Several years ago Braverman-Finkelberg-Nakajima proposed a mathematical definition of the Coulomb branch of a 4d N=2 or 3d N=4 gauge theory of cotangent type. The 4d Coulomb branch is defined as the spectrum of the equivariant K-theory of a certain ind-scheme, the space of triples, associated to a compact Lie group and a complex representation. On the other hand, following work of Kapustin-Saulina and Gaiotto-Moore-Neitzke it is anticipated that expectation values of Wilson-'t Hooft operators endow the coordinate ring of the 4d Coulomb branch with a positive integral basis, and that a similar statement holds for the irreducible half-BPS line operators in a general 4d N=2 field theory. In this talk we discuss work in progress with Sabin Cautis which proposes a mathematical definition of this category of half-BPS line operators (hence also the basis of irreducible objects) for gauge theories of cotangent type. The definition takes the form of a nonstandard t-structure on the dg category of equivariant coherent sheaves on the space of triples. In the case of pure gauge theory, this is the perverse coherent t-structure, which in previous work we showed provides a good definition in so far as it is nontrivially consistent with the expectations of framed BPS counting. However, since the natural strata of the space of triples are not finite-dimensional in the presence of matter, the perverse t-structure is not a priori well-defined, and new ideas are required to generalize it.