Historically, quantum field theory emerged as a unification of relativity and quantum mechanics. The former can be seen as a categorical structure, and the latter as a noncommutative probability structure. In this talk, we will introduce an attempt at quantum field theory based on the concept of "category algebras," which are convolution algebras defined on categories as spacetime, and "states on categories," which are positive unital linear functionals on them. (Based in part on joint research with Hiroshi Ando, Soichiro Fujii, Takahiro Hasebe, Kazuya Okamura, and Izumi Ojima.)