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I edited the formatting of internal category a bit and added a link to internal infinity-groupoid
it looks like the first query box discussion there has been resolved. Maybe we can remove that box now?
I think that similar ideas for algebraic Galois group are used in Grothendieck's still unfullfilled ideas in his 1980s manuscripts and is in general known in Galois theory (covering spaces and higher Postnikov fibers have natural common generalizations in this context).
not sure if this is relevant for what you have in mind, but taking path oo-groupoids can be made into a map from a lined oo-topos to itself. For instance taking a topological oo-groupoid (might be just a topological space) to the topological oo-groupoid .
Same for smooth oo-groupoids. For the smooth model of String, one can start with the Lie 2-algebra , then form the smooth space (sheaf) which is such that its plots are precisely the flat -valued differential forms on . Then one can form the smooth oo-groupoid of that space. Finally, we can take from this simplicial sheaf degreewise the underlying concrete sheaf (=diffeological space) to get the smooth oo-groupoid . The claim is that that's a smooth version of .
This is discussed in some detail (though with slightly more antiquated tools than I have now) in section 5.2.3 here.
Concerning the categorical degree, I feel like remarking that it's only the homotopy groups (of an infinity-stack) that have intrinsic meaning, not the degrees of a truncated model.
Fur instance the goup looks like, well, a group, but it is equivalent to the 2-group that comes from the crossed module .
Similarly, the String 2-group is a -central extension of an ordinary group , hence a 2-group extension, but this is equivalent to a -central extension, which looks like a 3-group extension, but is really equivalent to the original 2-group extension.
@David: you wrote
In this context one needs to be au fait with Frechet manifolds, sadly an area where I am lacking.
I have a little facility with Frechet manifolds. Is there something here that I could help with?
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@David: <br/><br/><blockquote><br/>PI_1(X) can indeed be topologised, but composition is only continuous when X is locally nice - locally connected and semilocally simply connected.<br/></blockquote><br/><br/>This is nice: if I'm not wrong here these are exactly the conditions for X to have a universal cover; also I find it nice that one needs more and more local regularity as higher covers come into play. <br/><br/><blockquote><br/>An easy way to see that the universal cover of a topological group X is topological is that the universal cover = the homotopy fibre of X --> Pi_1(X) at a point x = the source fibre of Pi_1(X) at x. This is a subgroup of the group of arrows, as Pi_1(X) is a strict 2-group as shown by Brown-Higgins in the 70s.<br/></blockquote><br/><br/>That's exactly what I was trying to say. what i think is important to stress is that there is a subtle interplay here between topological and groupoidal aspects: from the topological perspective one ends up with an object which is clearly a simply connected cover and then has to show that this carries a natural group structure; from the groupoids point of view, the object one ends up with is clearly a group, but one now has to show that it is a simply conneced cover. And an incredible elegant way to prove both things at a time is to perform the two constructions independently, and to see that they produce the same object. here I think there should be some very general argument a priori telling me that the two constructions will lead to teh same object, but I'm still missing this argument.<br/><br/>@Urs:<br/><blockquote><br/>Concerning the categorical degree, I feel like remarking that it's only the homotopy groups (of an infinity-stack) that have intrinsic meaning, not the degrees of a truncated model.<br/></blockquote><br/><br/>edit: let me see if I correctly undersatnd this. if we start with the action groupoid of on what we are dealing with is not only a groupoid, but a groupal groupoid (since is a group and the -action is compatible with the group operation of ). so the 0-th truncation of this groupoid is a group. this 0-th truncation is evidently , so we should think of not as our basic object but rather as . right?
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Just a comment, probably no news to anyone, but just for the record:
to realize the equivalence as a homotopy equivalence (morphisms going back and forth, being weak inverses) one needs to find suitable ana-2-functors, inverting the evident morphism from one 2-groupoid to the other, but for just knowing that the two are equivalent it is sufficient to have that one morphism and checking that it is a weak equivalence.
This is true for topological infinity-groupoid realizations of the situation and even for Lie-infinity-groupoid realizations.
For that one may observe that the functor is k-surjective for all k on all stalks of Top or Diff: these are the germs of n-dimensional disks (as described by Dugger in the reference cited at topological infinity-groupoid.) So for the standard n-disk of radius r, we form the groups of group-valued maps , and and then check that the functor of ordinary 2-groupoids is k-surjective for all k for some r small enough. And it clearly is: on any disk, any U(1)-valued function (continuous or even smooth) may be lifted to an -valued function, and the nonuniquenss of the lifts are precisely given by -valued functions on the disk.
but on second thought, I think a plausible perspective could be
Yes, precisely: very different looking objects may all be equivalent to the same 2-group, (or the same oo-groupoid). The highest degree of nondegenerate cells is not an invariant under this. Instead, the main invariants are the intrinsically defined homotopy groups
I find the use of AUT(K) something which works, but which is chosen in quite an arbitrary way. maybe it's worth creating a Schreier theory entry to discuss it
Yes, I agree with all you say here. As far as I am concerned: I don't have the energy and time this particular project right now, though. Probably later sometime.
Schreier theory is included in group extension entry.
It is instructive the way Faro et al. pharse in their article, placing the role of AUT K via interpreting the Grothendieck construction. I will once write about it.
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<blockquote>The point of nonabelian cohomology is to _not_ use stable theory</blockquote><br/><br/>so that's why it's called "nonabelian" or "unstable"! I should have known, but actually it was this remark above that really made me clear this fact, thanks!<br/><br/>still, I'd like to think of going from to as a suspension. in other words, when dealing with a cohomology with degrees I'd like to have two functors raising and lowering the degrees. in the stable theory these are looping and delooping, but one could have a nice theory in any case one has a nice shift. <br/><br/>this is somehow reminiscent of "old way" triangulated categories stuff. there teh shift functor is a basic datum; then in the stable infinity-category approach the shift is no more basic but it is recovered from homotopy pullbacks and pushforwards to the zero object. but we could still think of "unstable cohomology with degrees" as something coming from a shift functor which is not defined in terms of homotopy pullbacks/pushforwards to the zero object.<br/><br/>clearly, in order to have a nice and quite general theory, one would presumibly need not only the shift but also some good functorial behaviour. and then one could check that to indeed provides an example of this structure.
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There are ways to define homotopy groups for topological groupoids
There is, by the way, something even more general: the notion of homotopy group (of an infinity-stack)
a classical example is . this suggests that the natural setting for cohomology in different degrees is a stable -category.
It is true that only if the coefficient object has arbitrary delooping that the above definition of makes sense for all . But there is a priori no requirement that such a definition needs to make sense. For general nonabelian coefficients degree cohomology is defined only up to some finite . This is a standard thing in the literature on nonabelian cohomology.
But you can always form loops to go don do lower degrees.
Am hereby moving an old Discussion-section from internal category to here
[begin forwarded discussion]
Previous versions of this entry led to the following discussions
+–{: .query} I think things are mutliply inconsistent in this entry. I do not want to change as I do not know what the intentional notation to start with was. If $p_1; s = p_2; t$ that mean that target is read at the left-hand side (composition as o, not as ;), while the diagrams before that suggest left to right composition. Then finally the diagram for groupoids has $s$ both for source and inverse, and there is only for right inverse, and one should also check convention, once it is decided above.-Zoran
You're right; I think that I caught all of the inconsistencies now. Incidentally, one needs only inverses on one side (as long as all such inverses exist), although it's probably best to put both in the definition. (For groupoids, one also needs only identities on that same side too! This proof generalises.) —Toby =–
+–{: .query}
Question: I’ve looked at the definition of category in $A$ for a while and still haven’t been able to absorb it. Could we walk through an explicit example, e.g. “This is exactly what $C_0$ is, this is exactly what $C_1$ is, this is exactly what $s,t,i$ are, and this is how it relates to the more familiar context”? For example, an algebra is a monoid in $Vect$. I’ll try to step through it myself, but it will probably need some correcting. - Eric
Eric, one example to ponder is: how is an internal category in Grp the “same” as a crossed module? As a partial hint, try to convince yourself that given a internal category, part of whose data is $(C_1, C_0, s, t)$, the group $C_1$ of arrows can be expressed as a semidirect product with $C_0$ acting on $\ker(s)$. The full details of this exercise may take some doing, but it might also be enjoyable; if you get stuck, you can look at Forrester-Barker. - Todd
Urs: I don’t know, but maybe Eric should first convince himself of what Todd may find too obvious: how the above definition of an internal category reproduces the ordinary one when one works internal to Set. Eric, is that clear? If not, let us know where you get stuck!
Tim: I have just had a go at 2-group and looked at the relationship between 2-groups and crossed modules in a little more detail, in the hope it will unbug the definition for those who have not yet ’groked’ it.
Eric: Oh thanks guys. I will try to understand how a small category is a category internal to Set first and then move on to category in Grp and the stuff Tim wrote. I’m sure this is all obvious, but don’t underestimate my ability to not understand the obvious :)
Eric: Ok. Duh. It is pretty obvious for Set EXCEPT for pullback. Pullbacks in Set are obvious, but what about other cases? Why is that important and what is an example where there are not pullbacks? In other words, is there an a example of something that is ALMOST a category in some other category except it doesn’t have pullbacks, so is not?
Tim: If I remember rightly the important case is when trying to work on ’smooth categories’, that is, general internal categories in a category of smooth manifolds. Unless you take care with the source and target maps, the pullback giving the space of composible pairs of arrows may not be a manifold. (I remember something like this being the case in Pradines work in the area.) The point is then that one works with internal categories with extra conditions on $s$ and $t$ to ensure the pullback is there when you need it.
Toby: Usually in the theory of Lie groupoids, they require $s$ and $t$ to be submersions, which guarantees that the pullback of any map along them exists. =–
[end forwarded discussion]
Reference
at internal category has a bogus pdf link which redirects to internal category! Is somebody having a correct pdf link ?
Urs 25, when you delete a discussion and archive it, then please leave the backlink to the archived version from the old place, otherwise is essentially lost. I done it this time (into references).
I have created a new entry locally internal category and listed it under related notions at internal category.
@Zoran I was not sure what you meant by the first sentence of locally internal category. I have amended it to mean what I think you meant but please check.
I added a bit more.
Very nice, Mike !
On the other hand, why saying “in the sense of the appendix of (Johnstone)” at stack semantics, rather than more correctly attributing phrase “in the sense of Penon 1974”. Is there a slight difference ? (I did not look at Penon’s article yet; strangely I can find CR Acad Paris at the partly free gallica repository till 1973 and then again from 1979, but not for 1974-1978)
I resolved the bogus pdf link which was asked about it 26 and will correct it at internal category in a minute:
I didn’t write that phrase at stack semantics. Feel free to correct it if you know the correct reference.
“in the sense of the appendix of (Johnstone)” was written by Ingo Blechschmidt.
It is not a mistake, just more people together know more about the history :) and it is good when we agree (as often mathematicians do not agree on history).
I have brushed-up the entry internal category a little. Added the remark on cartesian closure discussed in another thread to a Properties-section
I just noticed how hard it is (or was) to find the crucial discussion at 2-topos – In terms of internal categories if all one does is search the nLab for “internal category”.
So I have now added a pointer to that at internal category by way of a brief paragraph a Properties – In a topos.
Should be expanded.
added one more item to the list of examples:
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There are more articles by Ehresmann that would deserve to be cited here, but I am out of patience now and will leave it at that.
Added the real original reference:
(Jean Pradines said in his talk at the New Spaces conference that the concept of Lie groupoid was already used, but not formally defined, by Ehresmann in earlier 1958 work on finite pseudogroups a la Cartan, but I haven’t yet found it. And possibly it is in Gattungen von lokalen Strukturen, but I wouldn’t be able to tell)
I found Ehresmann’s 1958 paper in Compte Rendus on pseudogroups, and he uses the Lie groupoid of germs and certain Lie subgroupoids, but is really still just doing rather formal differential geometry, rather than thinking of Lie groupoids per se.
re #42 Thanks! That’s the way to go.
I am producing a standalone pdf of that article. In process…
I can have a look at “Gattungen…” but I haven’t gotten hold of that one yet, as it doesn’t seem to be in the oeuvres pdf
So I uploaded a pdf copy of Ehresmann’s Catégories topologiques et categories différentiables.
Looking at it, I find it may be a little anachronistic to attribute the notion of internalization to this article: It defines topological and Lie categories by explicit description of the conditions, not by appeal to a general concept of internalization. The would-be ambient categories of $TopologicalSpaces$ and of $SmoothManifolds$ are not even being mentioned, are they? (I have only skimmed over it, I admit. If the actual notion of internalization is in there, let’s extract the page number and point to it.)
It seems to be the actual notion of internal categories only appears later in
where it is probably Definition 3 in part II on p. 36.
(Not sure, though. I find reading this article is dizzying. Not the French, but the mathematical notation and terminology. So I don’t claim to have penetrated what it’s saying. If anyone has, let’s add specific pointers.)
Oh, I don’t claim the 1959 paper has any abstract notion of internalisation, but that’s not what the text was saying when I made the edit. But this paper gave what was they key motivation for the definitions of internal category and groupoid, rather than just a purely formal idea, and in categories without all pullbacks, to boot (a generality many other authors don’t consider).
Ehresmann is hard to read for many reasons, not least the non-Bourbaki-ized mathematical style (probably more influenced by Cartan) and the isolation from other category theorists leading to non-standard terminology.
I adjusted the text after I suspected that your “real original reference” (#42) is not really about internal categories. It’s instead probably the original reference on topological categories and smooth categories , and I did copy it to there.
So possibly the first mentioning of internalization of categories is the reference given in #41. If you have the energy, you might sanity check whether we could quote Def. 3 on p. 63 there as the original definition (as suggested in #47).
I was looking at it, but I’ll have to come back. It seems that either Ehresmann is defining something only equivalent to what we would call an internal category, like maybe an externalisation, or else he’s only working in a concrete setting, since the definition of internal groupoid, for instance, just says that some data is a (plain!) groupoid, on top of having a structured category.
Okay, thanks.
It seems then my impression (here) is correct that the credit for understanding and articulating the concept of internalization goes to Eckmann-Hilton 1961.
Oh the mysterious ways of attribution. This could well have been “Eckmann-Hilton theory”, but instead their names are hardly mentioned at all in this context, outside of their one eponymous example (of many they gave), and all authors use their term “group object” as if it was always called that way, since the beginning of time.
This could well have been “Eckmann-Hilton theory”
to be honest, that’s a terrible way to name something. Giving people credit is not the same as using their names as the name of a thing. Imagine if we had “Eilenberg–Mac Lane theory” instead of “category theory”…
Maybe it’s a citation culture difference. Bénabou once tried publicly shaming me because my anafunctor paper had so many citations, whereas his bicategory paper had so few. Not to mention the whole issue of there not being category theory journals for so long that results were transmitted at meetings and conferences, and sometimes published in Springer’s LNM.
Hi David, don’t let yourself be distracted that easily. I was hoping you would come back with a closer reading of Ehresmann, as promised in #50. We are still looking for where in his writings we can first recognize the notion of internalization. I count on you.
Yeah, sorry :-). I’m going to have to ask on the categories mailing list, I think, or else take some more time to digest.
Okay. If you do ask on the mailing list, please make clear that we are looking for the notion of internalization as such, not just for some construction that we can recognize, after the fact, as equivalent to an internalized structure.
moved a section “Internal vs. enriched categories” from internalization to here
The import from internalization needed some adjustment for its new home.
All right, thanks. And maybe we should have the corresponding comment and cross-link also at enriched category. (Can’t edit myself right now.)
We already have this section at enriched category. There are a lot of ideas there. Maybe even we could have a page on enrichment/internalization comparisons, and then link from each side. But that’s beyond my skills to synthesize.
Bénabou once tried publicly shaming me because my anafunctor paper had so many citations, whereas his bicategory paper had so few.
This is weird, since
16 citations: Roberts, David Michael Internal categories, anafunctors and localisations. Theory Appl. Categ. 26 (2012), No.29, 788–829. (Reviewer: Enrico Vitale) 18D05 (18F10 22A22)
336 Citations: Bénabou, Jean Introduction to bicategories. 1967 Reports of the Midwest Category Seminar pp. 1–77 Springer, Berlin (Reviewer: J. R. Isbell) 18.10
Clearly he meant items listed in the article’s bibliography, not citations to/of the article.
I guess it refers to the idea that the less you cite the more of a bigshot you must be.
@Dmitri,
Urs has it. Apparently it meant my ideas must have been less original (though that’s a shallow reading of bibliographic entries). But this is, as Urs pointed out, off the thread :-)
It is somewhat on-topic in that it illustrates which social mechanisms are behind the desaster we have been struggling with above, of a whole field citing so unprofessionally as to forget the origin even of its most basic notions (here: internalization in general, which we discovered is due to Eckmann-Hilton, who are never credited for it, their peers possibly fearing to compromise their own originality if they did; and internal categories in particular, where tradition decided to attribute it to the article Catégories structurées which, however, on actual inspection, is at best a big mess).
I’m still interested in finding, and recording here, which reference first articulated the notion of internal categories, clearly.
Proper citation and attribution is part of professional academia. Not citing your precursors is not a sign of originality but of fraudulency.
I’ve added in a link to the parallel treatment of the internalization/enrichment comparison at enriched category.
At long last, we have found the origin of the definition of internal categories (thanks to Dmitri here!):
I have added that reference now, together with the precursor
where the general definition of internalization is given.
So then, to the reference of Ehresmann’s “Catégories structurées” – which most authors cite as the origin of internal cateories – I have added the comment that
the definition is not actually contained in there, certainly not in its simple and widely understood form due to Grothendieck61.
And, interesting to note, that second, precursor, reference also has the Yoneda embedding, and the fact it preserves finite products!
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Absolutely!
[deleted]
Thanks. But checking out the web version on my system, it appears broken: Most of the pages I see there appear empty except for a section headline, and those that are not empty break off in the middle of a sentence after a few lines. (using Firefox 89.0.1 (64-bit) on Windows 10)
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Oh, I see. Great.
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I think the s and t in the 2nd-5th pullback diagrams in the Internal categories section are inconsistent with those in the first pullback diagram and the laws specifying the source and target of composite morphisms; the earlier diagrams have $p_1$ being the first of the morphisms and $p_2$ being the second in the composition (so $s\circ c=s\circ p_1$ for example), whereas the 2nd-5th pullback diagrams seem to have them the other way round. But I may be wrong, so I am hesitant about making this edit.
On further thought, I’m pretty sure I’m right so I’ll make the changes. Feel free to undo them if I’m wrong!
The definition of internal category is due to Grothendieck. However, what’s the earliest reference for internal functors, natural transformations, and profunctors?
For what it’s worth, Johnstone’s “Topos theory” (1977) considers internal functors in section 2.1 and internal profunctors in section 2.4. That seems to be the earliest mentioning of these concepts among the references already collected in the entry (here), though I have no idea if there is an earlier one.
Internal profunctors and 2-cells between them are already present in §5.1 of Bénabou’s Les distributeurs (1973), which must be the earliest definition for those. I don’t see a definition of internal functor or natural transformation there, though I would imagine it to be known earlier.
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for the definition of internal profunctors (to readers who already know all about internal categories?).
We have a page for internal profunctor, but it would seem reasonable to me to collapse that page into the internal category entry. There doesn’t seem to be an advantage to having two different pages. Would anyone object if I made this change?
Well, I think there may be many places where someone would want to link directly to internal profunctor. It’s a different concept, so why not have a different page for it? The page internal category is already quite long.
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