Reddit reviews Topoi: The Categorial Analysis of Logic (Dover Books on Mathematics)

Preach it, brother!

Let me highly recommend Conceptual Mathematics: A First Introduction to Categories and Topoi: The Categorial Analysis of Logic as introductions to the topic requiring no more than a completely rudimentary grasp of set theory to get started--and really, they even motivate the rudimentary set theory. These are basically "Category Theory for Dummies," or at least the closest things that I've found so far.

If you know topology and algebra then I think the most fruitful way of approaching categories is by picking up a book on algebraic topology. Hatcher is a canonical reference and although it doesn't really introduce the formal language of category theory, it is shown through most of the book. If you're not that patient and have more mathematical maturity then just pick up May's concise course in algebraic topology which is a wild ride but will get you there.

The most canonical textbook is Mac Lane's categories for the working mathematician but it's kind of dry so you'll need to provide your own class of examples every time.

If you just want to take a look into the topic and see how it is like then I would recommend you to read the first three chapters of this book. The main topic of the book is a little bit advanced so you can ignore any mention of topoi but those first three capters are a very brief introduction to category theory through a couple of examples and as far as I remember it doesn't expect you to know the stuff beforehand. It is however very basic and it won't cover a lot of the useful constructions insipired in algebraic topology/geometry but I still believe it's a pretty nice summary for the language.

Category theory is not easy to get into, and you have to learn quite a bit and use it for stuff in order to retain a decent understanding.

The best book for an introduction I have read is:

Algebra (http://www.amazon.com/Algebra-Chelsea-Publishing-Saunders-Lane/dp/0821816462/ref=sr_1_1?ie=UTF8&qid=1453926037&sr=8-1&keywords=algebra+maclane)

For more advanced stuff, and to secure the understanding better I recommend this book:

Topoi - The Categorical Analysis of Logic (http://www.amazon.com/Topoi-Categorial-Analysis-Logic-Mathematics/dp/0486450260/ref=sr_1_1?ie=UTF8&qid=1453926180&sr=8-1&keywords=topoi)

Both of these books build up from the basics, but a basic understanding of set theory, category theory, and logic is recommended for the second book.

For type theory and lambda calculus I have found the following book to be the best:

Type Theory and Formal Proof - An Introduction (http://www.amazon.com/Type-Theory-Formal-Proof-Introduction/dp/110703650X/ref=sr_1_2?ie=UTF8&qid=1453926270&sr=8-2&keywords=type+theory)

The first half of the book goes over lambda calculus, the fundamentals of type theory and the lambda cube. This is a great introduction because it doesn't go deep into proofs or implementation details.

Depends on your background. Mac Lane is the standard text and he is a phenomenal author in general, but it builds off knowledge of concepts such as modules, tensor products and homotopy (I still don't have a sufficient background in AT to be honest though). For a more modest background, I would recommend the book "Sets for Mathematics" by Lawvere and Rosebrugh. The book is entirely on category theory, the title is because there is a focus on the category of sets. The first chapter or so is deceptively simple, it gets very difficult as it goes on, but still doesn't require much specific background.

I'll also note that I first got into the subject through a whim purchase in a local Borders of a cheap dover book Topoi by Robert Goldblatt when I was very into mathematical logic. It's 500 pages and requires pretty much no background (I'd know what a topological space is, but I can't think of anything else). It gets very challenging though, and I never got more than 250 pages in before getting overwhelmed, but the first hundred pages really sparked my interest in category theory. Functors (and especially adjoint functors) are postponed much later than you will see in many other sources though. You can find a link to an online version free from the author's webpage too.

Here is a link to John Baez's overview of what Topos theory is.

As /u/ziggurism has already said, you don't need to understand algebraic geometry to understand the theory of elementary toposes but some of the early motivating examples, the category of sheaves on a grothendieck site, are heavily steeped in the language of modern algebraic geometry.

A good, non algebro-geometric introduction to topos theory for those seeking to understand its place in logic is the book 'Topoi: The Categorial Analysis of Logic by Goldblatt. It also serves as a great introduction to the ideas of category theory.

Another fantastic book is Lawvere/Roserugh's Sets for Mathematics. This book seeks to explain the axioms of set theory using the language of category theory. It's not a book on arbitrary topos but it does serve to give you an idea of how topos theory axioms serve to build the logical system that every mathematician is familiar with, the logic in the category of sets. It's a good idea to have this 'concrete' application of topos axioms in Set under your belt before you tackling a book that seeks to explain how an arbitrary topos gives you a more abstract and unfamiliar logical system.

Edit: Also worth looking into is how topos theory can be used in the foundations of physics

For anyone else, I assume the specific book is Goldblatt's "Topoi: The Categorical Analysis of Logic"

Reading a bunch about Turbulence and Topoi, mostly Tsinober and Goldblatt. Working on a proof about the relation of two manifolds also, basicly a lot of Jacobians.

I did an independent study of category theory from Goldblatt's Topoi: The Categorical Analysis of Logic. Having not studied category theory further than that, I can't offer much comparison, but I found it just barely accessible (which is generally about the best I hope for) and pretty cheap to boot.