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.
$\square$