Let $V$ be an object of some stable infinity category (nothing is lost by taking spectra but I see no reason to state the question in this way as it is irrelevant) and suppose we have a two step filtration:


$$V\_0 \subset V\_1 \subset V\_2$$


Lets denote $A=V\_0$, $B=V\_1/V\_0$, $C= V\_2/V\_1$, $D=V\_1$, $E=V\_2/V\_0$, $F=V\_2$.


The data of the filtration gives us two different ways to build $F$ by iterated extensions. 


First option:


$$D \longrightarrow B \overset{\alpha}{\longrightarrow} A[1]$$


$$F \longrightarrow C \overset{\gamma}{\longrightarrow} D[1]$$


Second option:


$$E \longrightarrow C \overset{\beta}{\longrightarrow} B[1]$$


$$F \longrightarrow E \overset{\delta}{\longrightarrow} A[1]$$


Of course there must be relations between $\{\alpha,\beta,\gamma,\delta\}$ since they came from a single 2-step filtration.



> 
> **Question:** Is there a general set of relations (expressed via cup product massey products and perhaps even some extra coherence data) one can define on collections of the form $\{\alpha,\beta,\gamma,\delta\}$ which will give a bijective correspondence between 2-step filtered objects with subquotients $A,B,C,D,E,F$ and collections of the form $\{\alpha,\beta,\gamma,\delta\}$ which satisfy these relations?
> 
> 
> 



Spelling out explicitly the data of a two step filtration in terms of pieces and gluing data
Technically speaking the answer to your question is no, in the sense that the data of $(\alpha,\beta,\gamma,\delta)$ alone does not determine the filtered object $V\_0 \subseteq V\_1 \subseteq V\_2$. However, a variant on your construction does have a positive answer, and this is closely related to Massey products. Recall first that if $(A\_\bullet,d)$ is a (homologically graded) dg-algebra then an **$n$-fold Massey system** on $A$ is a collection of elements $a\_{i,j} \in A\_{-1}$ for $0 \leq i < j \leq n$ with $(i,j) \neq (0,n)$ satisfying the relation $\partial a\_{i,j} = \sum\_{k=i+1}^{j-1} a\_{i,k}a\_{k,j}$ (in particular, $\partial a\_{i,i+1} =0$). In this case, the element $b := \sum\_{k=1}^{n-1}a\_{0,k}a\_{k,n} \in A\_{-2}$ is closed and its class $[b] \in H\_{-2}(A)$ is called the value of the Massey system $\{a\_{i,j}\}$. Classically this value is often considered as the $n$-fold Massey product of the classes $[a\_{i,i+1}] \in H\_{-1}(A)$, although one should remember that this value depends on the choice of a Massey system, and not just on the $[a\_{i,i+1}]$. A **trivialization** of a Massey system $\{a\_{i,j}\}$ is an element $a\_{0,n} \in A\_{-1}$ satisfying $\partial a\_{0,n} = \sum\_{k=1}^{n-1}a\_{0,k}a\_{k,n}$, i.e., witnessing the vanishing of the associated $n$-fold Massey product.


The main observation (which I believe, under a somewhat different formulation, is essentially due to Dwyer), is that the data $(\{a\_{i,j}\},a\_{0,n})$ of an $n$-fold Massey system equipped with a trivialization classifies (right) $A$-modules $M$ equipped with an ascending filtration 
$$0 = M\_{-1} \to M\_0 \to M\_1 \to ...\to M\_n=M $$ 
such that each homotopy cofiber $M\_{i}/M\_{i-1}$ is (identified with) $A$. For example, suppose that $n=1$. Then the data of a trivialized Massey system is just the data of a closed element $a\_{0,1} \in A\_{-1}$, which can be considered as a map of $A$-modules $a\_{0,1}: A \to A[1]$. One then defines $M$ to be the homotopy fiber of this map, so that $M$ sits in a homotpy fiber sequence $A \to M \to A$. In particular, $M$ is equipped with a $2$-step filtration 
$$ 0 \to A \to M $$ 
whose successive cofibers are $A$. Now consider the case $n=2$. Then we are given closed elements $a\_{0,1},a\_{1,2} \in A\_{-1}$ and a trivialization $a\_{0,2} \in A\_{-1}$ satisfying $\partial a\_{0,2} = a\_{0,1}a\_{1,2}$. We may then consider $a\_{0,1}$ and $a\_{1,2}$ as $A$-module maps $a\_{0,1}: A[1] \to A[2]$ and $a\_{1,2}: A[0] \to A[1]$, and consider $a\_{0,2}$ as a null-homotopy of their composition $a\_{0,1} \circ a\_{1,2}$. Let $M\_{0,1}$ and $M\_{1,2}$ be the homotopy fibers of $a\_{0,1}$ and $a\_{1,2}$ respectively. Then the null-homotopy $a\_{0,2}$ determines a lift of $a\_{1,2}: A \to A[1]$ to a map $\tilde{a}\_{1,2}: A \to M\_{0,1}$. Let $M\_{0,2}$ be the homotopy fiber of $\tilde{a}\_{1,2}$. Then $M\_{0,2}$ is equipped with a $3$-step filtration
$$ 0 \to A \to M\_{0,1}[-1] \to M\_{0,2} $$ 
whose successive cofibers are $A$.


This idea can be generalized to describe filtered $A$-modules with other types of successive cofibers. Indeed, if $B\_0,...,B\_n$ are fixed $A$-modules, then we can define the notion of a **Massey system** on $B\_0,...,B\_n$ to be a collection of elements $a\_{i,j} \in \underline{\rm Hom}\_{-1}(B\_j,B\_i)$ (where $\underline{\rm Hom}\_\bullet$ means the mapping chain-complex of $A$-modules) for $(i,j) \neq (0,n)$ such that $\partial a\_{i,j} = \sum\_{k=i+1}^{j-1} a\_{i,k}\circ a\_{k,j}$, and define a trivialization of $\{a\_{i,j}\}$ to be an element $a\_{0,n} \in \underline{\rm Hom}\_{-1}(B\_n,B\_0)$ such that $\partial a\_{0,n} = \sum\_{k=1}^{n-1}a\_{0,k} \circ a\_{k,n}$. Then the data of a Massey system on $B\_0,...,B\_n$ equipped with a trivialization then classifies $A$-modules equipped with an $(n+1)$-step filtration whose successive cofibers are $B\_0,...,B\_n$. Let's call such objects $(B\_0,...,B\_n)$-filtered $A$-modules. You can also ask, what about Massey systems $\{a\_{i,j}\}$ not-necessarily trivialized? Well, these will classify triples $(M\_{0,n-1},M\_{1,n},f)$ where $M\_{0,n-1}$ is a $(B\_0,...,B\_{n-1})$-filtered $A$-module, $M\_{1,n}$ is a $(B\_1,...,B\_n)$-filtered $A$-module and $f$ is an equivalence of $(B\_1,...,B\_{n-1})$-filtered $A$-modules between ${\rm cofib}(B\_0 \to M\_{0,n-1})$ and ${\rm fib}(M\_{1,n}\to B\_n)$. The associated $n$-fold Massey product $[\sum\_{k=1}^{n-1}a\_{0,k} \circ a\_{k,n}] \in H\_{-2}(\underline{\rm Hom}(B\_n,B\_0))$ (which can be considered as a homotopy class of maps $B\_n \to B\_0[-2]$) is then the obstruction to finding a $(B\_0,...,B\_n)$-filtered $A$-module $M\_{0,n}$ such that ${\rm fib}(M\_{0,n}\to B\_n) \simeq M\_{0,n-1}$ and ${\rm cofib}(B\_0 \to M\_{0,n}) \simeq M\_{1,n}$.


Of course, what can be done for modules can usually be done for an arbitrary stable $\infty$-category. Indeed, if $B\_0,B\_1,...,B\_n$ are fixed objects in a stable $\infty$-category ${\cal C}$ then we can classify $(B\_0,B\_1,...,B\_n)$-filtered objects by suitable Massey systems $\{a\_{i,j}\}$, where the $a\_{i,i+1}$ are maps of the form $a\_{i,i+1}: B\_{i+1}[n-i-1] \to B\_{i}[n-i]$, $a\_{i,i+2}$ is a null-homotopy of the composition $a\_{i,i+1} \circ a\_{i+1,i+2}$, etc. The case you are asking about is the case of a $3$-step filtration with successive cofibers $A,B,C \in {\cal C}$. In this case $(A,B,C)$-filtered objects are classified by the data $(\alpha',\beta,h)$ consisting of maps $\alpha': B[1] \to A[2]$ and $\beta: C \to B[1]$ and a null-homotopy $h$ of $\alpha' \circ \beta$. Here $\alpha'$ is a shift of your $\alpha$. The null-homotopy $h$ is actually determined by either a lift of $\beta$ to $\gamma: C \to D[1]:= {\rm fib}(\alpha')$ or an extension of $\alpha'$ to $\delta': E[1] := {\rm cofib}(\beta) \to A[2]$. This is essentially what you did with your $\gamma$ and $\delta$. However, to get the null-homotopy $h$ it is not sufficient to remember just $\gamma$ (or $\delta$), one also need to keep the homotopy relating the composition $C \stackrel{\gamma}{\to} D[1] \to B[1]$ to $\beta$. Of course if you do this for both $\gamma$ and $\delta$ you end up encoding this null-homotopy twice, so you will need to add a homotopy relating the two choices. Alternatively, just drop one of $\gamma,\delta$ (and add the missing homotopy instead).


