If you are anything like me you might sometimes struggle with remembering the fine details of a definition. One example of this for me is the notion of recollement in the world of triangulated categories: this is a decomposition of a triangulated category in two smaller triangulated categories with properties that are reminiscent of the decomposition of a (deliberately left vague) triangulated category on a topological space into a piece associated to an open set and a piece associated to its complement.
After reading its definition for the umpteenth time I realised it is easy to reconstruct all parts of the definition, assuming you can remember the shape of the diagram:
- each pair of (vertically) adjacent arrows is an adjoint pair
- the (unique) composition from left to right is zero
- functors towards the big category are fully faithful
- the four non-equivalence (co)unit transformations are used to decompose objects of the big category
From this set of four ingredients you can deduce the explicit axioms for a recollement. Otherwise you can just use the little cheat sheet that I decided to write.