Sheldon at some point in his life had several cats.After terminating his relationship with Amy, who was not his girlfriend at the time, in season 4, episode 3, "The Zazzy Substitution", Sheldon began to collect cats as an affection substitution though he denied the connection. His first cat was named Dr. J. Robert Oppenheimer, whom Sheldon carried around with him including into the Caltech cafeteria. Later Leonard comes home and finds the living room full of cats. Sheldon explains that Dr. Robert Oppenheimer was lonely. “So you decided to get the whole Manhattan Project?” asked an exasperated Leonard. He named his other cats Enrico Fermi, Richard Feynman, Edward Teller, Otto Frisch, and Zazzles. The last cat was going to be named Hermann von Helmholtz, but Sheldon found him so zazzy that he named him Zazzles. Leonard again tried to tell Sheldon that he was replacing Amy with a bunch of cats, though Sheldon corrected him that a group of cats is a clowder or a glaring. Leonard just walked away after telling him about his Aunt Nancy, who had a lot of cats that ate her after she died on them.
A few days later after Leonard called Sheldon's mother, Mary Cooper, she arrived trying to assure Leonard that she has seen her son at his best and at his worst. Opening Sheldon's Bedroom door she found the smell quite pungent and all the cats a bit creepy. She felt that Leonard should have called her earlier. Sheldon introduces his friends as Oppenheimer, Frisch, Panofsky, Feynman, Weisskopf and Zazzles. Mary Cooper stops him and tells him that they have a lot of cute Jewish names. Sheldon continues that cats make wonderful companions that don't argue or question your intellectual authority like Amy did.
At suppertime, Mary Cooper calls Sheldon in telling him to leave the cats in his room. Sheldon comes out and finds Amy sitting the couch, invited by his mother. Amy explains that his mother thinks he might be losing his mind over her and as a neurobiologist, she was curious. Sheldon objects that he is in full possession of his faculties, to which Leonard pretends to sneeze and exclaimed “25 cats!” After Mary Cooper uses reverse psychology to repair the break between Amy and her son, Sheldon invites Amy to see his clowder. Amy agrees and tells him: I would. I love cats. They're the epitome of indifference.
Finally Sheldon and Amy are seen giving away his cats with a twenty dollar bonus with each cat.
It should be noted that after Leonard was rejected by Leslie Winkle, he also brought home some cat paraphernalia prior to picking up a cat. At the time Sheldon said that he was allergic to cats.