# Exploring Categories Without Sets: A Comprehensive Guide to Category Theory

• Reedeegi
In summary, the conversation revolved around finding resources for learning category theory without relying on set theory as a foundation. The participants mentioned a book by H. Simmons and also a section in MacLane's Categories for the Working Mathematician that directly presents category theory without relying on set theory. The idea of directly axiomatizing the 2-category of large categories and the possibility of finding it interesting was also mentioned. The conversation concluded with references to other books on topos theory, including Mac Lane's Sheaves in Geometry and Logic and Goldblatt's Topoi, and the possibility of the referenced article being Colin McLarty's work.
Reedeegi
Does anyone know of a book on Category Theory that purposely attempts to teach category theory without explicitly basing it upon set theory?

See http://www.cs.man.ac.uk/~hsimmons/BOOKS/CatTheory.pdf" . Also the appendix on Foundations in MacLane's Categories for the Working Mathematician is a direct presentation.

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Ah, yes, I see that MacLane's book does indeed include such a section; thank you!

What is the intent of your question? Maybe there are other interesting things out there you would find interesting -- topos theory comes to mind.

I know that someone (don't remember who) wrote a paper on directly axiomatizing the (super-large) 2-category of large categories, without reference to a formal set theory. Of course, his axioms provide for the construction of a large category Set, but I don't remember how that turns out to look. Maybe you'd find that interesting if you can find it?

Yes, I've been able to track down and purchase books on topos theory; including Mac Lane's Sheaves in Geometry and Logic and Goldblatt's Topoi; also, that article you referenced may be Colin McLarty's one on axiomatizing the category of categories; does this ring a bell?

## 1. What are categories without sets?

Categories without sets are mathematical structures used to organize and classify objects based on their relationships and properties. Unlike traditional set theory, categories do not require a collection of elements or a predetermined set membership, making them a more flexible and abstract way of organizing information.

## 2. How are categories without sets different from traditional set theory?

Categories without sets differ from traditional set theory in several ways. First, categories do not rely on a collection of elements or set membership, instead emphasizing the relationships and structures between objects. Additionally, categories allow for more flexibility and abstraction, allowing for a more general and intuitive way of organizing information.

## 3. What are the main applications of categories without sets?

Categories without sets have a wide range of applications in mathematics, computer science, and other fields. They are often used in abstract algebra, topology, and category theory to study mathematical structures and relationships. In computer science, categories are used in programming languages and software design to organize and structure data and algorithms.

## 4. Can categories without sets be visualized?

While categories without sets may seem abstract and difficult to visualize, they can be represented graphically using diagrams such as commutative diagrams, which illustrate the relationships between objects in a category. These diagrams can be helpful in understanding and visualizing the structure of a category.

## 5. What are the advantages of using categories without sets?

One of the main advantages of using categories without sets is their flexibility and applicability to a wide range of mathematical and scientific fields. Categories allow for a more abstract and intuitive way of organizing information, making them useful in studying complex structures and relationships. Additionally, categories can help identify common patterns and connections between seemingly unrelated concepts, leading to new insights and discoveries.

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