I would like to know which is the real difference between the Recursive topos (in the sense of Mulry) and the Effective topos (in the sense of Hyland). Especially what is related to recursive functions. Do they have the same semantic power?
I will be gratefull with some hints about texts related to this.


Thanks in advance.


PS: I don't give definitions because of their big extension, but I could give them if anybody wants.



The difference between the Recursive and the Effective topos.
Your question is a bit unclear, but an obvious difference between these two toposes is that the Recursive Topos is a topos of sheaves, hence cocomplete, wherease the Effective topos only has finite (non-trivial) coproducts. For example, the natural numbers object in the Effective topos is *not* a countable coproduct of 1's.


If you are looking for a deeper explanation, then perhaps it is fair to say that the Recursive Topos models computability a la Banach-Mazur (a map is computable if it takes computable sequences to computable sequences) and the Effective topos models computability a la Kleene (a map is computable if it is realized by a Turing machine). In many respects Kleene's notion of computability is better, but you'll have to ask another question to find out why :-)


