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In mathematics, the effective topos is a topos introduced by Martin Hyland (1982), based on Kleene's notion of recursive realizability, that captures the idea of effectivity in mathematics.

References
Hyland, J. M. E. (1982), "The effective topos", in Troelstra, A. S.; Dalen, D. van (eds.), The L.E.J. Brouwer Centenary Symposium (Noordwijkerhout, 1981), Studies in Logic and the Foundations of Mathematics, 110, Amsterdam: North-Holland, pp. 165–216, doi:10.1016/S0049-237X(09)70129-6, ISBN 978-0-444-86494-9, MR 0717245
Kleene, S. C. (1945). "On the interpretation of intuitionistic number theory". Journal of Symbolic Logic. 10 (4): 109–124. doi:10.2307/2269016. JSTOR 2269016.
Phoa, Wesley (1992), An introduction to fibrations, topos theory, the effective topos and modest sets
Bernadet, Alexis; Graham-Lengrand, Stéphane (2013). "A simple presentation of the effective topos". arXiv:1307.3832.