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Proposition: In $Fun(\mathcal C, Ch_k)$, a diagram is quasi-isomorphic to its homology. Consequently, if the homologies of two such diagrams are isomorphic, then the diagrams are quasi-isomorphic.

Proof: By Proposition 1, this is true for $Fun(T,Ch_k)$. Starting with an equivalence in the $\infty$-category from $D \leftarrow F \to B$ to its homology, we gradually construct more and more equivalences across the above 3x3 square by taking various fibers and cofibers.

Proposition 4: In $Fun(\mathcal C, Ch_k)$, a diagram is quasi-isomorphic to its homology. Consequently, if the homologies of two such diagrams are isomorphic, then the diagrams are quasi-isomorphic.

Proof: By Proposition 1, this is true for $Fun(T,Ch_k)$. Start with a model of the top right square such that $F \to D$, $F \to B$, and $B \oplus D \to Q$ are monomorphisms. By Proposition 1, we have a commuting square of quasi-isomorphisms from $F \to D \oplus B$ to the corresponding map of homologies. By Lemma 2, we may extend this to a quasi-isomorphism from $Q$ to its homology, in a natural way. We think of this as a natural transformation from the top-right square of the above grid to the corresponding grid of homologies, and we seek to fill out the remaining components of a natural transformation between grids.

Since $D \to Q$ is a monomorphism, we may use Lemma 2 again to extend this natural family of quasi-isomorphisms to $C$, and hence to the lower-right square. Modeling $A$ by the dual mapping cone of $F \to D$, we obtain a model of $A \to F$ which is epic. By the dual of Lemma 2, we may extend our natural family of quasi-isomorphisms to $A$ and hence to the upper-left square. To model the lower-right square, take an arbitrary factorization through an acyclic complex, which comes with a unique quasi-isomorphism to its homology. The remaining naturality squares all necessarily commute, and the naturality of the quasi-isomorphisms out of the lower-left square is precisely what we are looking for.

Let $T$ be a well-founded poset and $k$ a field. Let $H: Ch_k \to Ho(Ch_k) = Gr_k$ be the homology functor and $\iota: Gr_k \to Ch_k$ be the canonical section which sends a graded vector space to the corresponding chain complex with zero differential.

Proposition: For any $F: T \to Ch_k$, $F$ and $\iota H(F)$ are quasi-isomorphic.

In particular, if $H(F) \cong H(G)$, then $F \simeq \iota H(F) \cong \iota H(G) \simeq G$.

Proof: In the Reedy = injective model structure on $Fun(T,Ch_k)$, fibrations and weak equivalences are levelwise, and an object is cofibrant (and thus bifibrant) if and only if each morphism of $T$ is carried to a monomorphism. We may assume that $F \in Fun(T,Ch_k)$ is cofibrant and prove the more specific claim that there is a quasi-isomorphism $F \to \iota H(F)$. By induction on the structure of $T$, we are reduced to the following:

Lemma: Let $X \to Y$ be a monomorphism in $Ch_k$. If $X \to \iota H(X)$ is a quasi-isomorphism, then there exists a quasi-isomorphism $Y \to \iota H(Y)$ forming a commutative square.

Proof: We may decompose $X \to Y$ into a series of cell attachments, and construct the map $Y \to \iota H(Y)$ by induction, one cell at a time. There are two cases. In the first case we have $Y = X \oplus k[n]$, in which case we have $H(Y) = H(X) \oplus k[n]$, and we extend using the identity on $k[n]$. In the second case we have $Y = X \cup_z w$ where $z$ is a cycle representing a nonzero homology class of $X$ and $w$ is a cell we are attaching to kill it. In this case we have $H(Y) = H(X)/v$, and we can extend via the zero map on $w$.

Let $T$ be a well-founded poset and $k$ a field. Let $H: Ch_k \to Ho(Ch_k) = Gr_k$ be the homology functor and $\iota: Gr_k \to Ch_k$ be the canonical section which sends a graded vector space to the corresponding chain complex with zero differential.

Proposition 1: For any $F: T \to Ch_k$, $F$ and $\iota H(F)$ are quasi-isomorphic.

In particular, if $H(F) \cong H(G)$, then $F \simeq \iota H(F) \cong \iota H(G) \simeq G$.

Proof: In the Reedy = injective model structure on $Fun(T,Ch_k)$, fibrations and weak equivalences are levelwise, and an object is cofibrant (and thus bifibrant) if and only if each morphism of $T$ is carried to a monomorphism. We may assume that $F \in Fun(T,Ch_k)$ is cofibrant and prove the more specific claim that there is a quasi-isomorphism $F \to \iota H(F)$. By induction on the structure of $T$, we are reduced to the following:

Lemma 2: Let $X \to Y$ be a monomorphism in $Ch_k$. If $X \to \iota H(X)$ is a quasi-isomorphism, then there exists a quasi-isomorphism $Y \to \iota H(Y)$ forming a commutative square.

Proof: We may decompose $X \to Y$ into a series of cell attachments, and construct the map $Y \to \iota H(Y)$ by induction, one cell at a time. There are two cases. In the first case we have $Y = X \oplus k[n]$, in which case we have $H(Y) = H(X) \oplus k[n]$, and we extend using the identity on $k[n]$. In the second case we have $Y = X \cup_z w$ where $z$ is a cycle representing a nonzero homology class of $X$ and $w$ is a cell we are attaching to kill it. In this case we have $H(Y) = H(X)/v$, and we can extend via the zero map on $w$.

Added in response to comment: It's worth noting that although the the proposition only applies directly to tree-shaped diagrams, we can leverage it to understand certain more complicated diagrams. For example:

Let $\mathcal C$ be the "marked category" $$\require{AMScd} \begin{CD} a @>>> b\\ @VVV @VVV\\ 0 @>>> c \end{CD}$$

with the understanding that $Fun(\mathcal C,Ch_k)$ means functors $\mathcal C \to Ch_k$ carrying $0$ to an acyclic chain complex. Then $Fun(\mathcal C,Ch_k)$ with levelwise weak equivalences models "maps $A \xrightarrow f B \xrightarrow g C$ equipped with a nullhomotopy of $gf$".

Lemma 3: If $\mathcal D$ is/presents a stable $\infty$-category, then the $\infty$-category (presented by) $Fun(\mathcal C, \mathcal D)$ is equivalent to the $\infty$-category (presented by) $Fun(T,\mathcal D)$ where $T$ is the tree $D \leftarrow F \to B$.

Proof: Consider the following diagram, where the rows and columns are fiber sequences:

$$\require{AMScd} \begin{CD} A @>>> F @>>> D \\ @V{=}VV @VVV @VVV \\ A @>>> B @>>> Q \\ @VVV @VVV @VVV \\ 0 @>>> C @>{=}>> C \end{CD}$$

The diagram can be uniquely reconstructed from either the bottom left square, or the top right square subject to the requirement that it be an exact square, or from the part $D \leftarrow F \to B$ subject to no requirements.

Proposition: In $Fun(\mathcal C, Ch_k)$, a diagram is quasi-isomorphic to its homology. Consequently, if the homologies of two such diagrams are isomorphic, then the diagrams are quasi-isomorphic.

Proof: By Proposition 1, this is true for $Fun(T,Ch_k)$. Starting with an equivalence in the $\infty$-category from $D \leftarrow F \to B$ to its homology, we gradually construct more and more equivalences across the above 3x3 square by taking various fibers and cofibers.

Let $T$ be a well-founded poset and $k$ a field. Let $H: Ch_k \to Ho(Ch_k) = Gr_k$ be the homology functor and $\iota: Gr_k \to Ch_k$ be the canonical section which sends a graded vector space to the corresponding chain complex with zero differential.

Proposition: For any $F: T \to Ch_k$, there is a quasi-isomorphism $F \to \iota H(F)$.

In particular, if $H(F) \cong H(G)$, then $F \simeq \iota H(F) \cong \iota H(G) \simeq G$.

Proof: In the Reedy = injective model structure on $Fun(T,Ch_k)$, fibrations and weak equivalences are levelwise, and an object is cofibrant (and thus bifibrant) if and only if each morphism of $T$ is carried to a monomorphism. We may assume that $F \in Fun(T,Ch_k)$ is cofibrant. By induction on the structure of $T$, we are reduced to the following:

Lemma: Let $X \to Y$ be a monomorphism in $Ch_k$. If $X \to \iota H(X)$ is a quasi-isomorphism, then there exists a quasi-isomorphism $Y \to \iota H(Y)$ forming a commutative square.

Proof: We may decompose $X \to Y$ into a series of cell attachments, and construct the map $Y \to \iota H(Y)$ by induction, one cell at a time. There are two cases. In the first case we have $Y = X \oplus k[n]$, in which case we have $H(Y) = H(X) \oplus k[n]$, and we extend using the identity on $k[n]$. In the second case we have $Y = X \cup_z w$ where $z$ is a cycle representing a nonzero homology class of $X$ and $w$ is a cell we are attaching to kill it. In this case we have $H(Y) = H(X)/v$, and we can extend via the zero map on $w$.

Let $T$ be a well-founded poset and $k$ a field. Let $H: Ch_k \to Ho(Ch_k) = Gr_k$ be the homology functor and $\iota: Gr_k \to Ch_k$ be the canonical section which sends a graded vector space to the corresponding chain complex with zero differential.

Proposition: For any $F: T \to Ch_k$, $F$ and $\iota H(F)$ are quasi-isomorphic.

In particular, if $H(F) \cong H(G)$, then $F \simeq \iota H(F) \cong \iota H(G) \simeq G$.

Proof: In the Reedy = injective model structure on $Fun(T,Ch_k)$, fibrations and weak equivalences are levelwise, and an object is cofibrant (and thus bifibrant) if and only if each morphism of $T$ is carried to a monomorphism. We may assume that $F \in Fun(T,Ch_k)$ is cofibrant and prove the more specific claim that there is a quasi-isomorphism $F \to \iota H(F)$. By induction on the structure of $T$, we are reduced to the following:

Lemma: Let $X \to Y$ be a monomorphism in $Ch_k$. If $X \to \iota H(X)$ is a quasi-isomorphism, then there exists a quasi-isomorphism $Y \to \iota H(Y)$ forming a commutative square.

Proof: We may decompose $X \to Y$ into a series of cell attachments, and construct the map $Y \to \iota H(Y)$ by induction, one cell at a time. There are two cases. In the first case we have $Y = X \oplus k[n]$, in which case we have $H(Y) = H(X) \oplus k[n]$, and we extend using the identity on $k[n]$. In the second case we have $Y = X \cup_z w$ where $z$ is a cycle representing a nonzero homology class of $X$ and $w$ is a cell we are attaching to kill it. In this case we have $H(Y) = H(X)/v$, and we can extend via the zero map on $w$.