The mod p Langlands correspondence for GL_2(Q_p) relates irreducible mod p representations of this group, to mod p representations of the absolute Galois group of Q_p. In contrast with the situation over the complex numbers, all attempts at extending this correspondence to more general p-adic Lie groups have been so far unsuccessful, due to the lack of a classification of irreducible representations. In this talk we will discuss what can be gained by shifting attention from the irreducible objects to the whole category of representations. The results are joint work in progress with Matthew Emerton and Toby Gee, and with Bao V. Le Hung.