added pointer to:

- John Michael Boardman,
*Algebraic objects in categories*, Chapter 7 of:*Stable Operations in Generalized Cohomology*(pdf) in: Ioan Mackenzie James (ed.)*Handbook of Algebraic Topology*Oxford 1995 (doi:10.1016/B978-0-444-81779-2.X5000-7)

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

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?

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

]]>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)

]]>

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.

]]>since we need an empty set to define a group

I don’t get this, can you say more?

]]>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:

- we choose ahead of time a cartesian monoidal structure on the category $\mathcal{C}$, 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 $\mathcal{J}$; 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.

]]>(double post)

]]>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 :

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

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

]]>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.

]]>Added the second paper by Eckmann and Hilton:

- Beno Eckmann, Peter J. Hilton,
*Group-like structures in general categories II. Equalizers, limits, lengths*. Mathematische Annalen 151:2 (1963), 150–186. doi:10.1007/bf01344176.

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.

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

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.

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’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.

]]>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.

]]>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.

]]>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.

]]>Perhaps from the perspective of this page, internal categories and enriched categories are just two

differentways of internalizing the notion of category in twodifferentdoctrines?

Twelve years on since this was added, can we say something more definite?

]]>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.

]]>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…

]]>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.

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