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I had started an entry “exponentiation” but then thought better of it and instead expanded the existing exponential object: added an examples-section specifically for $Set$ and made some remarks on exponentiation of numbers.
There is an entry exponential map as well. It should stay separate -- he continuous context is different from the arithmetics of cardinals which may fall into decategorification of the situation at exponential object. So the case of cardinals where the exponentiation is by the counting of the size of power set related etymologically to exponential object is the intersection of two notions which would be overstretched to unify -- solutions of ODEs leading to exponential functions, operators, maps, unipotent groups and so on, and the case of generalizing counting power sets to exponential objects.
It would be nice to have something about the unipotent groups, as it is related to many cases where Feynman integrals appear. By opening the entry on free Lie algebra I mean one should look at the exponent of this Lie algebra via ordered products...
I added many more references to free Lie algebra including Kapranov and also Schneps.
I agree with Zoran here, and I didn’t see what you were getting at in the final paragraph of the section on exponentiation of sets and numbers (in particular, I didn’t know what “it” in “It yields for instance” was supposed to refer to).
Cardinal numbers as isomorphism classes of objects in Set is OK, but more general numbers, operators etc. indeed belong to different kind of exponentiation. I guess Urs was looking at cardinals (including nonnegative integers) only what is OK, but somehow wanted to think beyond what is not that apt.
I added to exponential object an example that a natural transformation is exponentiable in $D^C$ if it is cartesian and pointwise-exponentiable. Has anyone seen this before?
Added to exponential object the usage exponential transpose (which is frequently used, but somewhat surprisingly was found on three pages on the nLab only) and also the lambda-notation, with a reference, and, confined to the footnote, the rather rare alternative “flat”-notation.
It’s also known as currying, and I’m sure other names are in usage. I added a note on that.
I made exponential transpose a redirect to currying, although I suppose one might argue that it should redirect to exponential object instead.
Since to me it seems useful for readers I added, confined to footnotes and with references, some notational remarks to exponential object and adjunct.
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