Probability theory and statistics are usually developed based on Kolmogorov's axioms of probability space as a foundation. In this talk, I will present an alternative foundation that is currently under development, and I will argue that it improves over the traditional one in various respects. This approach is probabilistic in terms of category theory, and it makes Markov kernels instead of spaces into the fundamental primitives. Its abstract nature also implies that no measure theory is needed. After introducing the approach itself, I will summarize our proof of the de Finetti theorem in terms of it.