Fix typo: exponentiable => exponential
]]>I have fixed the typesetting of the formula and touched the wording (here).
]]>Elaborate on the formulation of exponential objects that doesn’t require cartesian products.
]]>Mention that exponentiable objects can be defined in categories without finite products.
]]>Added a sentence to the idea section, to remove any temporary impression that a cartesian closed category is prerequisite to speaking of exponential objects.
]]>Added to the description that the definition of exponential object is equivalent to that of a representation of .
]]>have hyperlinked the term relational structure
]]>Added a reference to
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 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:
added pointer to:
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.
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