  We develope $\mathbb{C}^{\ast}$-equivariant categorical Donaldson-Thomas
theory for local surfaces, i.e. the total spaces of canonical line bundles on
smooth projective surfaces. We introduce $\mathbb{C}^{\ast}$-equivariant DT
categories for local surfaces as Verdier quotients of derived categories of
coherent sheaves on derived moduli stacks of coherent sheaves on surfaces, by
subcategories of objects whose singular supports are contained in unstable
loci. Via Koszul duality, our construction may be regarded as certain gluing of
$\mathbb{C}^{\ast}$-equivariant derived categories of factorizations.
  We also develope $\mathbb{C}^{\ast}$-equivariant DT theory for stable
D0-D2-D6 bound states on local surfaces, including categorical
Pandharipande-Thomas theory. The key result toward the construction is the
description of the stack of D0-D2-D6 bound states on the local surface as the
dual obstruction cone over the moduli stack of pairs on the surface. We propose
several conjectures on wall-crossing of PT categories, motivated by
categorifications of wall-crossing formula of PT invariants and d-critical
analogue of D/K equivalence conjecture in birational geometry.
  We establish three ways toward the categorical wall-crossing conjecture:
semiorthogonal decomposition via linear Koszul duality, window theorem for DT
categories, and categorified Hall products. These techniques indicate several
implications, e.g. rationality of generating series of PT categories,
wall-crossing equivalence of DT categories for one dimensional stable sheaves,
and categorical MNOP/PT correspondence for reduced curve classes.
