2025-12-02

This following is a very brief summary of what happened in class on 2025-12-02.

We started by recalling the six properties of an abelian category. We then re-introduced the idea of a sequence in a category, and then a chain complex, which makes sense in any category that contains a null object. We then recalled the definition of an exact sequence in a category of modules, and noted that the definition required the concept of an image of an arrow.

We then took an aside to give an element-free definition of the image of a morphism.

We noted that the category of chain complexes is itself an abelian category, so it made sense to talk about chain complexes in that category. This led to the idea of a double complex.

If all of these element-free definitions (of kernels and images and bears, oh my!) are driving you crazy, keep in mind that the "official" definitions are being made so as to be as broadly applicable as possible; i.e., in "non-concrete" categories where notions of "elements" is not officially available. However, in our context (of abelian categories), it turns out that every abelian category is (equivalent to) a subcategory of a category of modules. This means we're free to treat objects and arrows in an abelian category as we would modules and module morphisms. We can refer to elements and view arrows as (particular) set maps between sets of elements. We'll do just that in our final two classes of the quarter.

Concepts

• Abelian categories
• Chain complexes
• Exact sequences and chain homology

References

nLab, perhaps?