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- Discussion Type
- discussion topicquotient space
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by TobyBartels
- Last Active Feb 24th 2011

Stub for quotient space.

- Discussion Type
- discussion topicKhovanov homology
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by Urs
- Last Active Feb 24th 2011

have created an entry Khovanov homology, so far containing only some references and a little paragraph on the recent advances in identifying the corresponding TQFT. I have also posted this to the $n$Café here, hoping that others feel inspired to work on expanding this entry

- Discussion Type
- discussion topicMilnor mu-bar invariants
- Category Latest Changes
- Started by Andrew Stacey
- Comments 4
- Last comment by Tim_Porter
- Last Active Feb 22nd 2011

I wanted to understand Milnor’s paper on Link Groups, so I basically rewrote the main bits in to Milnor mu-bar invariants. (I don’t understand the difference between $\mu$-invariants and $\bar{\mu}$-invariants, but I was only working on the original paper so presumably haven’t gotten that far yet.)

I even put a TOC in so Urs will be happy!

- Discussion Type
- discussion topicLisbon meeting notes
- Category Latest Changes
- Started by Tim_Porter
- Comments 10
- Last comment by Tim_Porter
- Last Active Feb 22nd 2011

I have just added a link to the notes that I prepared for the Lisbon meeting on my personal page. I would love to have some feedback, and in particular suggestions for incorporating some more of this in the nLab. The new material also forms part of the extended version of the Menagerie (which is now topping 800 pages.)

- Discussion Type
- discussion topicString 2-group
- Category Latest Changes
- Started by Guest
- Comments 7
- Last comment by Urs
- Last Active Feb 21st 2011

- At string 2-group it is claimed that the sequence of classifying spaces ending --> BSO(n) --> BO(n) is the Whitehead tower of O(n). Also mentioned is the version for smooth infinity groupoids (so I assume it is Urs who put that there). It is certainly not true that the sequence of classifying spaces so stated is the Whitehead tower for O(n), but the details for groups considered as one-object infinity groupoids are open to interpretation, so I haven't changed anything. Just a heads up.

-David Roberts

- Discussion Type
- discussion topiclocally equi-connected space
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 21st 2011

stub for locally equi-connected space

- Discussion Type
- discussion topicsimplicial topological group
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Feb 19th 2011

have started simplicial topological group

- Discussion Type
- discussion topicimpredicativity
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Feb 19th 2011

I have added a brief note about type-theoretic polymorphism to the list of impredicative axioms at predicative mathematics.

- Discussion Type
- discussion topicstructured ring spectrum
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 18th 2011

I see there is quite a bit of room for improvement of the $n$Lab material on ring spectra.

I noticed that smash product of ring spectra still pointed to a stub entry, while in parallel we have a fairly good beginning of a genuine entry at symmetric monoidal smash product of spectra. So I blanked the former and made it redirect to the latter.

I also made structured ring spectrum a redirect to this for the moment.

- Discussion Type
- discussion topicconcrete category
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Feb 18th 2011

Added the statement of the Isbell-Freyd characterization of concrete categories, in the special case of finitely complete categories for which it looks more familiar, along with the proof of necessity.

- Discussion Type
- discussion topiceffects of foundations
- Category Latest Changes
- Started by DavidRoberts
- Comments 4
- Last comment by DavidRoberts
- Last Active Feb 16th 2011

At effects of foundations on “real” mathematics I’ve put in the example of Fermat’s last theorem as being potentially derivable from PA, and pointed to two articles by McLarty on this topic.

(Edit: the naive wikilink to the given page breaks, due to the ” ” pair)

- Discussion Type
- discussion topicQuiver: discussion; Directed graph: discussion
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Feb 15th 2011

I’ve removed old discussion from quiver and directed graph. They can be found at revision #20 and revision #24, respectively.

- Discussion Type
- discussion topicSegal refined Lie group cohomology as intrinsic cohomology in SmoothooGrpd
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 13th 2011

You may or may not recall the observation, recorded at Lie group cohomology, that there is a natural map from the Segal-Blanc-Brylinski refinement of Lie group cohomology to the intrinsic cohomology of Lie groups when regarded as smooth infinity-groupoids.

For a while i did not know how to see whether this natural map is an equivalence, as one would hope it is. The subtlety is that the Cech-formula that Brylinski gives for refined Lie group cohomology corresponds to making a degreewise cofibrant replacement of $\mathbf{B}G$ in $Smooth \infty Grpd$ and then taking the diagonal, and there is no reason that this diagonal is itself still cofibrant (and I don’t think it is). So while Segal-Brylinski Lie group cohomology is finer and less naive than naive Lie group cohomology, it wasn’t clear (to me) that it is fine enough and reproduces the “correct” cohomology.

So one had to argue that for certain coefficients the degreewise cofibrant resolution in $[CartSp^{op}, sSet]_{proj,loc}$ is already sufficient for computing the derived hom space. It was only yesterday that I realized that this is a corollary of the general result at function algebras on infinity-stacks once we embed smooth infinity-groupoid into synthetic differential infinity-groupoids.

So I believe I have a proof now. I have written it out in synthetic differential infinity-groupoid in the section Cohomology and principal $\infty$-bundles.

- Discussion Type
- discussion topic(infinity,n)-category with duals
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Feb 13th 2011

stub for (infinity,n)-category with duals

- Discussion Type
- discussion topichigher and derived geometry
- Category Latest Changes
- Started by Urs
- Comments 24
- Last comment by zskoda
- Last Active Feb 12th 2011

in reply to Jim's question over on the blog, I was looking for a free spot on the nLab where I could write some general motivating remarks on the point of "derived geometry".

I then noticed that the entry higher geometry had been effectively empty. So I wrote there now an "Idea"-section and then another section specifically devoted to the idea of derived geometry.

(@Zoran: in similar previous cases we used to have a quarrel afterwards on to which extent Lurie's perspective incorporates or not other people's approaches. I tied to comment on that and make it clear as far as I understand it, but please feel free to add more of a different point of view.)

- Discussion Type
- discussion topiccategory of cobordisms
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 12th 2011

I hadd added a little bit of this and that to category of cobordisms earlier today in a prolonged coffee break.

This was in reaction to learning about the work by Ayala, now referenced there, whou considers categories of cobordisms equipped with

*geometric structure*given by morphisms into an $\infty$-stack $\mathcal{F}$.

- Discussion Type
- discussion topicBord(X) as free symmetric monoidal on Pi(X)
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by domenico_fiorenza
- Last Active Feb 11th 2011

A while back I had a discussion here with Domenico on how the framed cobordim $(\infty,n)$-category $Bord^{fr}_n(X)$ of cobordisns

*in*a topological space $X$ should be essentially the free symmetric monoidal $(\infty,n)$-category on the fundamental $\infty$-groupoid of $X$.This can be read as saying

Every flat $\infty$-parallel transport of fully dualizable objects has a unique $\infty$-holonomy.

(!)

Some helpful discussion with Chris Schommer-Pries tonight revealed that this is (unsurprisingly) already a special case of what Jacob Lurie proves. He proves it in more generality, which makes the statement easy to miss on casual reading. So I made it explicit now at cobordism hypothesis in the new section For cobordisms in a manifold.

- Discussion Type
- discussion topicWeil algebra
- Category Latest Changes
- Started by Urs
- Comments 30
- Last comment by Urs
- Last Active Feb 9th 2011

edited Weil algebra a bit. More to come.

- Discussion Type
- discussion topicmonoidal Dold-Kan correspondence
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Urs
- Last Active Feb 8th 2011

the invaluable Denis-Charles Cisinski provided a useful reference with a bit on cosimplicial algebras at MO (here). I added that reference to monoidal Dold-Kan correspondence and to cosimplicial algebra.

- Discussion Type
- discussion topicde Rham theorem for smooth oo-groupoids
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by zskoda
- Last Active Feb 8th 2011

At synthetic differential infinity-groupoid I have entered statement and detailed proof that flat and infinitesimally flat real coefficients are equivalent in $SynthDiff\infty Grpd$

$\mathbf{\flat}_{inf} \mathbf{B}^n \mathbb{R} \simeq \mathbf{\flat} \mathbf{B}^n \mathbb{R} \,.$The proof proceeds by presentation of $\mathbf{\flat}_{inf} \mathbf{B}^n \mathbb{R}$ by essentially (a cofibrant resolution of) Anders Kocks’ s infinitesimal singular simplicial complex. In this presentation cohomology with coefficients in this object is manifestly computed as in de Rham space/Grothendieck descent-technology for oo-stacks.

But we also have an intrinsic notion of de Rham cohomology in cohesive $\infty$-toposes, and the above implies that in degree $n \geq 2$ this coincides with the de Rham space presentation as well as the intrinsic real cohomoloy.

All in all, this proves what Simpson-Teleman called the “de Rham theorem for $\infty$-stacks” in a note that is linked in the above entry. They consider a slightly different site of which I don’t know if it is cohesive, but apart from that their model category theoretic setup is pretty much exactly that which goes into the above proof. They don’t actually give a proof in this unpublished and sketchy note and they work (or at least speak) only in homotopy categories. But it’s all “morally the same”. For some value of “morally”.

- Discussion Type
- discussion topicformal smooth manifold
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 8th 2011

have created an entry formal smooth manifold, but without much beyond references for the moment.

- Discussion Type
- discussion topiccohomology localization
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Feb 8th 2011

quick entry for cohomology localization, but have to interrupt now

- Discussion Type
- discussion topicgroupoid of spin structures
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Feb 7th 2011

A manifold has

a set of orientations;

an xyz of topological spin structures

a 3-groupoid of topological string structures;

a 7-groupoid of topological fivebrane stuctures, etc.

and for some reason it is common in the literature (which of course is small in the last cases) to speak of these $n$-groupoids, but not so common to speak of the xyz here:

- A manifold has a
*groupoid*of spin structures.

Namely the homotopy fiber of the second Stiefel-Whitney class

$Spin(X) \to Top(X,B SO) \stackrel{(w_2)_*}{\to} Top(X, B^2 \mathbb{Z}_2) \,.$I have added one reference that explicitly discusses the groupoid of spin structures to spin structure.

Do you have further references?

- Discussion Type
- discussion topic2-out-of-3
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 7th 2011

I have split off an entry 2-out-of-3 property

- Discussion Type
- discussion topicdifferential fivebrane structures
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Feb 6th 2011

created stub for differential fivebrane structure

sounds easy, but due to lots of software trouble that took me a good bit of the afternoon! :-(

- Discussion Type
- discussion topicchain homopy
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 6th 2011

wrote something at chain homotopy

- Discussion Type
- discussion topicline Lie n-algebra
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 4th 2011

I had created line Lie n-algebra, just for the sake of completeness and so that I know where to link to when I mention it

- Discussion Type
- discussion topicquantum sheaf cohomology
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 4th 2011

stub for quantum sheaf cohomology

- Discussion Type
- discussion topicdifferential characteristic class
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 4th 2011

I have created an entry differential characteristic class.

I felt need for this as the traditional term secondary characteristic class first of all has (as discussed there) quite a bit of variance in convention of meaning in the established literature, and secondly it is unfortunately undescriptive (which is probably the reason for the first problem, I guess!).

Moreover, I felt the need for a place to discuss the concept “differential characteristic class” in the fully general abstract way in the spirit of our entry on cohomology, whereas “secondary characteristic class” has a certain association with concrete models. Some people use it almost synonymously with “Cheeger-Simons differential character”.

Anyway, so I created a new entry. So far it contains just the “unrefined” definition. I’ll try to expand on it later,

- Discussion Type
- discussion topicproblem at "essential image"
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Feb 4th 2011

I notice that the entry essential image is in a bad state:

it starts out making two statements, the first of which is then doubted by Mike in a query box, the second doubted by Zoran in a query box.

If there is really no agreement on what should go there, we should maybe better clear the entry, and discuss the matter here until we have a minimum of consensus.

But I guess the problems can easily be dealt with and somebody should try to polish this entry right away.

- Discussion Type
- discussion topicProfunctors (and anafunctors)
- Category Latest Changes
- Started by DavidRoberts
- Comments 7
- Last comment by Urs
- Last Active Feb 2nd 2011

I have taken this opportunity to update the references section at profunctor, based on recent emails from Marta Bunge and Jean Benabou.

I have added a little detail to the comment at anafunctor that Kelly considered anafunctors without naming them, namely the paper and the year, and also a small concession to Jean Benabou who wanted it widely known that he recently discovered the equivalence between anafunctors and representable profunctors viz, naming him explicitly at the appropriate point of the discussion.

(I do not want to drag the recent discussion held on and off the categories mailing list here - I just wanted to make the changes public)

- Discussion Type
- discussion topicconnection on a smooth principal infinity-bundle
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 2nd 2011

I have renamed the entry formerly called (and still redirecting) “connection on a principal infinity-bundle” into connection on a smooth principal infinity-bundle.

I will now start with bringing that entry into shape.

In the same vein I have renamed the entry formerly titled (and still redirecting) “infinity-Chern-Weil theory” into Chern-Weil theory in Smooth∞Grpd.

This way things are set up well for when the legions of students arrive who will do all the analogous discussion in other cohesive $(\infty,1)$-toposes such as $Algebraic \infty Grpd$, $ComplexAnalytic \infty Grpd$ as well as the derived version of all of these. ;-)

- Discussion Type
- discussion topicfull sub-2-category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 2nd 2011

have created full sub-2-category

also reworked full subcategory a little

- Discussion Type
- discussion topicHodge theory
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by Urs
- Last Active Feb 1st 2011

Expanded Hodge theory

- Discussion Type
- discussion topicvolume form
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 31st 2011

quick stub for volume form, as I need the link somewhere for completeness

- Discussion Type
- discussion topicsemilattice of commutative subalgebras
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jan 31st 2011

- Discussion Type
- discussion topicFinite regular cardinals
- Category Latest Changes
- Started by TobyBartels
- Comments 23
- Last comment by TobyBartels
- Last Active Jan 31st 2011

I’ve decided that these shouldn’t exist (making me agree with the standard terminology) and explained why at regular cardinal.

- Discussion Type
- discussion topiccanonical presentation
- Category Latest Changes
- Started by Yaron
- Comments 1
- Last comment by Yaron
- Last Active Jan 29th 2011

Added canonical presentation.

- Discussion Type
- discussion topic(1,1)Cat and (2,1)Cat inside (oo,1)Cat
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Mike Shulman
- Last Active Jan 28th 2011

Do we have a discussion anywhere that 2-limits in the (2,1)-category of categories as defined in the 2-category-literature do coincide with the coresponding limits computed inside the $(\infty,1)$-category of $(\infty,1)$-categories?

I thought we had, but maybe we don’t. If not, I’ll try to add some discussion.

- Discussion Type
- discussion topic(2,1)-algebraic theory of E-infinity-algebras
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jan 28th 2011

I split off (2,1)-algebraic theory of E-infinity algebras, but it’s still the same stubby context as before.

(I will probably/hopefully fill in more details in two weeks, as preparation for one of the sessions of our derived geometry semninar)

- Discussion Type
- discussion topicsesquicategory
- Category Latest Changes
- Started by FinnLawler
- Comments 22
- Last comment by FinnLawler
- Last Active Jan 28th 2011

New page at sesquicategory.

- Discussion Type
- discussion topictopos theory
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jan 27th 2011

we are lacking content in the entry topos theory.

I added a one-line Idea and then expanded the list of references.

- Discussion Type
- discussion topicseparated (2,1)-presheaf
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Jan 26th 2011

have created an entry separated (2,1)-presheaf

- Discussion Type
- discussion topic2-monads
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Jan 24th 2011

Added to 2-monad a remark about Power’s result that any monad on the underlying category of a strict 2-category with powers or copowers has at most one enrichment to a strict 2-monad.

- Discussion Type
- discussion topicnatural model structure on groupoids
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jan 23rd 2011

have created an entry natural model structure on groupoids

- Discussion Type
- discussion topicmodel structure for (2,1)-sheaves
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 23rd 2011

have created an entry model structure for (2,1)-sheaves

- Discussion Type
- discussion topicTopMfd
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 20th 2011

Have created an entry TopMfd

(this is supposed to be in the tradition that with the entry topological manifold that discusses the properties of the objects we also have an entry that discusses the properties of the category that these objects form).

- Discussion Type
- discussion topicessentially small site
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jan 20th 2011

have created essentially small site

- Discussion Type
- discussion topicmodel structure on simplicial sheaves
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jan 20th 2011

expanded and polished the entry model structure on simplicial sheaves (to be distinguished from the one of simplicial pre-sheaves!)

Made explicit the little corollary that for $D \to C$ a dense sub-site, the corresponding hypercompleted $\infty$-sheaf $\infty$-toposes are equivalent.

- Discussion Type
- discussion topicLocally contractible locales
- Category Latest Changes
- Started by TobyBartels
- Comments 13
- Last comment by Urs
- Last Active Jan 19th 2011

I added a definition to locally contractible space, but is it correct?

- Discussion Type
- discussion topiccoordinate system
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by TobyBartels
- Last Active Jan 19th 2011

created stub for coordinate system (redirecting also coordinate chart and chart)

- Discussion Type
- discussion topicsheafification
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by Urs
- Last Active Jan 19th 2011

there is some confusion on this MO thread about sheafification, with the $n$Lab entry sheafification somehow involved. I had a look at the entry and find that it can do with lots of polishing, but that the statement discussed over there is clearly right. (the misleading answer on MO that seems to claim a problem on the nLab page gets twice as many votes as the good answer by Clark Barwick, which confirms the statement) I have tried to edit it a bit to make things clearer, but don’t have the leisure for that now.

Given the recent success with the polishing of the entry on geometric realization, maybe I should announce that sheafification is going to be submitted for $n$Journal peer-review soon, so that everybody here will jump on it to brush it up ;-)

- Discussion Type
- discussion topicpro-set
- Category Latest Changes
- Started by Mike Shulman
- Comments 5
- Last comment by Mike Shulman
- Last Active Jan 19th 2011

Created pro-set with an adjunction and a counterexample.

- Discussion Type
- discussion topicoo-Chern-Weil theory
- Category Latest Changes
- Started by Urs
- Comments 199
- Last comment by Urs
- Last Active Jan 19th 2011

Behind the scenes Domenico Fiorenza is having a long discussion with me and Jim Stasheff on the matters that are being discussed at differential cohomology in an (oo,1)-topos – examples. It seems we want to work on this together. Accordingly, I have now moved at least parts of this to the main nLab in the new entry

I added a remark right at the beginning that is supposed to indicate the nature of this material.

- Discussion Type
- discussion topicSchlessinger's Criterion
- Category Latest Changes
- Started by hilbertthm90
- Comments 6
- Last comment by hilbertthm90
- Last Active Jan 19th 2011

Hello. I’ve taken up a new cause. I made an article about schlessinger’s criterion. There seems to be very little about the higher category perspective on deformation theory. This is what I’m really interested in as a grad student, so I thought I’d try to fill in a few holes.

- Discussion Type
- discussion topicsheaf
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 19th 2011

I rewrote a good bit of the entry sheaf, trying to polish and strengthen the exposition.

The rewritten material is what now constituttes the section “Definition”. This subsumes essentially everything that was there before, except for some scattered remarks which I removed and instad provided hyperlinks for, since they have meanwhile better discussions in other entries.

I left the discussion of sheaves and the general notion of localization untouched (it is now in the section “Sheaves” and localization”). This would now need to be harmonized notationally a bit better. Maybe later.

- Discussion Type
- discussion topicTesting guest posting
- Category Latest Changes
- Started by Guest
- Comments 4
- Last comment by TobyBartels
- Last Active Jan 18th 2011

- Had a problem with guest posting, just testing ... nothing to see here ...

(Andrew Stacey)

- Discussion Type
- discussion topicGerbe (as a stack)
- Category Latest Changes
- Started by hilbertthm90
- Comments 17
- Last comment by Urs
- Last Active Jan 17th 2011

I’ve added a section called $\mathcal{A}-gerbes$ at gerbe (as a stack) in an attempt to add something about the differential geometry question that was raised. I’m just a lowly grad student so be gentle if I’ve accidentally written something crazy.

- Discussion Type
- discussion topicn-connected spaces
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by Urs
- Last Active Jan 16th 2011

New: n-connected space.

- Discussion Type
- discussion topicAxioms of ZFC
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Jan 16th 2011

Every axiom listed at ZFC now has its own article, except for axiom of separation. These are:

- axiom of extensionality (already existed);
- axiom of the null set (now redirects to empty set, where additional material has just been added);
- axiom of pairing (created recently);
- axiom of union (created recently);
- axiom of separation (needs to be written);
- axiom of replacement (already existed);
- axiom of power sets (now redirects to power set, where a few remarks have just been added);
- axiom of infinity (already existed);
- axiom of choice (already existed);
- axiom of foundation (already existed).