The family of Akai Gurley, the unarmed man shot by police in an unlit Brooklyn housing project stairwell in November 2014, has filed paperwork that begins the process of suing the city, the housing authority, the NYPD and the two officers involved in his death.

Gurley was shot in the stairwell of the Louis H Pink houses in East New York, Brooklyn, by officer Peter Liang. Liang and his partner, Shaun Landau, who were both rookie officers, were conducting a routine patrol of the block when they encountered Gurley in the stairwell.

The case added to heightened tensions across the US in 2014 over police brutality, after the shooting of Michael Brown in Ferguson, Missouri, and the death of Eric Garner in Staten Island.

On Thursday, lawyers representing the family filed a notice of claim with the city comptroller’s office, the first step toward a lawsuit. The city can now choose whether to opt for a hearing or move straight to the lawsuit. A separate notice was filed against the housing authority.

The suit is seeking damages totalling $50m, but Scott Rynecki, the attorney representing the family, told the Guardian that the suit “isn’t just about the monetary value”.

Rynecki said the family was calling for changes in the NYPD training academy regarding how officers are trained in the use and discharge of firearms, and for an independent review of police training procedures.

“In this case, with no crime in progress, there was no reason why a police officer would have his weapon out and his finger on the trigger,” he said.

New York police commissioner Bill Bratton proposed a departmental review of procedures at the time of the shooting, but Rynecki believes an internal investigation would be insufficient.

“We think it shouldn’t be done in-house; it should be done by an independent agency who can review what goes on at the police academy,” he said.

The housing authority is being targeted by the suit because, according to Rynecki, “there are certain codes and statutes that require a certain amount of lighting, and from what we’ve devised, these were darkened stairwells. Obviously they were not properly maintained by the housing authority.”

The housing authority declined to comment on the suit, and the mayor’s office and the NYPD did not respond to requests for comment.