  We begin the study of categorifications of Donaldson-Thomas invariants
associated with Hilbert schemes of points on the three-dimensional affine
space, which we call DT categories. The DT category is defined to be the
category of matrix factorizations on the non-commutative Hilbert scheme with a
super-potential whose critical locus is the Hilbert scheme of points. The first
main result in this paper is the construction of semiorthogonal decompositions
of DT categories, which can be regarded as categorical wall-crossing formulae
of the framed triple loop quiver. Each summand is given by the categorical Hall
product of some subcategories of matrix factorizations, called quasi-BPS
categories. They are categories of matrix factorizations on twisted versions of
noncommutative resolutions of singularities considered by \v{S}penko-Van den
Bergh, and were used by the first author to prove a PBW theorem for K-theoretic
Hall algebras. We next construct explicit objects of quasi-BPS categories via
Koszul duality equivalences, and show that they form a basis in the torus
localized K-theory. These computations may be regarded as a numerical
K-theoretic analogue in dimension three of the McKay correspondence for Hilbert
schemes of points. In particular, the torus localized K-theory of DT categories
has a basis whose cardinality is the number of plane partitions, giving a
K-theoretic analogue of MacMahon's formula.
