I've heard that someone's thought up a way of using category theory, involving something called topoi, as a foundation for mathematics. If this is true then are there any texts which explain such a categorical approach to foundations? Preferably one which is self-contained, as while I'm sure I could, with time, teach myself the category theory required...I'd rather have it bundled up in a package for me.
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2$\begingroup$ To start, see ncatlab.org/nlab/show/ETCS. $\endgroup$ – Qiaochu Yuan May 10 '15 at 7:32
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$\begingroup$ Somewhat related: Foundations book using category theory for student embarking on PhD in mathematical biology?. $\endgroup$ – Martin Sleziak May 10 '15 at 7:32
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$\begingroup$ I looked at that, but there was only a single recommendation that looked like it might actually be suitable $\endgroup$ – Nethesis May 10 '15 at 7:34
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$\begingroup$ The original paper of Lawvere is actually readable with a small categorical background. $\endgroup$ – Pece May 11 '15 at 10:00
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The best book for your purposes would be McLarty, Elementary Categories, Elementary Toposes.
