2025-12-01

This following is a very brief summary of what happened in class on 2025-12-01.

Our last topic for the quarter is a survey of abelian categories. We started by stating the six properties a given category might additionally satisfying. This broke down as follows:

• Two properties that qualified a category to be called a preadditive category (or $\mathbf{\text{Ab}}$-category)
• Two additional properties that qualified a preadditive category to be called an additive category
• Two last additional properties that qualified an additive category to be called an abelian category

Some of these properties involved terms we hadn't defined yet, so then we started filling in those details. We defined

• null objects
• biproducts
• kernels and cokernels

Next class we will pick back up with kernels and cokernels, and then proceed to:

• Chain complexes
• Double complexes

Concepts

• Ab-categories
• Additive categories
• Abelian categories

References

No good ones at the moment.