Suppose that we wanted to construct a mathematical universe where all objects were computable in some sense. How would we do it? Well, we could certainly allow the set \mathbb{N} into our universe: natural numbers are the most basic computational objects there are. (Notation: I’ll use N to refer to \mathbb{N} when we’re considering it as part of the universe we’ll building, and just \mathbb{N} when we’re talking about the set of natural numbers in the “real” world.) What should we take as our set of functions N^N from N to N? Since we want to admit only computable things, we should let N^N be the set of computable functions from \mathbb{N} to \mathbb{N}, which we can represent non-uniquely by their indices (i.e., by the programs which compute them).
J. Hyland. The L. E. J. Brouwer Centenary Symposium Proceedings of the Conference held in Noordwijkerhout, том 110 из Studies in Logic and the Foundations of Mathematics, Elsevier, (1982)
B. Toen, и G. Vezzosi. (2002)cite http://arxiv.org/abs/math/0212330arxiv:math/0212330Comment: 22 pages. Slightly enlarged. Clarified the part on homotopy types of Segal topoi.