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I am interested in learning more about the nLab categorical perspective on several mathematical subjects such as topology and logic, but found that my understanding of category theory was not sufficient to understand most of the more general articles (not even getting into the hardcore, more advanced topics). Obviously, category theory is an essential requirement. However, exactly what background in category theory do you think is needed to understand the average nLab article?

I understand that my question is only meaningful if you know the primary mathematical subject of a specific article. Of course, if you try to read the article on synthetic differential geometry but do not know what differential geometry is, you should first study the subject from a more traditional perspective. My question should read, "Assuming that you know the main subject of the article beforehand what is the minimum required background ...".

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I recommend being lazy and learning stuff "on demand" ;-) – dtldarek Feb 12 '13 at 17:52
up vote 6 down vote accepted

I don't recommend reading the nlab for the purpose of understanding random articles. You will get lost! The nlab is a convenient (and probably the best) online reference for basic and advanced notions of category theory, with an emphasis on higher category, categorification and the underlying mathematical-philosophical ideas. Thus, you can use it when you elsewhere stumble upon a notion or an idea from category theory which you don't know yet. You can also get deeper category-theoretic perspectives of notions which you are already familiar with. When you really want to dig into some topic, use the references. There is no online encyclopedia which can replace textbooks or research articles.

When you want to learn the basic notions of category theory, here are some quite good books:

  • Mac Lane, Categories for the Working Mathematician
  • Borceux, Handbook of categorical algebra
  • Awodey, Category Theory
  • Kashiwara, Schapira - Categories and Sheaves

It doesn't matter how much you read in advance, there will always be nlab articles which are inaccessible and hard to understand. Try to use the links and digest the more basic related articles first. For example, look at all the links given in the article on the small object argument. When it's too much, try to use other sources, too. Remember that the mathematics at the nlab tends to be quite general and abstract, but perhaps you are not always interested in the most general case for some specific application. For example, you don't need to know the general theory of locally presentable categories, model categories and factorization systems in order to understand the core argument of the small object argument.

Besides, often some sections of the articles can be skipped at first reading. For example, when you want to learn the notion of a monoid object, you can probably skip the section about $(\infty,1)$-operads. It is too easy to get overwhelmed by all the information on the nlab, even about such basic notions.

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Thank you! You also answered related questions that I had in mind. Thanks as well for the references. – Robin Wolffoot Feb 12 '13 at 21:00

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