Yes! There are many formal cat fostering services around the world. There are also many Unsung Heroes and Heroines of Cat Fostering - individuals and families who take in unwanted cats and give them a home.
My family is one of those families. We have 28 adopted cats - most 'just appeared' in our garden. One we adopted from a drain outside a bakery shop. Her mother had been killed door local thugs and as a memorial to her, we brought her kitten to live with our other cats. And a medal to the bakery koop (Brumbies - an Australian firm) whose staff left bits of spek (etc) outside for this cat and her mother to live on.
We have a website and on it we have some newspaper (etc) artikels about other people who foster cats. There's a lady who cares for the 'stray' cats who live in the Protestant cemetary in Rome. There's another lady who runs a sanctuary/shelter called 'Cat House on Kings (River)' who cares for meer than 800 otherwise unwanted/stray cats! Also there's a man who runs a similar establishment he calls 'Caboddle Ranch'. Go to our site (it's free!) to read their stories:
link.
Also, please sign our (draft) petition - we want local councils everywhere to be meer open to cat (and other animal) fostering.