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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?

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What about cwru.edu/artsci/math/wells/pub/ttt.html ? (Warning: not tested.) –  darij grinberg Feb 18 '11 at 20:02
    
It would be nice if someone comes with an answer having a whiff of computer science in it (none of the first 6 answers has). I can think of no book and not example. –  Jérôme JEAN-CHARLES Feb 19 '11 at 1:18
    
The 'effective topos' is also an object you must look at as a computer scientist. –  Jérôme JEAN-CHARLES Feb 19 '11 at 1:40
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8 Answers 8

Sheaves in Geometry and Logic, by MacLane and Moerdijk, is a beautifully written book on the subject. It's one of the rare texts on such formal material that is fun to read, and is relatively easy from start to finish.

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I second this suggestion. Mac Lane and Moerdijk's book is a pleasure to read, and really has no prerequisites besides a basic knowledge of category theory. –  Daniel Miller Feb 18 '11 at 21:37
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I'm not sure how appropriate this question is for MO, but a clear candidate would be:

Topoi: A Categorial Approach to Logic, by Goldblatt.

It's free for download online, and it is pretty much perfect for what you're describing.

Another option could be Awodey's book.

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Have you seen the article An informal introduction to topos theory by Tom Leinster?

http://arxiv.org/abs/1012.5647

The abstract says:

"This short expository text is for readers who are confident in basic category theory but know little or nothing about toposes. It is based on some impromptu talks given to a small group of category theorists."

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I discovered these Tom Leinster notes from this post via a Google search. They are fantastic, thanks. There is an even shorter (a few pages) "informal introduction" to topos theory in Tom's category theory course notes (see pg 110), which can be seen as an "introduction to this introduction": maths.gla.ac.uk/~tl/msci/all.pdf –  Bruce Bartlett Mar 4 '11 at 10:38
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Introduction to Higher Order Categorical Logic by Joachim Lambek and P. J. Scott. Try Google Books for this. The point of view is much more suitable for the functional programming aspects, even though the words "computer science" may never appear in the book. (Historically it is quite impossible to understand where toposes came from without sheaves, but technically starting from cartesian closed categories and adding bells and whistles is a shortcut.)

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If your computer science background includes any type theory (especially typed functional programming, etc.), then I’d highly recommend this book. And if it doesn’t… then it should, since if you’re a computer science student who enjoys category theory and is excited by toposes, I’d guess you may well love type theory — a beautiful crossover zone between proof theory, category theory, and theory of programming languages. –  Peter LeFanu Lumsdaine Feb 19 '11 at 4:21
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Along the lines of "Sheaves in Geometry in Logic", Ieke Moerdijk (co-author of that book) also wrote these lecture notes with Jaap van Oosten:

http://www.staff.science.uu.nl/~ooste110/syllabi/toposmoeder.pdf

I found them very good when I was first learning.

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Many people would say this is a terrible suggestion, I think, but depending on your tastes and style, Peter Johnstone’s 1971 book “Topos Theory” might be good.

…true, it’s exceedingly dry, and has been described as “famously impenetrable”, and I certainly wouldn’t recommend it as an only text to try to learn about toposes from. But I actually found it very helpful when I was first learning Topos Theory — first and foremost because it has really, really excellent exercises, with a big range of subjects and difficulties. Secondarily, I also found that once I’d struggled tooth and nail to understand a construction elsewhere, I could come back to Johnstone and appreciate a really clear, neat, perfectly tuned presentation — albeit one I wouldn’t have been able to get anywhere with on its own.

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Johnstone's book was my first introduction to the subject as well, and I found it extremely helpful. I like this book a lot! –  Ramin Feb 19 '11 at 6:30
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Elementary Categories, Elementary Toposes by Colin McLarty seems like it is what you want.

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Ncat has a bunch of links: http://ncatlab.org/nlab/show/topos

In particular an outline of Johnstone's book is here: http://ncatlab.org/nlab/show/Elephant

I've been wanting to read it...

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