In the theory of open games, there has been a longstanding search for a good theory of “dependent optics”, that is a common generalization of Riley’s theory of optics and the dependent lenses of Spivak. We develop such a theory in the case of Markov categories, by introducing a novel object which we call Markov fibrations. Given an indexed family of objects (Xi)i∈I and a function f : J → I, we can construct a new family (Xf(j))j∈J.
This structure is captured by the classical notion of Grothendieck fibration. If f is instead a probability kernel, this reindexing no longer makes sense, yet there is still a relation between stochastic maps between indexing sets and indexed stochastic maps between the
indexed families. Markov fibrations generalize Grothendieck fibrations in a way which captures this relationship, and they admit a notion of fiberwise opposite which gives a useful notion of stochastic optic. In addition to introducing this concept and proving its basic properties, we also give applications to the theory of open games, and to Myers’ categorical theory of systems. Along the way we also give a detailed treatment of the so-called Para-construction, including how to generalize it to arbitrary 2-categories.
| Date of Award | 25 Feb 2026 |
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| Original language | English |
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| Awarding Institution | - University Of Strathclyde
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| Sponsors | University of Strathclyde |
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| Supervisor | Bob Atkey (Supervisor) |
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