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.)

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

]]>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.

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.

]]>added pointer to:

- Francis Borceux, Section 7.1 of:
*Categories and Structures*, Vol. 2 of:*Handbook of Categorical Algebra*, Encyclopedia of Mathematics and its Applications**50**Cambridge University Press (1994) (doi:10.1017/CBO9780511525865)

added pointer to:

- Susan B. Niefield, Dorette A. Pronk,
*Internal groupoids and exponentiability*, Cahiers de topologie et géométrie différentielle catégoriques,**LX**4 (2019) (pdf)

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.

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

]]>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>
```

]]>
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.

]]>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.

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!

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

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

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

]]>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.

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?

]]>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 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).

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

]]>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 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.

]]>