We consider data-processing of Markov chains through the lens of information geometry. We first develop a theory of congruent Markov morphisms in the context of Markov kernels that we show to correspond to the congruent embeddings with respect to the lumping operation. Furthermore, we inspect information projections onto geodesically convex sets of Markov kernels, and show that under some conditions, m-projecting onto doubly convex submanifolds can be regarded as a data-processing operation. Finally, we show that the family of lumpable kernels can be meaningfully endowed with the structure of a foliated manifold.
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