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In Hairer's notes A Theory of Regularity Structures he defines automorphisms of a regularity structure on page 28. I will recall the definition here:

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Is there any way of extending this to morphisms between different regularity structures? For example, one would probably want the polynomial regularity structure in $d$ variables to embed into the polynomial regularity structure of $d+1$ variables, right?

Is there a coherent way to define morphisms between different regularity structures? If so, what does the category of regularity structures look like? Does it have any interesting properties?

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