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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2011

    started to add to internalization a list of links to examples. Probably we have much more.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2011
    • (edited Jul 5th 2011)

    here is the list of examples of internalization currently listed at internalization:

    Which entries to link to have I still forgotten?

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeJul 5th 2011
    • (edited Jul 5th 2011)

    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).

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2011

    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!

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeJul 5th 2011
    • (edited Jul 5th 2011)

    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 ?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2011

    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.

    • CommentRowNumber7.
    • CommentAuthorTobyBartels
    • CommentTimeJul 7th 2011

    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.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 13th 2019

    Added a couple more examples.

    diff, v54, current

    • CommentRowNumber9.
    • CommentAuthorTaro
    • CommentTimeMay 27th 2020
    In the entry , I found
    <<Like categorification or oidification, there is currently no completely general formal definition of this process, although there are one or two fairly general theorems.>>.
    I would like to learn this fairly general theorems.
    Could anyone teach me details on them or references on them ?
    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMar 21st 2021
    • (edited Mar 21st 2021)

    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 CC 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.

    diff, v60, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMar 21st 2021

    added pointer to:

    diff, v60, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMar 21st 2021

    added pointer to:

    diff, v61, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMar 21st 2021

    added pointer to:

    diff, v61, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMar 22nd 2021
    • (edited Mar 22nd 2021)

    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):

    Highlighting the role of the Yoneda lemma in internalization:

    Moreover with discussion of action objects:

    More general internalization via sketches:

    For more see at:


    diff, v65, current

    • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 22nd 2021

    That choice of the Lu and Zhu reference is curious. What in particular is in there that made you use it?

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeMar 22nd 2021

    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.

    • CommentRowNumber17.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 22nd 2021

    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.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeMar 22nd 2021

    Okay. The hh is clearly a typo for mm etc., but sure, I am removing it.

    diff, v67, current

    • CommentRowNumber19.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 22nd 2021

    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.

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2021
    • (edited Mar 23rd 2021)

    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.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2021
    • (edited Mar 23rd 2021)

    For internal categories I have now added pointer to

    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…

    diff, v68, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2021

    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.

    • CommentRowNumber23.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 23rd 2021

    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?

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2021

    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.

    • CommentRowNumber25.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 24th 2021

    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.

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2021

    The nLab.

    Then I’d like to ask you to reconsider. The idea that on nnLab 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.

    • CommentRowNumber27.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 24th 2021

    I’ve added a gloss from the point made in #24. I don’t know if there is a reading in terms of doctrines, as still currently stated.

    diff, v70, current

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2021

    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.

    diff, v71, current

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2021

    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.

    diff, v71, current

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeMar 25th 2021

    So I am (re-)moving the section “Internal vs. enriched categories” to internal categories.

    diff, v73, current

    • CommentRowNumber31.
    • CommentAuthorzskoda
    • CommentTimeMar 25th 2021
    • (edited Mar 25th 2021)

    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.

    • CommentRowNumber32.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 26th 2021

    Added the second paper by Eckmann and Hilton:

    diff, v74, current

    • CommentRowNumber33.
    • CommentAuthorUrs
    • CommentTimeMar 27th 2021

    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.

    • CommentRowNumber34.
    • CommentAuthorGE
    • CommentTimeMay 29th 2021
    • (edited May 29th 2021)

    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 :

    𝔾=(G,σ, G)\mathbb{G} = (G, \sigma, \mathcal{I}_{G})

    Which denotes the algebraic structure of a group, where :

    • There is GG which is the domain of discourse,
    • There is σ\sigma which is the group signature
    • There is G\mathcal{I}_{G} which is the interpretation function of the signature over the domain domain of discourse

    So why don’t we see :

    𝔾=(G,*,σ, G)\mathbb{G} = (G, *, \sigma, \mathcal{I}_{G})

    (Where ** is also a subset of the domain of discourse) instead?

    Thanks

    • CommentRowNumber35.
    • CommentAuthorHurkyl
    • CommentTimeMay 29th 2021
    • (edited May 29th 2021)

    (double post)

    • CommentRowNumber36.
    • CommentAuthorHurkyl
    • CommentTimeMay 29th 2021

    For the same reason we don’t include in the tuple a choice for the set we describe as G×GG \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:

    • we choose ahead of time a cartesian monoidal structure on the category 𝒞\mathcal{C}, which tells us what functors to use to define products, what natural transformations give the projections, et cetera.
    • the choice of products is part of 𝒥\mathcal{J}; a choice is part of the definition of “interpretation”

    For example, the latter is what we use if we formalize this construction by means of a finite product sketch.

    • CommentRowNumber37.
    • CommentAuthorLiamJackson
    • CommentTimeMay 31st 2021
    • (edited May 31st 2021)
    Hi, oh I didn't know that, I read a lot of useful stuff here, thanks!
    • CommentRowNumber38.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 31st 2021

    since we need an empty set to define a group

    I don’t get this, can you say more?

    • CommentRowNumber39.
    • CommentAuthorHurkyl
    • CommentTimeMay 31st 2021
    • (edited May 31st 2021)

    I understood that comment to be referring to the domain of the nullary operation e:*Ge : * \to G, thinking of the “usual” point *={}* = \{ \varnothing \} in the case of sets. I could be wrong, of course.

    • CommentRowNumber40.
    • CommentAuthorUrs
    • CommentTimeJun 18th 2021

    Added:


    The general notion of internalization is due to

    • Alexander Grothendieck, p. 340 (3 of 23) in: Technique de descente et théorèmes d’existence en géométrie algébriques. II: Le théorème d’existence en théorie formelle des modules, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195 (numdam:SB_1958-1960__5__369_0, pdf)

    with specialization to internal groups, internal actions, internal categories and internal groupoids made explicit in:

    • Alexander Grothendieck, Section 4 of: Techniques de construction et théorèmes d’existence en géométrie algébrique III: préschémas quotients, Séminaire Bourbaki: années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212, (numdam:SB_1960-1961__6__99_0, pdf)

    diff, v78, current

    • CommentRowNumber41.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 6th 2021

    The page speaks of ’deinternalization’, so I added externalization as a synonym.

    diff, v81, current

    • CommentRowNumber42.
    • CommentAuthorUrs
    • CommentTimeJul 6th 2021

    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?

    diff, v82, current

    • CommentRowNumber43.
    • CommentAuthorUrs
    • CommentTimeJul 22nd 2021

    added pointer to:

    Am adding this pointer now also to internal group, etc.

    diff, v83, current

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