Metrics on stable infinity-categories

Proof For stability under cofibers, let us consider $a_*, b_*:\mb{N}^{\leq}\to\mc{C}$ two Cauchy sequences in $\mc{C}$ for the metric $\mc{B}$, together with $f_n:a_n\to b_n$ a collection of maps in $\mc{C}$ indexed by $n\in\mb{N}$. Let $c_n$ be the cofibre of $f_n$. We wish to show that the assignment $c_*:\mb{N}^{\leq}\to\mc{C}$, given by $n\mapsto c_n$, defines a Cauchy sequence. Fix $i\in\mb{N}$ and $j\in\mb{Z}$. Since $a_*$ and $b_*$ are Cauchy sequences, by considering $i+1\in\mb{N}$ and $j\in\mb{Z}$ we can find an $M>0$ such that the the $j$-suspension of cofibers $a_{mm'}$ and $b_{mm'}$ belong to $B_{i+1}$ for every $m'>m\geq M$, that is $\susp^ja_{mm'},\susp^j b_{mm'} \in B_{i+1}$. Now the cofibre $c_{mm'}$ of $c_m\to c_{m'}$ fits in the cofibre sequence $a_{mm'}\to b_{mm'}\to c_{mm'}$, that is in the cofibre sequence $b_{mm'}\to c_{mm'}\to\susp a_{mm'}$. By suspending, we get the cofibre sequence $\susp^j b_{mm'}\to\susp^j c_{mm'}\to \susp^{j+1} a_{mm'}$. Since $\susp^jb_{mm'}\in B_{i+1}$ and $\susp^{j+1} a_{mm'}\in \susp B_{i+1}$ are both contained in $B_{i+1}$, hence in $B_i$, the triangle inequality implies that $\susp^jc_{mm'}\in B_{i}$. Stability under shifts is then immediate, since the zero object of $\mc{C}$ defines a Cauchy sequence $0_*:\mb{N}^{\leq}\to \mc{C}$.

Proof By [(Lurie 2017), Lemma 1.1.3.3], to show that the full subcategory $\Li(\mc{C})\subseteq \Ind(\mc{C})$ is stable it is sufficient to check that it is pointed, it admits cofibers and that the suspension functor restricts to an equivalence $\susp : \Li(\mc{C})\to\Li(\mc{C})$. We do it step by step.

1. First of all, let us show that $\Li(\mc{C})$ is pointed. Let $0\in\mc{C}$ be a zero object and consider the constant Cauchy sequence $0_*:\mb{N}^{\leq}\to \mc{C}$. We claim that
2. We now show that $\Li(\mc{C})$ is stable under cofibers. Let $f:A\to B$ be a map between limits, and consider its cofiber $C$ computed in $\Ind(\mc{C})$. By \cite{Lurie-HTT}, Proposition 5.3.5.15, the map $f:A\to B$ is obtained as the ind-completion of maps $f_i:a_i\to b_i$ in $\mc{C}$. By \autoref{lemma: stability of Cauchy sequences}, the cofibers $c_i$ of $f_i$ define a Cauchy sequence $c_*:\mb{N}^{\leq}\to \mc{C}$. By [(Lurie 2009), Proposition 5.3.5.14], we have that $C\simeq \colim \yo(c_*)$ exhibits the cofiber of $f$ as a colimit of a Cauchy sequence.
3. To conclude, we only need to show that $\susp:\Li(\mc{C})\to\Li(\mc{C})$ is an equivalence. Being $\Li(\mc{C})\subseteq\Ind(\mc{C})$ a full subcategory, the suspension is clearly fully faithful. Let us show that it is essentially surjective. Let $A\simeq\colim\yo(a_*)$ be the colimit of a Cauchy sequence $a_*:\mb{N}^{\leq}\to \mc{C}$. Consider the limit $\susp^{-1}A:= \colim \yo(\susp^{-1}a_*$. By \autoref{lemma: stability of Cauchy sequences}, $\susp^{-1}A\in\Li(\mc{C})$ is still a limit of a Cauchy sequence. Since $\yo:\mc{C}\to\Ind(\mc{C})$ commutes with colimits by [(Lurie 2009), Proposition 5.3.5.14], we have that \begin{align*} \susp \susp^{-1}A &= \susp \colim\yo(\susp^{-1}a_*)\\ &\simeq \colim \susp\yo(\susp^{-1}a_*)\\ &\simeq \colim \yo(\susp\susp^{-1}a_*)\simeq A. \end{align*}

This concludes the proof.

Proof We apply again \cite{Lurie-HA}, Lemma 1.1.3.3. Let us spell every detail.

1. It is clearly pointed, since the representable functor $y(0):\C^{\op}\to\Spc$ is compactly supported. Indeed, if we fix $j\in\mb{Z}$, then for every integer $i>0$ we have that \[ \pi_0\Hom_{\Ind(\mc{C})}(\yo(\susp^jM_i),\yo(0)) \simeq \pi_0\Hom_{\Ind(\mc{C})}(\yo(\susp^jM_i),0)\simeq 0 \] since $\yo:\mc{C}\to\Ind(\mc{C})$ preserves limits.
2. Let $A,B:\mc{C}^{\op}\to\Spc$ be two compactly supported presheaves, and let $f:A\to B$ be a map. To show that its cofiber $C$ is again compactly supported, let us fix $j\in\mb{Z}$ and find an integer $i>0$ such that \[ \pi_0\Hom_{\Ind(\mc{C})}(\yo(\susp^jM_i),A)\simeq 0 \simeq \pi_0\Hom_{\Ind(\mc{C})}(\yo(\susp^jM_i),B). \] Since the cofiber sequence $A\to B\to C$, which is also a fiber sequence by stability of $\Ind(\mc{C})$, induces the fiber sequence of spaces
   by the long exact sequence on homotopy groups we conclude that also \[\pi_0\Hom_{\Ind(\mc{C})}(\yo(\susp^jM_i),C)\simeq 0.\] Hence $C\in\CS(\mc{C})$.
3. To show that the suspension functor $\susp :\CS(\mc{C})\to\CS(\mc{C})$ is an equivalence, let us immediately note that it is fully-faithful, by fullness of $\CS(\mc{C})$ in $\Ind(\mc{C})$. To show that it is essentially surjective, let us consider a compactly supported presheaf $A:\mc{C}^\op\to\Spc$. We define $\susp^{-1}A:\C^\op\to\Space$ by $\susp^{-1}A(x)= A(\susp^{-1}x)$. Then it is immediate to check that $\susp\susp^{-1}A\simeq A$. So we are left to check that $\susp^{-1}A\in\CS(\mc{C})$. Fix $j\in \mb{Z}$. By definition of $A$, we find an integer $i>0$ such that $\pi_0\Hom_{\Ind(\mc{C})}(\yo(\susp^{j}M_i),A)\simeq 0$. However, since $yo:\C\to\Ind(\mc{C})$ preserves colimits, we have that \begin{align*} \Hom_{\Ind(\mc{C})}(\yo(\susp^{j}M_i),\susp^{-1}A) &\simeq \Hom_{\Ind(\mc{C})}(\susp^{-1}\yo(\susp^{j+1}M_i),\susp^{-1}A)\\ &\simeq \Hom_{\Ind(\mc{C})}(\yo(\susp^{j}M_i),A) \end{align*} and so by taking $\pi_0$ we conclude.