf{Z}}$ generated by $p^{n}$, so that we have a tower of classifying simplicial sets

7.74

\begin{equation} \begin{gathered}\label{equation:bad-inverse-limit} \xymatrix@C =50pt@R=50pt{ \cdots \ar [r] & B_{\bullet }( p^{3} \operatorname{\mathbf{Z}}) \ar [r] & B_{\bullet }( p^2 \operatorname{\mathbf{Z}}) \ar [r] & B_{\bullet }( p \operatorname{\mathbf{Z}}) \ar [r] & B_{\bullet }(\operatorname{\mathbf{Z}}). } \end{gathered} \end{equation}

Then:

The tower (7.74) has a limit in the ordinary category of simplicial sets, given by the simplicial set $\Delta ^0$ (which we can identify with the classifying simplicial set for the trivial group $(0) = \bigcap _{n \geq 0} p^{n} \operatorname{\mathbf{Z}}$).
The simplicial set $\Delta ^0$ is also a limit of the tower (7.74) in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.
In the $\infty $-category $\operatorname{\mathcal{S}}$, the tower (7.74) has a different limit, which has uncountably many connected components (see Remark ).

We now give some easy examples of sequential limits and colimits.

Example 7.6.5.8 (Sequential Colimits in $\operatorname{\mathcal{QC}}$). Suppose we are given a collection of $\infty $-categories $\{ \operatorname{\mathcal{C}}(n) \} _{n \geq 0}$ and functors $F_ n: \operatorname{\mathcal{C}}(n) \rightarrow \operatorname{\mathcal{C}}(n+1)$, which we view as a diagram

\[ \operatorname{\mathcal{C}}(0) \xrightarrow { F_0 } \operatorname{\mathcal{C}}(1) \xrightarrow { F_1} \operatorname{\mathcal{C}}(2) \xrightarrow { F_2} \operatorname{\mathcal{C}}(3) \rightarrow \cdots \]

Let $\varinjlim _{n} \operatorname{\mathcal{C}}(n)$ denote the colimit of this diagram (formed in the ordinary category of simplicial sets). Then $\varinjlim _{n} \operatorname{\mathcal{C}}(n)$ is also an $\infty $-category, which is also a colimit of the associated diagram $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ) \rightarrow \operatorname{\mathcal{QC}}$. This is a special case of Corollary 9.1.7.3.

Variant 7.6.5.9 (Sequential Colimits in $\operatorname{\mathcal{S}}$). Suppose we are given a collection of Kan complexes $\{ X(n) \} _{n \geq 0}$ and morphisms $f_ n: X(n) \rightarrow X(n+1)$, which we view as a diagram

\[ X(0) \xrightarrow { f_0 } X(1) \xrightarrow { f_1} X(2) \xrightarrow { f_2} X(3) \rightarrow \cdots \]

Let $\varinjlim _{n} X(n)$ denote the colimit of this diagram (formed in the ordinary category of simplicial sets). Then $\varinjlim _{n} X(n)$ is also a Kan complex, which is also a colimit of the associated diagram $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ) \rightarrow \operatorname{\mathcal{S}}$. See Variant 9.1.7.4.

Example 7.6.5.10 (Towers of Isofibrations). Suppose we are given a collection of $\infty $-categories $\{ \operatorname{\mathcal{C}}(n) \} _{n \geq 0}$ and functors $F_ n: \operatorname{\mathcal{C}}(n+1) \rightarrow \operatorname{\mathcal{C}}(n)$, which we view as a tower

\[ \cdots \rightarrow \operatorname{\mathcal{C}}(4) \xrightarrow { F_3 } \operatorname{\mathcal{C}}(3) \xrightarrow { F_2 } \operatorname{\mathcal{C}}(2) \xrightarrow { F_1} \operatorname{\mathcal{C}}(1) \xrightarrow { F_0 } \operatorname{\mathcal{C}}(0) \]

If each of the functors $F_{n}$ is an isofibration, then the limit $\varprojlim _{n} \operatorname{\mathcal{C}}(n)$ (formed in the ordinary category of simplicial sets) is also an $\infty $-category, which can be also be viewed as a limit of the associated tower $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{QC}}$. This follows by combining Example 4.5.6.8, Example 7.5.5.3, and Proposition 7.5.5.7.

Variant 7.6.5.11 (Towers of Kan Fibrations). Suppose we are given a collection of Kan complexes $\{ X(n) \} _{n \geq 0}$ and morphisms $f_ n: X(n+1) \rightarrow X(n)$, which we view as a tower

\[ \cdots \rightarrow X(4) \xrightarrow { f_3 } X(3) \xrightarrow { f_2 } X(2) \xrightarrow { f_1} X(1) \xrightarrow { f_0 } X(0). \]

If each of the morphisms $f_ n$ is a Kan fibration, then the limit $\varprojlim _{n} X(n)$ (formed in the ordinary category of simplicial sets) is also a Kan complex, which can be also be viewed as a limit of the associated tower $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{S}}$ (combine Example 7.6.5.10 with Remark 7.4.4.4).

Variant 7.6.5.12 (Limits of General Towers). Suppose we are given a sequence of $\infty $-categories $\{ \operatorname{\mathcal{C}}(n) \} _{n \geq 0}$ and functors $F_{n}: \operatorname{\mathcal{C}}(n+1) \rightarrow \operatorname{\mathcal{C}}(n)$, which we view as a tower

7.75

\begin{equation} \label{equation:sequential-limits-in-QCat} \cdots \rightarrow \operatorname{\mathcal{C}}(4) \xrightarrow { F_3 } \operatorname{\mathcal{C}}(3) \xrightarrow { F_2 } \operatorname{\mathcal{C}}(2) \xrightarrow { F_1} \operatorname{\mathcal{C}}(1) \xrightarrow { F_0 } \operatorname{\mathcal{C}}(0) \end{equation}

If the functors $F_{n}$ are not assumed to be isofibrations, then the limit $\varprojlim _{n} \operatorname{\mathcal{C}}(n)$ (formed in the ordinary category of simplicial sets) might not be a limit of the associated tower in $\operatorname{\mathcal{QC}}$ (for example, $\varprojlim _{n} \operatorname{\mathcal{C}}(n)$ might fail to be an $\infty $-category). Nevertheless, we can always compute the relevant limit in $\operatorname{\mathcal{QC}}$ by replacing (7.75) by a levelwise equivalent diagram of $\infty $-categories in which the transition functors are isofibrations. For example, we can replace (7.75) by the isofibrant tower of iterated homotopy fiber products

\[ \cdots \rightarrow \operatorname{\mathcal{C}}(2) \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}(1)} (\operatorname{\mathcal{C}}(1) \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}(0)} \operatorname{\mathcal{C}}(0)) \rightarrow \operatorname{\mathcal{C}}(1) \times _{\operatorname{\mathcal{C}}(0)}^{\mathrm{h}} \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(0). \]

Let us denote the limit of this tower (in the category of simplicial sets) by

\[ \cdots \times _{ \operatorname{\mathcal{C}}(3) }^{\mathrm{h}} \operatorname{\mathcal{C}}(3) \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}(2)} \operatorname{\mathcal{C}}(2) \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}(1) } \operatorname{\mathcal{C}}(1) \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}(0)} \operatorname{\mathcal{C}}(0). \]

It is an $\infty $-category whose objects can be identified with sequences of pairs $\{ (C_ n, \alpha _ n) \} _{n \geq 0}$, where each $C_{n}$ is an object of the $\infty $-category $\operatorname{\mathcal{C}}(n)$ and each $\alpha _{n}: F_{n}( C_{n+1} ) \xrightarrow {\sim } C_ n$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}(n)$. Combining Example 7.6.5.10 with Remark 7.1.1.8, we see that it can be identified with a limit of the diagram (7.75) in the $\infty $-category $\operatorname{\mathcal{QC}}$.

Sequential limits are useful for building more complicated types of limits.

Proposition 7.6.5.13. Suppose we are given a diagram of simplicial sets

\[ K(0) \rightarrow K(1) \rightarrow K(2) \rightarrow K(3) \rightarrow \cdots \]

having colimit $K$. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, corresponding to a compatible sequence of diagrams $f_ n: K(n) \rightarrow \operatorname{\mathcal{C}}$. Suppose that each of the diagrams $f_ n$ admits a limit in $\operatorname{\mathcal{C}}$. Then there exists a tower $X: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ with the following properties:

$(1)$: For each $n \geq 0$, the object $X(n) \in \operatorname{\mathcal{C}}$ is a limit of the diagram $f_ n$.
$(2)$: An object of $\operatorname{\mathcal{C}}$ is a limit of the diagram $f$ if and only if it is a limit of the tower $X$. In particular, the diagram $f$ has a limit if and only if the tower $X$ has a limit.
$(3)$: Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which preserves the limits of each of the diagrams $f_ n$. Then $F$ preserves limits of the diagram $f$ if and only if it preserves limits of the tower $X$.

Proof. Combine Propositions 7.5.8.12 and 9.1.7.1. $\square$

Corollary 7.6.5.14. Suppose we are given a diagram of simplicial sets

\[ K(0) \rightarrow K(1) \rightarrow K(2) \rightarrow K(3) \rightarrow \cdots \]

having colimit $K$, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits sequential limits and $K(n)$-indexed limits, for each $n \geq 0$. Then $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits. If $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories which preserves sequential limits and $K(n)$-indexed limits for each $n \geq 0$, then it also preserves $K$-indexed limits.

Example 7.6.5.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits finite products. If $\operatorname{\mathcal{C}}$ admits sequential limits, then it also admits countable products. More precisely, for any countable collection of objects $\{ X_ n \} _{n \geq 0}$ of $\operatorname{\mathcal{C}}$, the product ${\prod }_{n \geq 0} X_{n}$ can be computed as the limit of a tower

\[ \cdots \rightarrow X_2 \times X_1 \times X_0 \rightarrow X_1 \times X_0 \rightarrow X_0. \]

We now establish a partial converse to Example 7.6.5.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits countable products, and suppose that we are given a tower

\[ \cdots \rightarrow X(3) \xrightarrow { f_2 } X(2) \xrightarrow { f_1 } X(1) \xrightarrow { f_0 } X(0) \]

in $\operatorname{\mathcal{C}}$. Then the collection of morphisms $\{ f_ n \} _{n \geq 0}$ determine an endomorphism $f$ of the product $P = {\prod }_{n \geq 0} X(n)$, given informally by the composition

\begin{eqnarray*} P & = & {\prod }_{n \geq 0} X(n) \\ & \rightarrow & {\prod }_{n > 0 } X(n) \\ & = & {\prod }_{m \geq 0} X(m+1) \\ & \xrightarrow { {\prod }_{m \geq 0} f_ m } & {\prod }_{m \geq 0} X(m) \\ & = & P. \end{eqnarray*}

In this case, we can identify limits of the tower $X$ with equalizers of the pair of morphisms $f, \operatorname{id}_ P: P \rightarrow P$. We can formulate this assertion more precisely as follows:

Proposition 7.6.5.16 (Sequential Limits as Equalizers). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ be a tower, which we identify with the diagram

\[ \cdots \rightarrow X(3) \xrightarrow { f_2 } X(2) \xrightarrow { f_1 } X(1) \xrightarrow { f_0} X(0). \]

Suppose that there exists an object $P \in \operatorname{\mathcal{C}}$ equipped with morphisms $\{ q_ n: P \rightarrow X(n) \} _{n \geq 0}$ which exhibits $P$ as a product of the collection $\{ X(n) \} _{n \geq 0}$. Then:

$(1)$

There exists a morphism $f: P \rightarrow P$ with the property that, for each $n \geq 0$, the diagram

7.76

\begin{equation} \begin{gathered}\label{equation:sequential-limit-as-equalizer} \xymatrix@R =50pt@C=50pt{ P \ar [r]^-{ [f] } \ar [d]^-{ [ q_{n+1} ]} & P \ar [d]^-{ [q_ n] } \\ X(n+1) \ar [r]^-{ [f_ n] } & X(n) } \end{gathered} \end{equation}

commutes in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. Moreover, the morphism $f$ is uniquely determined up to homotopy.

$(2)$

An object of $\operatorname{\mathcal{C}}$ is a limit of the tower $X$ if and only if it is an equalizer of the pair of morphisms $f, \operatorname{id}_{P}: P \rightarrow P$.

Proof. Assertion $(1)$ follows immediately from the definitions (see Warning 7.6.1.12). To prove $(2)$, let $M = \operatorname{\mathbf{Z}}_{\geq 0}$ denote the set of nonnegative integers, which we regard as a commutative monoid with respect to addition. Let $BM$ denote the associated category (consisting of a single object $E$ having endomorphism monoid $\operatorname{Hom}_{ BM}(E,E) = M$) and let $B_{\bullet } M$ denote the nerve of $BM$ (Construction 1.3.2.5). There is a functor of ordinary categories $(\operatorname{\mathbf{Z}}_{\geq 0}, \leq )^{\operatorname{op}} \rightarrow BM$ which is characterized by the requirement that, for every pair of nonnegative integers $m \leq n$, the induced map

\[ \operatorname{Hom}_{\operatorname{\mathbf{Z}}_{\geq 0} }( m, n ) \rightarrow \operatorname{Hom}_{BM}(E,E) = M \]

carries the unique element of $\operatorname{Hom}_{\operatorname{\mathbf{Z}}_{\geq 0} }( m, n)$ to the difference $n-m \in M$. Passing to nerves, we obtain a functor of $\infty $-categories $U: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} ) \rightarrow B_{\bullet } M$. The functor $U$ is a cartesian fibration, whose fiber over the vertex $E \in B_{\bullet } M$ can be identified with the discrete simplicial set $\{ 0, 1, 2, \cdots \} $. Applying Corollary 7.3.4.8, we deduce that there exists a functor $Y: B_{\bullet } M \rightarrow \operatorname{\mathcal{C}}$ and a natural transformation $\alpha : Y \circ U \rightarrow X$ which exhibits $Y$ as a right Kan extension of $X$ along $U$.

For every nonnegative integer $n$, $\alpha $ induces a morphism $\alpha _{n}: Y(E) \rightarrow X(n)$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Using the criterion of Proposition 7.3.4.1, we see that the collection of morphisms $\{ \alpha _ n \} _{n \geq 0}$ exhibit $Y(E)$ as a product of the collection of objects $\{ X(n) \} _{n \geq 0}$. We may therefore assume without loss of generality that $P = Y(E)$ and $q_{n} = \alpha _ n$, for each $n \geq 0$. Let $f: P \rightarrow P$ be the morphism obtained by evaluating the functor $Y$ on the generator $1 \in M$. For each $n \geq 0$, the natural transformation $\alpha $ carries the edge $n+1 \rightarrow n$ of $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )$ to a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ P \ar [r]^-{ f } \ar [d]^-{ q_{n+1} } & P \ar [d]^-{ q_ n } \\ X(n+1) \ar [r]^-{ f_ n } & X(n) } \]

in the $\infty $-category $\operatorname{\mathcal{C}}$, which witnesses the commutativity of the diagram (7.76) in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. Moreover, an object $C \in \operatorname{\mathcal{C}}$ is an equalizer of the pair of morphisms $f, \operatorname{id}_{P}: P \rightarrow P$ if and only if it is a limit of the diagram $Y$ (Variant 7.6.4.9). To prove $(2)$, it suffices to observe that this is equivalent to the requirement that $C$ is a limit of the tower $X$, which follows from Corollary 7.3.8.20. $\square$

Remark 7.6.5.17. In the situation of Proposition 7.6.5.16, suppose that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories which preserves the product of the collection $\{ X(n) \} _{n \geq 0}$. Then $F$ preserves limits of the tower $X$ if and only if it preserves equalizers of the pair of morphisms $f, \operatorname{id}_{P}: P \rightarrow P$.