In this talk, for finite $T_0$-spaces $X$ and $Y$ satisfying certain conditions, I will introduce reflection functors between the categories $\mathfrak{C^*sep}(X)$ and $\mathfrak{C^*sep}(Y)$ of separable C*-algebras over $X$ and $Y$. By the universal properties of KK-theory and E-theory, these functors induce reflection functors between the categories $\mathfrak{KK}(X)$ and $\mathfrak{KK}(Y)$ of Kirchberg's ideal-related KK-theory, as well as between the categories $\mathfrak{E}(X)$ and $\mathfrak{E}(Y)$ of Dadarlat and Meyer's ideal-related E-theory. I will show that the reflection functors between $\mathfrak{KK}(X)$ and $\mathfrak{KK}(Y)$ define an equivalence between the bootstrap categories $\mathcal{B}(X)$ and $\mathcal{B}(Y)$. I will also show that the categories $\mathfrak{E}(X)$ and $\mathfrak{E}(Y)$, and the E-theoretic bootstrap categories $\mathcal{B}_{\mathrm{E}}(X)$ and $\mathcal{B}_{\mathrm{E}}(Y)$ are equivalent via the reflection functors. The reflection functors introduced here are reminiscent of the BGP-reflection functors in the representation theory of quivers. As a consequence, I will show that $\mathcal{B}(X)$ and $\mathcal{B}(Y)$, $\mathfrak{E}(X)$ and $\mathfrak{E}(Y)$, and $\mathcal{B}_{\mathrm{E}}(X)$ and $\mathcal{B}_{\mathrm{E}}(Y)$ are respectively equivalent whenever the undirected graphs associated with $X$ and $Y$ are the same tree. If time permits, I will also discuss how reflection functors can be applied to construct K-theoretic invariants that satisfy the Universal Coefficient Theorems for ideal-related KK-theory and E-theory.
