Discussion Overview
The discussion revolves around the concept of categories in category theory, exploring the nature and properties of various types of functors, including fully faithful functors, contravariant functors, and the notion of star categories. Participants engage in defining these concepts, referencing existing literature, and debating the nuances of definitions and properties.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- Some participants assert that there is an infinite number of categories and discuss the implications of this statement.
- There is a proposal to use the thread as a general purpose discussion for basic category theory, including definitions of functors and their properties.
- Definitions of fully faithful functors and contravariant functors are provided, with references to Saunders MacLane's work.
- Participants express uncertainty about the completeness of definitions regarding star categories and challenge each other's interpretations.
- One participant suggests that the definition of a star category may not fully capture Baez's concept, indicating a need for further refinement.
- There is a discussion about the relationship between different categories, such as Hilb and vector spaces, and the potential for infinite variations of categories.
- Some participants express a desire for clarification on specific terms, such as sheaves, indicating varying levels of familiarity with the subject matter.
- There is mention of the complexity of defining "kinds" of categories, suggesting that the discussion may hinge on specific definitions and contexts.
Areas of Agreement / Disagreement
Participants exhibit a mix of agreement and disagreement, particularly regarding the definitions and properties of star categories and functors. There is no clear consensus on the completeness of definitions or the interpretation of certain concepts.
Contextual Notes
Participants reference various definitions and properties from literature, indicating that the discussion may depend on specific interpretations and assumptions about category theory. Some definitions may not be universally accepted or may require additional context to be fully understood.
Who May Find This Useful
This discussion may be of interest to those studying category theory, particularly in understanding the nuances of functors and the definitions of various types of categories. It may also benefit individuals seeking clarification on foundational concepts in mathematics.