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Register a new userThe linear-non-linear substitution 2-monad
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Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We introduce a general construction on 2-monads. We develop background on maps of 2-monads, their left semi-algebras, and colimits in 2-category. Then, we introduce the construction of a colimit induced by a map of 2-monads, show that we obtain the structure of a 2-monad and give a characterisation of its algebras. Finally, we apply the construction to the map of 2-monads between free symmetric monoidal and the free cartesian 2-monads and combine them into a linear-non-linear 2-monad.
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We provide new categorical perspectives on Jacobs notion of hypernormalisation of sub-probability distributions. In particular, we show that a suitable general framework for notions of hypernormalisation is that of a symmetric monoidal category endowed with a linear exponential monad---a notion arising in the categorical semantics of Girards linear logic. We show that Jacobs original notion of hypernormalisation arises in this way from the finitely supported probability measure monad on the category of sets, which can be seen as a linear exponential monad with respect to a monoidal structure on sets arising from a quantum-algebraic object which we term the Giry tricocycloid. We give many other examples of hypernormalisation arising from other linear exponential monads.
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