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started to add to internalization a list of links to examples. Probably we have much more.
here is the list of examples of internalization currently listed at internalization:
Which entries to link to have I still forgotten?
Why the default first choice of terminology is not chosen uniformly ? I mean if internal groupoid is the default, then internal group would be also default. By the way, I was surprised that the latter was not even a redirect (I added it).
Why the default first choice of terminology is not chosen uniformly ?
Because nobody took care to make it uniform when these entries were created over the last years!
I made some changes to the section on presheaf approach in group object aka internal group, including formulating a proposition. The way it is organized now is pretty confusing, that is why I prefer to have a clear proposition statement and then the rest is kind of proof. But it is a bit strange, e.g. there is a mention of supermanifolds in the proof! Could you look at my changes and see if this can be further improved ? Also did I correctly use the word “creates” in the proposition ?
Thanks, Zoran, for looking into this.
I have removed the weird “of supermanifolds”. Clearly this is a leftover of some copy-and-pasting from the discussion of supergroups.
Mostly, it seems that ‘X object’ is for things that are normally just structured sets, while ‘internal X’ is for things that are a bit fancier (higher dimensional in the categorial sense). The distinction on the Lab probably comes from (at least historical, and as far as I know contemporary) practice in the literature.
Coming to this old entry to check for citable references, I see that – besides the complete lack of these – there is lots of room for improvement.
Am on my phone and can’t edit much right now, but I did touch the very first sentence, trying to remove various distractions and circularities.
Now it reads as follows:
Any mathematical structure whose traditional Bourbaki-style definition is formulated within set theory can be formulated internally to any category $C$ that admits all those types of operations (usually: universal constructions) on its objects that the traditional definition applies to sets, hence to the objects of the base category of Sets.
added pointer to:
added pointer to:
added pointer to:
Okay, now I have the following list of References here. Maybe it can be expanded much further (?) but I’ll leave it at that now, have spent too much time with this already:
On internalization with focus on H-spaces, internal monoids and internal groups (and proving the Eckmann-Hilton argument):
Beno Eckmann, Peter Hilton, Structure maps in group theory, Fundamenta Mathematicae 50 (1961), 207-221 (doi:10.4064/fm-50-2-207-221)
Beno Eckmann, Peter Hilton, Group-like structures in general categories I multiplications and comultiplications, Math. Ann. 145, 227–255 (1962) (doi:10.1007/BF01451367)
Beno Eckmann, Peter Hilton, Group-like structures in general categories III primitive categories, Math. Ann. 150 165–187 (1963) (doi:10.1007/BF01470843)
Highlighting the role of the Yoneda lemma in internalization:
Moreover with discussion of action objects:
Saunders Mac Lane, Ieke Moerdijk, Section V.6 of: Sheaves in Geometry and Logic, Springer 1992 (doi:10.1007/978-1-4612-0927-0)
Francis Borceux, George Janelidze, Gregory Maxwell Kelly, around p. 8 of: Internal object actions, Commentationes Mathematicae Universitatis Carolinae (2005) Volume: 46, Issue: 2, page 235-255 (dml:249553)
More general internalization via sketches:
Andrée Bastiani, Charles Ehresmann, Categories of sketched structures, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 13 (1972) no. 2, pp. 104-214 (numdam:CTGDC_1972__13_2_104_0)
Michael Barr, Charles Wells, Section 4 of: Toposes, Triples, and Theories, Originally published by: Springer-Verlag, New York, 1985, republished in: Reprints in Theory and Applications of Categories, No. 12 (2005) pp. 1-287
For more see at:
That choice of the Lu and Zhu reference is curious. What in particular is in there that made you use it?
A clear up-front statement of internal group actions.
It is hard to find a reference on this that one could show with a straight face to educated but non-category-expert readers. This one eventually showed up in searching.
I haven’t read it beyond the definition (also the symbols don’t render properly in my viewer). If you have and find it’s wrong or silly, please let me know.
Re #16: I’ve looked at this paper by Lu and Zhu, and it simply does not make sense.
Already the first commutative diagram uses symbols j and g that are not defined.
The diagonal morphism goes G⨯G→G, according to them.
The morphisms in the commutative diagram do not make any sense at all. Already the top map should be m⨯id, not e⨯h, whereas e⨯h would be a totally different morphism, of the form G⨯G→G⨯G.
None of this makes any sense at all.
I seriously don’t think we need a citation for the definition of an internal group action, just give it outright, and point to a standard reference (Borceux?) if you feel the itch.
Who is your “we”?
If you ever write an article to a non-category theoretic audience then you may feel the itch to cite basic facts of category theory, and then it may become apparent how most of the literature is speaking just to an inner circle.
We didn’t haven any reference whatsoever here for a decade. Now it took me half a day to track down those I did collect above. None of which one can really hand to an educated mathematical physicist and say: Just see there.
But I’ll check again if Borceux gives a decent account, thanks for the suggestion.
For internal categories I have now added pointer to
Francis Borceux, Chapter 8 in Vol 1 Basic Category Theory of: Handbook of Categorical Algebra, Cambridge University Press (1994) (doi:10.1017/CBO9780511525858)
Peter Johnstone, Chapter B2 in: Volume 1 of Sketches of an Elephant – A Topos Theory Compendium, Oxford University Press (2002)(ISBN:9780198534259)
Neither of these seem to mention actions nor the special case of internal groups. (MacLane-Moerdijk talk about internal category actions, but that item is already in the list.)
I’ll check Borceux’s vol 2 now…
It doesn’t seem to be in Borceux’s Vol. 2 and 3 either.
I’ll leave it at that now. But if anyone has their “canonical reference” to add to the list, please do!
One insight I take from this goose hunt is that the concept of internalization really seems to be due to Eckmann-Hilton 1961, 62. They state it first (at least two years before Ehresmann), generally, clearly and straightforwardly (in contrast to Ehresmann, I find); they understand and amplify the dependence on the available ambient co-limits (their whole volume II is devoted only to this aspect) and they bring out a list of key examples of practical relevance, with their eponymous theorem only being one of these.
They also introduce the terminology “X-object”, notably “group object”. It’s their term.
At the same time, no other reference I have seen ever cites them for any of this.
Perhaps from the perspective of this page, internal categories and enriched categories are just two different ways of internalizing the notion of category in two different doctrines?
Twelve years on since this was added, can we say something more definite?
I’d say this depends instead on which of two possible definitions of “category” one internalizes: at category the two possibilities are called 1) “with one collection of morphisms” and 2) “with a family of collections of morphism”.
The first of these definitions generalizes to internal categories, the second to enriched categories.
Who is your “we”?
The nLab. I was not thinking of potential readers of your papers. I too have a fondness for tracking down original sources for things (like getting John Baez to get me a copy of an article from a seminar from the 1950s by Mac Lane so I could check if it predated something May published). But in this instance, I think the original choice was ill-considered, despite its surface appearance. I’m glad you found the Eckmann–Hilton paper on group objects, that’s rather nice to see.
It’s entirely possible that no category theorist sat down and considered the special case of internal group actions (and not groupoid actions), in enough detail to dedicate space to them over a general picture.
My own ahistorical inclination is that group actions are nice examples of multisorted Lawvere theories, and so one might wonder if they were ever considered from that point of view. But I don’t have a reference, and this probably never came up. Maybe some early operad people wrote it down, but they were so keen on symmetric operads, so maybe not there either.
The nLab.
Then I’d like to ask you to reconsider. The idea that on $n$Lab entries, of all places, we shouldn’t give comprehensive references just because some regular here might consider the topic too trivial to bother looking these up is absolutely ill-conceived.
The entry “internalization” had not a single reference for over a decade. That was a sorry and embarrassing state of affairs. For “us, if you wish. If we want to use “our” time efficiently, we stop debating this and instead continue – or else start, as the case may be – to contribute content here.
I have expanded-out the Idea section to something that might now qualify as providing the idea.
I removed the following sentence from the paragraph on doctrines, since it seems to say nothing that the previous sentence did not already say:
The question of what exactly a “doctrine” is is a tricky one, but for purposes of this page, we take a “doctrine” to mean a certain type of structure (or property-like structure) with which a category can be equipped.
I am not sure that the following paragraph on “dedoctrinization” is useful, but I didn’t touch it.
I have renamed the section “Internalization versus enrichment” to “Internal versus enriched categories” (here).
I find this is at the wrong spot here: This section should be at internal category, instead.
So I am (re-)moving the section “Internal vs. enriched categories” to internal categories.
If this remark is of some help, Grothendieck and his school, and Pierre Gabriel who collaborated with them, have taken the point of view that it is interesting always if some presheaves are representable (or pro-representable), and very often this was tested for presheaves of groups. Representability is especially important in FGA constructions where representing schemes are defined and constructed for a number of functors like for coherent sheaves (Quot scheme), construction of Hilbert scheme and so on. These were stemming from hard classical problems and people could not even formulate them without thorough understanding of considering representability questions very seriously. Representable presheaf of groups is precisely the internal group, and all of the equivalent descriptions were certainly used from the first usages of gaming with presheaves of groups in FGA (1957-1961) and SGA phase (1961 till mid 1960s) of the development of the school. Mumford also took up going back and forth from early 1960s, but still dominantly taking the presheaf point of view.
Added the second paper by Eckmann and Hilton:
I didn’t add that one here since it doesn’t seem to talk about internalization. Instead in this volume II they seemedd to have felt the need, at that time, to go on a detour and set up a whole lot of background theory of (co-)limits. Which is rather interesting, historically, as it signifies that this may be the original source of many of these baic results.
Therefore I was thinking we should instead cite this article at various of our entries in various kinds of (co-)limits. But I am out of energy, for the moment, to spend more time with this.
Hi,
I have a question outside of the main scope of this page : since we need an empty set to define a group, why do we don’t usually write it in the tuple which encode the formal theory? I mean usually we see :
$\mathbb{G} = (G, \sigma, \mathcal{I}_{G})$Which denotes the algebraic structure of a group, where :
So why don’t we see :
$\mathbb{G} = (G, *, \sigma, \mathcal{I}_{G})$(Where $*$ is also a subset of the domain of discourse) instead?
Thanks
(double post)
For the same reason we don’t include in the tuple a choice for the set we describe as $G \times G$ along with its projections, or whatever higher cartesian products we need to describe a variety of algebra. In fact, these should be literally the same case, since * is the empty product.
Depending on the precise details of how we’re setting things up, we could have:
For example, the latter is what we use if we formalize this construction by means of a finite product sketch.
since we need an empty set to define a group
I don’t get this, can you say more?
I understood that comment to be referring to the domain of the nullary operation $e : * \to G$, thinking of the “usual” point $* = \{ \varnothing \}$ in the case of sets. I could be wrong, of course.
Added:
The general notion of internalization is due to
with specialization to internal groups, internal actions, internal categories and internal groupoids made explicit in:
The page speaks of ’deinternalization’, so I added externalization as a synonym.
Thanks for highlighting. Looking at it, I see that I don’t know what that paragraph is saying. Is it meant to allude to a theorem or just meant to declare a way of speaking?
added pointer to:
Am adding this pointer now also to internal group, etc.
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