2025-12-03

This following is a very brief summary of what happened in class on 2025-12-04.

We began with the situation of a double complex and then considered the various ways to measure exactness in a chain complex. We noted that there are "horizontal" and "vertical" homology objects, as well as two competing homology objects for measuring "diagonal" exactness. We called these new objects the donor and receptor objects. We then noted the existence of intramural and extramural maps.

We then brought all of these objects together in the (soon-to-be?) celebrated Salamander Lemma. While we didn't prove that lemma, next time we will see how it immediately leads to situations in which the extramural maps are isomorphisms.

Tomorrow we will start by noting situations in which the Salamander Lemma guarantees mural maps are isomorphisms, and then we'll set the Salamander Lemma loose on a bunch of named diagram lemmas. Be warned, Snake Lemma! The Salamander Lemma is coming for you!

Concepts

• Exact sequences and chain homology
• Double complexes and mural maps
• The Salamander Lemma

References

nLab, perhaps?