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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 16th 2010

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.

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeOct 16th 2010

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...

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeOct 16th 2010

I added many more references to free Lie algebra including Kapranov and also Schneps.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeOct 16th 2010

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).

• CommentRowNumber5.
• CommentAuthorzskoda
• CommentTimeOct 16th 2010

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.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeAug 25th 2016

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?

• CommentRowNumber7.
• CommentAuthorPeter Heinig
• CommentTimeJun 19th 2017
• (edited Jun 19th 2017)

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.

• CommentRowNumber8.
• CommentAuthorTodd_Trimble
• CommentTimeJun 19th 2017

It’s also known as currying, and I’m sure other names are in usage. I added a note on that.

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeJun 19th 2017

I made exponential transpose a redirect to currying, although I suppose one might argue that it should redirect to exponential object instead.

• CommentRowNumber10.
• CommentAuthorPeter Heinig
• CommentTimeJun 24th 2017

Since to me it seems useful for readers I added, confined to footnotes and with references, some notational remarks to exponential object and adjunct.

• CommentRowNumber11.
• CommentAuthorDmitri Pavlov
• CommentTimeNov 29th 2019
Does the nLab have an article on exponentiable maps? Currently, only exponential object seems to exist, but I may be missing something.
• CommentRowNumber12.
• CommentAuthorMike Shulman
• CommentTimeNov 29th 2019

No, they’re just bullet point 3 under “Examples” in exponential object. In principle they certainly deserve their own page; if you want to create one that’d be great!

• CommentRowNumber13.
• CommentAuthorTim Campion
• CommentTimeFeb 21st 2021

I just added a few more examples, including examples of exponentiable morphisms. I’m not an expert in the literature here, but I couldn’t find a characterization of all exponentiable morphisms of locales or of toposes. I added several references, but I got a bit lazy at the end so some of the references are little more than just links.

• CommentRowNumber14.
• CommentAuthorTim Campion
• CommentTimeFeb 21st 2021
• (edited Feb 21st 2021)

I also just noticed that the doi of one of the references I added has parentheses in the URL, which are getting mis-parsed by the page render so that the link is not usable. I’m not sure what to do about that.

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeFeb 21st 2021

Yes, that’s an issue. Two ways to go about it:

either replace “(” by “%28” and “)” by “%29” (browsers may do that for you in the URL line when asked to open it).

or use HTML-syntax for the link

  <a href="url">text</a>

• CommentRowNumber16.
• CommentAuthorTim Campion
• CommentTimeFeb 22nd 2021

Thanks, Urs. I think I’ve fixed the URL (I used the html option.)

• CommentRowNumber17.
• CommentAuthorMike Shulman
• CommentTimeMar 20th 2021

Clarified the requirements of existence of products and pullbacks for exponentiability, and added the fact that exponentiable morphisms are pullback-stable in the presence of coreflexive equalizers.

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeAug 28th 2021
• (edited Aug 28th 2021)

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeAug 28th 2021

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeSep 8th 2021

Prompted by the thread for “residual” here I discovered that this entry here had a line saying (here):

When $C$ is not cartesian but merely monoidal, then the analogous notion is that of a left/right residual.

This seems strange to me, given that “residual” seems to be just a non-standard invention for the classical “internal hom”, and given that the entry here starts out saying in its first line that exponential objects are internal homs in cartesian monoidal categories.

So I have changed “residual” to “internal hom” in that line. But I think the whole line remains somewhat redundant in the present entry.

• CommentRowNumber21.
• CommentAuthorDavid_Corfield
• CommentTimeSep 8th 2021

I guess it’s coming from things like residuated lattices.

We have at closed monoidal category

The analogue of exponential objects for monoidal categories are left and right residuals.

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeSep 8th 2021

That line at closed monoidal category is from the one and same author who’s edits are being debated: rev 34 in 2017.

• CommentRowNumber23.
• CommentAuthorTim_Porter
• CommentTimeSep 8th 2021
• (edited Sep 8th 2021)

My impression is that there are several intuitions, examples and concepts, that are interlocking here. I knew of residuated lattices from seminars on semigroup theory, languages and automata, etc., and also on the idempotent semiring (max,plus)-algebra ideas modelling discrete event systems. The link with exponential objects sometimes is mentioned in the literature, but those seminars etc were 15 years ago, so (i) I probabnly have lost and forgotten the references, and (ii) there are probably more recent treatments more in line with the categorical viewpoint.

(I wonder if there is not more recent stuff under possibly yet another name, within the Applied Category Theory community.)