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--- blog:2024-01-14 [2024/01/15 07:29] – created pzhou
+++ blog:2024-01-14 [2024/01/16 04:21] (current) – pzhou
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   - category of set? then isomorphism, bijection
   - category of vector space? isomorphism
-  - of chain complex of abelian groups? Well, quasi-isomorphism may not be
+  - of chain complex of abelian groups? Well, quasi-isomorphism may not be.
+  - 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.
-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