I am looking through Persistent homology and microlocal sheaf theory to learn a bit on barcodes. They are require the notion of a microsupport of a sheaf, looks like it could be a rather concrete geometric object. In order to study this particular notion of "barcode" they require microlocal sheaves and derived categories.

The reference book of Kashiwara-Shapira Sheaves on Manifolds has a through general discussion, but now it's hard to extract only the relevant parts to this one application. As for the barcodes paper they say

(p. 2) The main goal of this paper is to describe constructible $\gamma$-sheaves on $\mathbb{V}$

(p. 9) The aim of this paper is to describe the category $D^b_{\mathbb{R}c, \gamma^{\circ a}}(\mathbf{k}_{\mathbb{V}})$.

Before we do any of that, we try to understand the notion of *microsupport* for a sheaf.

- $F \in D^b(\mathbf{k}_M)$ write as $\mu \text{supp}(F)$ it's microsupport, a closed conic isotropic subset of $T^*M$.
- Let $M$ be a real manifold of dimension $\text{dim} M$. Let $\text{Mod}(\mathbf{k}_M)$ be the abelian category of sheaves of $\mathbf{k}$-modules on $M$, and let $\text{D}^b(\mathbf{k}_M)$ denote it's bounded derived category.

And I think we need slightly more, the notion of a $\gamma$-sheaf:

$$ \text{D}^b_{\gamma^{\circ a}}(\mathbf{k}_{\mathbb{V}}) = \{ F \in \text{D}^b(\mathbf{k}_\mathbb{V}) ; \mu\text{supp} \subset \mathbb{V} \times \gamma^{\circ a} \} $$ Here $\mathbb{V}$ is a finite dimensional vector space (such as $\mathbb{R}$ or $\mathbb{R}^2$) and $\gamma^\circ$ define a certain cone in a vector bundle $\pi: E \to M$

So it seems in order to learn this particular theory of barcodes, we need to understand microlocal sheaves a bit better (and I do not). However, there is also other points of view:

- Robert Ghrist Barcodes: The Persistent Topology of Data
- Gunnar Carlson Topology and Data