Present Cisinski to reading group <2024-09-22 Sun 17:00-19:00>

I volunteered to present a chapter from Cisinski. We were working through 7.5, which covers Derived Functors.

DONE [#B] Prepare for reading group

Lemma 7.5.15

For any ∞-Category with weak equivalences and fibrations, the canonical functor $ho(L(C_{f}))\to ho(L(C))$ is an equivalence of categories.

Here, $ho(C)$ denotes the Homotopy category of $C$, $L(C)$ denotes the Localization of $C$, and $C_{f}$ denotes the full subcategory of Fibrant objects.

Note that this is really a defininition and two theorems in one:

• The definition of a cleavage
• Every ∞-Category with weak equivalences and fibrations can be equipped with a cleavage
• Every cloven ∞-Category with weak equivalences and fibrations satisfies the lemma.

1. Proof

   We start by choosing a cleavage, which consists of the following data:

   1. For each object $x:C$, a choice of fibrant object $R(x)$ and weak equivalence $j_{x}:x\to R(x)$. (If $x$ is already fibrant, then we choose $R(x)=x$ and $j_{x}=1_{x}$).

   2. For each map $u:x\to y$ a choice of a (potentially non-commutative) diagram

      where $\sigma(u)$ is a weak equivalence, $p(u)$ and $q(u)$ are fibrations, such that the image of the diagram in $ho(C)$ is commutative. (Whenever $u=1_{x}$, we choose $S(1_{x})=x$), and $p(u)=q(u)=1_{R(x)}$.

      What does this data give us? The name suggests that this is like a 1-categorical Cleavage, which gives us a choice of cartesian lifts for a fibration. This in turn lets us classify fibrations as Pseudofunctors $B\to\mathrm{Cat}$. Here, it seems like we are instead taking a choice of fibrant replacements (the weak equivalences $j_{x}$), along with choices of "weak" factorizations of arbitrary maps $u:x\to y$ into a weak equivalence $\sigma(u)$ followed by a fibration $p(u)$. Note that the triangle

      seems to mostly exist as a coherence thing. Moreover, note that $q(u)$ must be a trivial fibration by 2-Out-Of-3.

      Useful example to think about is iso-comma categories as doubly displayed widgets; these are isofibrations in both directions, so they fit the criteria we need.

   Existence of cleavages: We can obtain our choice of fibrant replacements by factoring (one of) the maps $x\to 1$ into the chosen final object of $C$ into $x\to R(x)\to 1$.

   The second property is a bit harder. Given a non-identity map $u:x\to y$, there is a canonical map $\langle j_{x},j_{y}u\rangle:x\to R(x)\times R(y)$ that we can factor as a weak equivalence $\sigma(u):x\to S(u)$ followed by a fibration $\pi(u):S(u)\to R(x)\times R(y)$. The maps $p(u)$ and $q(u)$ are obtained by postcomposing $\pi(u)$ with the projections out of $R(x)\times R(y)$.

   The actual proof: We need to promote a cleavage $R$ on $C$ to a functor $R:ho(C)\5oho(L(C_{f}))$. Given a map $u:x\to y$, we define $R(u)=p(u)q^{-1}(u)$. By definition, we have $R(1_{x})=1_{R(x)}$. Let $u:x\to y$ and $v:y\to z$ be a pair of maps. We start by forming a pullback in $C$.

   The following commutative square gives us a map $f:x\to T(v,u)$.

   Next, we choose a composite $w:x\to z$ of $u$ and $v$. This gives the following commutative diagram:

   This is a bit easier if you think about this as composition of spans, followed by finding an intermediate.

   We can then compose the functor $R:ho(C)\to ho(L(C_{f}))$ with the canonical functor $ho(\iota):ho(C_{f})\to ho(C)$, which is equal to the canonical functor $ho(\gamma_{f}):ho(C_{f})\to ho(L(C_{f}))$.

   Then, by construction, for any map $u:x\to y$ in $C$, the square

   commutes in $ho(L(C))$. This gives us an invertible natural transformation $j:ho(\gamma)\to ho(\iota)\circ R$, so the functor $ho(L(C_{f}))\to ho(L(C))$ induced by $\iota$ is an equivalence of categories.

Proposition 7.5.16

Let $x$ be a Fibrant object in a ∞-Category with weak equivalences and fibrations $C$. The induced functor $C_{f}/\gamma_{f}(x)\to C/\gamma(y)$ is final.

1. Motivation

   Taking colimits over slices like $C/F(x)$ is a classic tool of 1-category theory, and final functors let us compute colimits by restricting along them (See 6.4.5 for the higher case of this). So this theorem will let us do that same "final replacement" trick we do when we compute colimits: in particular, when computing these sorts of slice-shaped colimits, we can restrict our attention to only the fibrant objects.

2. Proof

   Recall that

   $$C_{f}/\gamma_{f}(x)=L(C_{f})/\gamma_{f}(x)\times_{L(C_{f})}C_{f}$$

   and

   $$C/\gamma(x)=L(C)/\gamma(x)\times_{L(C)}C$$

   Intuition here is that the following pullback square gives us the data of $\gamma(c)\to\gamma(x)$, as the object in $L(C)$ is in the image of $\gamma$.

   Why didn't he just write $\gamma/\gamma(x)$?

   Moreover, the comparison functor is induced by the functor $\overline{\iota}:L(C_{f})\to L(C)$. An object $\xi:C/\gamma(x)$ is determined by a pair $(c,u)$ where $c:C$ and $u:\gamma(c)\to\gamma(x)$ is a map in $L(c)$.

   We need to show that the coslice $\xi\backslash(C_{f}/\gamma_{f}(x))$ is weakly contractible. By lemma 4.3.15, it suffices to consider a functor of the form $F:E\to\xi\backslash(C_{f}/\gamma_{f}(x))$ where $E$ is a nerve of a finite partially ordered set, and to prove that $F$ is $\Delta^{1}$ homotopic to a constant functor.

   Let $X$ be a simplicial set such that for any finite partially ordered set $E$, any map $N(E)\to X$ is $\Delta^{1}$ homotopic to a constant map. Then the map $X\to\Delta^{0}$ is a weak homotopy equivalence.

   If $E$ is empty, then, then it suffices to show that $\xi\backslash(C_{f}/\gamma_{f}(x))$ is non-empty. Cisinski says to pick a weak equivalence $p:c\to d$ with $d$ fibrant, but we may as well just use $j_{c}:c\to R(c)$? Either way, we can use our previous lemma to turn the morphism $u$ from $\xi$ into a unique map $\delta:\gamma_{f}(d)\to\gamma_{f}(x)$ in $ho(L(C_{f}))$ such that $\overline{\iota}(\delta)\gamma(p)=u$.

   Now for the case when $E$ is non-empty. First, observe that a functor $F$ as above is essentially determined by a functor $\Phi:E\to C_{f}$ along with a map $\varphi:\gamma_{f}(\Phi)\to\gamma_{f}(x)$ and a map $\psi:c\to\Phi$ such that the diagram

   commutes in $\underline{\mathrm{Hom}}(E,L(C))$. Proposition 7.4.19 lets us perform a Reedy fibrant replacement to get a fibrewise equivalence $w:\Phi\to\Phi_{0}$ such that $\Phi_{0}$ is Reedy fibrant.

   Let $u:F\to G$ be a mrophism of $C$ valued presheaves on a finite direct category. There exists a Reedy fibration $p:H\to G$ and a fibrewise equivalence $w:F\to H$ such that $u$ is a composite of $w$ and $p$.

   Since $\gamma_{f}(w):\gamma_{f}(\Phi)\to\gamma_{f}(\Phi_{0})$ is invertible, one can find a map $\varphi_{0}:\gamma_{f}(\Phi_{0})\to\gamma_{f}(x)$ such that $\varphi$ is a composition of $\varphi_{0}$ and $\gamma_{f}(w)$. Moreover, we can compose $w$ with $\psi$ to get a map $\psi_{0}:c\to\Phi_{0}$. These 3 pieces of data give us a functor $F_{0}:E\to\xi\backslash(C_{f}/\gamma_{f}(x))$, and $w$ extends to a natural transformation $F\to F_{0}$.

   This process lets us replace $F$ with some $F_{0}$ whose underlying functor $\Phi_{0}$ is Reedy fibrant. We then invoke proposition 7.4.8 to show that the limit of $\Phi_{0}$ exists and is fibrant in $C$. Moreover, the map $\psi_{0}$ is a cone for $\Phi_{0}$, so we get a map $\pi:c\to\mathrm{lim}\ \Phi_{0}$.

   As $\mathrm{lim}\ \Phi_{0}$ is fibrant in $C$, we can factor $\pi$ into a weak equivalence $p:c\to d$ and a fibration $q:d\to\mathrm{lim}\ \Phi_{0}$. The canonical morphism $v:\mathrm{lim}\ \Phi_{0}\to\Phi_{0}$ along with a choice of composition with $\varphi_{0}$ defines a map $\lambda:\gamma_{f}(\mathrm{lim}\ \Phi_{0})\to\gamma_{f}(x)$. We can then compose $q$ and $\lambda$ to get a morphism $\delta:\gamma_{f}(d)\to\gamma_{f}(x)$ between constant functors.

   Now, let $e$ be an object, and $\delta_{e}:\gamma_{f}(d)\to\gamma_{f}(x)$ be a component of the aforementioned natural transformation. As $u$ (which, recall, came from $\xi$) is a composition of $\overline{\iota}(\delta_{e})$ with the inverse of $\gamma(p)$, the triple $(d,p,\delta_{e})$ defines a constant functor $E\to\xi\backslash(C_{f}/\gamma_{f}(x))$. A short diagram chose shows that any composition of $v$ and $q$ defines a natural transformation from $(d,p,\delta_{e})$ to $F$.

Corollary 7.5.17

Let $C$ be a $\mathcal{U}$ small ∞-Category with weak equivalences and fibrations. For any infinity category $D$ with $\mathcal{U}$ small colimits and any functor $F:C\to D$, the commutative square 7.5.11 induces an invertible map $(\gamma_{f})_{!}\iota^{*}(F)\simeq(\overline{\iota})^{*}\gamma_{!}(F)$

Recall that $\iota^{*}:[C,D]\to[C_{f},D]$ is precomposition with $\iota:C_{f}\to C$, and $\gamma_{!}:[C,D]\to[L(C),D]$ is the extension of a functor to the localization that we compute by taking a colimit.

1. Motivation

   This is sort of a nerve-and-realization result.

2. Proof

   We already have a canonical map $(\gamma_{f})_{!}\iota^{*}(F)\to(\overline{\iota})^{*}\gamma_{!}(F)$ that comes from the definition of $(\gamma_{f})_{!}(F\circ\iota)$ as a colimit. If we apply Proposition 6.4.9 twice to unfold the definition of $!$, we see that this is equivalent to a map between colimits being invertible. The previous result lets us do a "final replacement" of the side where we compute the colimit over $\gamma$, so this map must be invertible.