Learning Categorical Foundations

[Reedeegi]

Hello,
I am wondering what resources are most helpful in learning the progress made by category and topos theorists in creating a "new" foundation for mathematics. So far I have:

Categories for the Working Mathematician
Topoi by Goldblatt
First-Order Categorical Logic by Makkai
An Introduction to Higher-Order Categorical Logic by Lambek

After learning from these books, where would I go to learn about the mathematical applications and framework supplied by topos theory?

Uh-oh... This should be in General Math, not General Physics. Can someone move it?

 

[Valerie Park]

To answer your question though, I would suggest reading the books Theory and Applications of Categories by Borceux and Categorical Logic and Type Theory by Lambek and Scott. These books provide a good overview of the mathematical applications and framework supplied by topos theory. Additionally, there are a number of other books and articles available on the subject. You may also want to check out the websites for The n-Category Café and The nLab for further resources.

 

[sir-pinski]

Hi there,

Thank you for your question. It sounds like you are interested in learning more about the progress made by category and topos theorists in creating a new foundation for mathematics. The books you have listed are great resources for learning about these topics, and I would also recommend checking out the following resources:

1. "Sheaves in Geometry and Logic: A First Introduction to Topos Theory" by Saunders Mac Lane and Ieke Moerdijk
2. "Category Theory for the Sciences" by David Spivak
3. "Topos Theory" by Steve Awodey
4. "Conceptual Mathematics: A First Introduction to Categories" by F. William Lawvere and Stephen H. Schanuel

After learning from these resources, you may want to explore the applications of topos theory in various areas of mathematics, such as algebra, geometry, and set theory. Some specific topics you may want to look into include sheaf theory, algebraic geometry, and homotopy theory. Additionally, attending conferences and workshops on category theory and topos theory can also be a great way to learn about the latest developments in the field and to connect with other researchers and practitioners.

I hope this helps guide you in your learning journey. Best of luck!