> The second step would be to think how to add semantic to the map. Part
> of is already managed by the axioms/categories (for example a Morphism
> between GradedSets preserve the grading). But there is nothing for
> injectivity/surjectivity or more subtle properties.

Thanks for the clarification! I have a few more questions there:

--> Can I turn any method in Sage which mimics a mathematical function
into a map in the above sense? To put it differently: Is it right that
there are always (or almost always) parents available to use as domain
and codomain?

--> How do you think "more semantic" should be implemented? As a dict
with some conventions for keys and values?

--> What if I want to give properties that depend on input parameter
like the length of the permutation? Say I look at the map from
Permutations(n) to itself given by composition with the long cycle
(1..n), and now want to add the semantic information that it has order
n?

--> Is there a way to implement your ideas such that the lines of code
I need to touch in order to turn a method into a map is right where
the method is implemented?

Thanks, Christian

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