TITLE: Topos theory reference suitable for undergraduates
QUESTION [19 upvotes]: I am a third year undergraduate who has just learnt the rudimentals of category theory. 
My specialization is computer science, not mathematics. As part of my course work I want to write an essay on Topos theory. My professor says that it is possible to do so with my level (very little) of mathematical maturity, but I am not able to find any sources that treat this theory at anywhere near my level. Any suggestions?

REPLY [2 votes]: The internal language of the effective topos can be understood with requiring barely any technology and is lots of fun! For instance, Andrej Bauer observed that if you construct the effective topos using infinite-time Turing machines instead of ordinary Turing machines, then internal to the topos there is an injection $\mathbb{N}^\mathbb{N} \to \mathbb{N}$. See these slides for undergraduates.