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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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  • $\begingroup$ 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. $\endgroup$
    – Jérôme JEAN-CHARLES
    Commented Feb 19, 2011 at 1:18
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    $\begingroup$ The 'effective topos' is also an object you must look at as a computer scientist. $\endgroup$
    – Jérôme JEAN-CHARLES
    Commented Feb 19, 2011 at 1:40
  • $\begingroup$ Since you're a computer scientist, I would suggest Mclarty's "Elementary Categories, Elementary Toposes", Lambek&Scott's "Introduction to Higher Categorical Logic" and Jacobs' "Categorical Logic and Type Theory". The last one is not exactly about topoi, but it's related and is specially useful for computer scientists. These books I cited are more lambda calculus oriented although the first one is not too much. $\endgroup$
    – user40276
    Commented Nov 25, 2016 at 13:08
  • $\begingroup$ Do you have an idea what shape the essay might take? Would it be an exposition of theory, or history of ideas, or applications, or what? $\endgroup$
    – Todd Trimble
    Commented Nov 26, 2016 at 16:08

9 Answers 9

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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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    $\begingroup$ 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. $\endgroup$
    – Daniel Miller
    Commented Feb 18, 2011 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: The Categorial Approach to Logic, by Robert Goldblatt (another source).

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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    $\begingroup$ Is the Awodey book you mention Category theory (MR)? $\endgroup$
    – LSpice
    Commented Nov 24, 2016 at 20:31
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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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    $\begingroup$ 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 $\endgroup$
    – Bruce Bartlett
    Commented Mar 4, 2011 at 10:38
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    $\begingroup$ Bruce's link is dead. The notes appear to have transmuted into a book, here: maths.ed.ac.uk/~tl/bct $\endgroup$
    – Neil Strickland
    Commented Nov 24, 2016 at 19:28
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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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Elementary Categories, Elementary Toposes by Colin McLarty seems like it is what you want.

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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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    $\begingroup$ 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. $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Feb 19, 2011 at 4:21
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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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  • $\begingroup$ Johnstone's book was my first introduction to the subject as well, and I found it extremely helpful. I like this book a lot! $\endgroup$
    – Ramin
    Commented Feb 19, 2011 at 6:30
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    $\begingroup$ I too agree that it's an excellent resource for learning topos theory. It's not a terrible suggestion at all. $\endgroup$
    – Ingo Blechschmidt
    Commented May 3, 2017 at 13:25
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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.

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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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