a category with a notion of 'equal', or 'isomorphism', or 'quasi-isomorphism'.

1. category of set? then isomorphism, bijection
2. category of vector space? isomorphism
3. of chain complex of abelian groups? Well, quasi-isomorphism may not be

if you have a sheaf on $\R$, how would you define its singular support? well, at each point, we only have two directions, we can ask for before and after.

I want to assume that