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- 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 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).

- Discussion Type
- discussion topic[[range]]
- Category Latest Changes
- Started by TobyBartels
- Comments 3
- Last comment by TobyBartels
- Last Active Jan 16th 2011

We don’t use the term “range” much here, and I explained why.

- Discussion Type
- discussion topicEuclidean topology
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by TobyBartels
- Last Active Jan 15th 2011

have created Euclidean topology, for completeness

- Discussion Type
- discussion topicfiltered (infinity,1)-category
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Jan 14th 2011

added to filtered (infinity,1)-category the statement that these are precisely those shapes of diagrams such that $\infty$-colimits over them commute with finite $\infty$-limits.

- Discussion Type
- discussion topicalgebrad
- Category Latest Changes
- Started by zskoda
- Comments 11
- Last comment by zskoda
- Last Active Jan 14th 2011

algebrad and additions at Nikolai Durov. The movie starts slow and boring but gets very interesting after a while when the topic develops.

- Discussion Type
- discussion topiccohesive infinity-toposes - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 13th 2011

I have created a table of contents cohesive infinity-toposes - contents and have added it as a “floating TOC” to some relevant entries

- Discussion Type
- discussion topicFreedom vs fascism
- Category Latest Changes
- Started by TobyBartels
- Comments 27
- Last comment by David_Corfield
- Last Active Jan 13th 2011

Added a note about cofree functors to free functor.

- Discussion Type
- discussion topicposite
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by TobyBartels
- Last Active Jan 12th 2011

I have edited posite

In particular I tried to work the query box into the text. Mike and David R. please check if you agree with the result.

- Discussion Type
- discussion topichomotopy dimension
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jan 12th 2011

have created an entry homotopy dimension

- Discussion Type
- discussion topicthread
- Category Latest Changes
- Started by zskoda
- Comments 4
- Last comment by TobyBartels
- Last Active Jan 12th 2011

New entry thread needed at newly expanded pro-object. Adequate changes at filtered limit.

While threads are elements in the cofiltered limit of sets or spaces, germs are classes of equivalence which appear in the treatment of filtered colimits. However the entry germ is taking germs just in the special case of colimits forming stalks of an etale space. In my practice the notion of germ could be used more generally for directed colimits or even filtered colimits in groups, sets, and alike. Is it only me using the terminology in this extended sense ? I would like to know the opinions.

- Discussion Type
- discussion topiccohomological dimension
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 12th 2011

have started cohomological dimension

- Discussion Type
- discussion topicoo-cohesive site
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 11th 2011

added to infinity-cohesive site statement and proof that if all objects of the $\infty$-cohesive site have points, then the cohesive $\infty$-topos over it satisfies the axiomm

*pieces have points*.(Easy proof using the previous results and Dugger’s cofibrant replacement theorem for $[C^{op}, sSet]_{proj,loc}$).

- Discussion Type
- discussion topicover-(oo,1)-category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 9th 2011

added to over-(infinity,1)-category the statement that the hom-spaces are computes by homotopy fibers of hom-spaces in the underlying $\infty$-category in direct analogy to the 1-categorical case

- Discussion Type
- discussion topiclocal (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 9th 2011

added at local (infinity,1)-topos

statement and proof that over $\infty Grpd$ the $\infty$-connectivity condition is redundant;

statement and proof that for any $\infty$-topos $\mathbf{H}$ over $\infty Grpd$ and $X \in \mathbf{H}$ any small-projective object, the slice $\mathbf{H}/X$ is local.

- Discussion Type
- discussion topic(2,1)-site
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by TobyBartels
- Last Active Jan 8th 2011

because I needed the link I have created a stub (2,1)-site. I define it there as an (infinity,1)-site whose underlying category is a (2,1)-category. Don’t have the leisure right now to check that this is also a 2-site whose underlying 2-category is a $(2,1)$-category, as hopefully it is

- Discussion Type
- discussion topicQuestions by Jim
- Category Latest Changes
- Started by DavidRoberts
- Comments 3
- Last comment by Urs
- Last Active Jan 7th 2011

Lots of questions by Jim Stasheff at descent for L-infinity algebras. But they are not marked clearly, so have a look at the changes.

- Discussion Type
- discussion topicinfinity-connected site
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jan 6th 2011

have created infinity-connected site

- moved over from infinity-cohesive site the proof that the $\infty$-topos over a locally and globally $\infty$-connected site is locally and globally $\infty$-connected; (and had occasion to polish and streamline it a bit more)

- Discussion Type
- discussion topicstrongly infinity-connected site
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jan 6th 2011

created strongly infinity-connected site with the analogous definition, analogous proposition and analogous proof as at strongly connected site.

Only difference is that I define it as a cosifted $\infty$-connected site instead of as a cosifted

*locally*$\infty$-connected site, because I am currently not quite sure about the definition of the latter because my model-category theoretic proofs rely on the existence of a terminal object, without that my standard Quillen adjunction model for the terminal $\infty$-geometric morphism fails and I’d need to think harder.But all these entries of higher connected and local sites currently have the issue that they give sufficient conditions and don’t prove necessary conditions, so I think it’s okay, but we should keep in mind that there might be refinements of these definitions.

- Discussion Type
- discussion topicconnected site
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jan 6th 2011

created connected site and declared it to be a locally connected site with a terminal object.

That’s sufficient for its sheaf topos to be connected, I don’t now if it is necessary: If we find a weaker sufficient condition we should refine the entry.

- Discussion Type
- discussion topicinfinity-local site
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 6th 2011

created infinity-local site (by separating the old material at infinity-cohesive site into its local and its connected bits, the connected bits are now split off at ∞-connected site).

- Discussion Type
- discussion topictotally infinity-connected site
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 6th 2011

created totally infinity-connected site with definition, theorem and proof analogous to totally connected site.

- Discussion Type
- discussion topicsifted (oo,1)-category
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jan 6th 2011

created sifted (infinity,1)-category

- Discussion Type
- discussion topicstrongly connected site
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 5th 2011

created strongly connected site with the evident definition and proposition

- Discussion Type
- discussion topicstrongly/totally oo-connected (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 5th 2011

I am trying to polish some entries in the topic cluster of higher connected toposes. In the course of this I felt like creating

mostly for cosmetic reasons at the moment. Not non-trivial content there yes.

- Discussion Type
- discussion topicSkolem
- Category Latest Changes
- Started by zskoda
- Comments 6
- Last comment by zskoda
- Last Active Jan 4th 2011

I plan to write few foundations/set theory stubs including Skolem paradox. It will wait for a bit as the $n$Lab seems to be down at the moment.

The entry forcing has phrase

**downward**Löwenheim-Skolem theorem. What does it mean*downward*in this phrase ? Is it a modifier at all ?

- Discussion Type
- discussion topicfppf-site
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by zskoda
- Last Active Jan 4th 2011

have created fppf-site

- Discussion Type
- discussion topiclocal site
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 4th 2011

created local site

- Discussion Type
- discussion topicconnected topos
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by Urs
- Last Active Jan 3rd 2011

I added a number of facts and proofs to connected topos, and shortened the example since it is a special case of the general fact about connected locally connected sites.

- Discussion Type
- discussion topicn-localic (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jan 3rd 2011

I rewrote the Idea-section at n-localic (infinity,1)-topos (trying to make it more to the point) and added propositions in the Properties- and the Examples-section.

- Discussion Type
- discussion topicfpqc-site
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by zskoda
- Last Active Jan 3rd 2011

have created fpqc-site

- Discussion Type
- discussion topic(oo,1)-Yoneda lemma
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active Jan 2nd 2011

added the full statement to (infinity,1)-Yoneda lemma

It's an easy consequence of the sSet-enriched Yoneda lemma, using the theorem that oo-presheaves are presented by fibrant-cofibrant sSet enriched presheaves.

Notice that this appears without stating the name "Yoneda lemma" as Lemma 5.5.2.1 in HTT. (Beware, though, that there are typos in there. There is a general mix-up of ops already in the statement of the lemma and the first in the proof is a . )

- Discussion Type
- discussion topicstructured (infinity,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jan 2nd 2011

I rewrote the Idea-section of structured (infinity,1)-topos.

i tried to make it clearer and shorter. And I highlighter more the aspect that this is a way to equip

*little*$\infty$-toposes with geometric structure.

- Discussion Type
- discussion topichomotopy groups of a Lie groupoid
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 2nd 2011

created homotopy groups of a Lie groupoid

The definition is the one following from the general abstract $\infty$-topos theoretic notion applied in ooLieGrpd, but I wanted a separate entry for this in order to record the references for that special case

- Discussion Type
- discussion topicapplied mathematics
- Category Latest Changes
- Started by zskoda
- Comments 5
- Last comment by TobyBartels
- Last Active Dec 30th 2010

- Discussion Type
- discussion topiccohesive topos and cohesive site
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Dec 30th 2010

I expanded at cohesive site the example “Families of sets”.

I also started expanding the Examples-section at cohesive topos, but it remains stubby for the moment. I have to run now.

- Discussion Type
- discussion topicprojective stack
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by zskoda
- Last Active Dec 29th 2010

created projective stack