The US firm, which is owned by Nike, is demanding 2.7 million kronor ($400,000) in compensation from Coop. It is furthermore demanding that Coop be prevented from selling shoes which resemble the original.
Coop had declined to comment on the latest move in the dispute.
“We want to read through the writ first,” said press officer Christine Kullgren.
Converse trainers have become popular again in Sweden in recent years and can be seen adorning the feet of all from hipsters to teens to toddler-toting mums despite their high retail price.
The firm asked a Swedish court in May for permission to search Coop stores after it noticed that Coop was selling its wares for about 399 kronor, half of the usual retail price of 799 kronor.
The company ordered some pairs and found the replicas lacked a number code that Converse uses to brand its genuine goods.
But Coop later rejected Converse’s piracy accusations, claiming that the shoes on sale at Coop Forum stores in Bäckebol, Norrköping, Skövde and Skara were the genuine articles.
“We’re very surprised that they would go forward in this way and ignore the documentation we sent to them when we first heard about the accusations that we were selling pirated copies of Converse shoes,” Coop spokeswoman Marika Hjelm Siegwald told the Svenska Dagbladet (SvD) newspaper in May.
“It’s not true,” she added, explaining that a law firm has helped Coop obtain certificates of authenticity from the manufacturers.