2024-01-14

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.
4. the category of dg categories. we really just want to do equivalence of category, not isomorphism of categories. that requires that we have two ways functors.

Let's go simple. Suppose you have a sheaf on $\R$, and you want to test whether it is locally constant, so you do restriction. what's the condition? to test $(x,\xi)$ is not in the SS, it means there is an open set of $U$, such that $F(x-\epsilon, x+\epsilon) \to F(x-\epsilon, x)$ is a nice arrow.

Are we working in the homotopy category of dg categories?

Suppose the category $C$ have the notion of isomorphisms. Like set, or homotopy category of dg categories.

Then, we need to say, what does locally constant mean, indeed restriction to smaller open induces isomorphism in $C$.

Does Ho(dg-cat) admits arbitrary limit and colimit? Should be. The cat dg-cat embed into Ho(dg-cat), since it is a localization.