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I am trying to learn topos theory and I am finding a strong scarcity of resources. Is there any canonical textbook to refer someone to when learning this topic?

So far, I have only been able to find the following.

  • The Stacks project (Part 1, Chapter 7 gives an introduction)
  • "Théorie des Topos et Cohomologie Etale des Schémas" by Grothendieck, Artin and Verdier (yes, this one is quite old and in French but I have found all the necessary topics together in a better way than any other resource known to me)

In particular, I have only found a clear distinction between topology and pretopology in the second one, as well as clear guidance on how to work interchangeably between them.

Do people have any suggestions? Please list them below so other people trying to learn topos theory have a better idea of where to start!

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    $\begingroup$ I recommend the standard references "Sheaves in geometry and logic", by Maclane-Moerdijk, as well as the two volumes of Peter Johnstone "Sketches of an Elephant". $\endgroup$
    – godelian
    Commented Dec 25, 2020 at 14:10
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    $\begingroup$ Googling will immediately turn up at least four more good references. $\endgroup$
    – Kevin Carlson
    Commented Dec 25, 2020 at 14:38
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    $\begingroup$ E.g. ncatlab.org/nlab/show/topos#References $\endgroup$
    – Mike Shulman
    Commented Dec 25, 2020 at 15:25
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    $\begingroup$ Pretty much a duplicate of Topos theory reference suitable for undergraduates. Question not exactly the same, but close enough that there’s really no difference in what answers they invite. $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Dec 26, 2020 at 16:39
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    $\begingroup$ @PeterLeFanuLumsdaine: The OP used tags "sites" and "grothendieck-topology", which clearly point to Grothendieck toposes. The other question mentioned computer science, which clearly points to elementary toposes. Only the OP can offer clarifications, of course, but it would seem to me that the current question is already unambiguous, if not very explicit, and is certainly distinct for the other question. $\endgroup$
    – Dmitri Pavlov
    Commented Dec 26, 2020 at 18:32

3 Answers 3

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For a beginner, the more accessible textbooks seem to be the following two.

They both cover sheaves, Grothendieck topologies, locales, classifying toposes, and other classical topics.

The second book is a bit longer and covers some additional topics, e.g., the independence of the axiom of choice and continuum hypothesis.

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Dmitri has mentioned two fantastic references, which are very complete and well written.

I will mention two short references for those that want to get the general idea, before approaching a complete book.

  1. Leinster, An informal introduction to topos theory.
  2. Borceux, Some glances at topos theory.
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Here are some references on Steven Vickers works to look at topoi from other "geometric" and logical sides.

I think it also worth to mention the following

For some advanced matters I also suggest to read

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  • $\begingroup$ I’m not sure any of this seems relevant to a request for a “canonical textbook.” $\endgroup$
    – Kevin Carlson
    Commented Dec 27, 2020 at 16:07
  • $\begingroup$ @Kevin Arlin I agree, the refences on Vickers may be not canonical but, in my opinion good additional resources to look at topoi more widely in parallel with Johnston, Borceux, Maclane, Moerdijk and Leinster. Nonetheless Classifying Spaces and Classifying Topoi is quite canonical. $\endgroup$
    – Evgeny Kuznetsov
    Commented Dec 27, 2020 at 16:15

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