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- Discussion Type
- discussion topicfield net
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 14th 2012

I have created a stub

*field net*to go with*net of local observables*(for the moment mainly such as to record references)

- Discussion Type
- discussion topicDHR superselection theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 14th 2012

at

*DHR superselection theory*I have added the argument (here) for why every DHR representation indeed comes from a net-endomorphism, assuming Haag duality and that the net takes values in vN algebras.

- Discussion Type
- discussion topicquantum lattice system
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 14th 2012

started stub for

*quantum lattice system*, for the moment mainly as a reminder for me concerning the book by Bratteli now referenced there.

- Discussion Type
- discussion topicfull and faithful (infinity,1)-functor
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by TobyBartels
- Last Active May 14th 2012

- Discussion Type
- discussion topicRecursion
- Category Latest Changes
- Started by TobyBartels
- Comments 8
- Last comment by TobyBartels
- Last Active May 13th 2012

As reported elsewhere, Zhen Lin began recursion. I changed the section title “In classical mathematics” to “In general” since there didn’t seem to be anything inherently classical about it. But maybe I’m missing something.

- Discussion Type
- discussion topicpresentation of homotopy type theory
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by TobyBartels
- Last Active May 13th 2012

I am working on prettifying the entry

*contractible type*and noticed that where in*Categorical semantics*it says “Let … with sufficient structure…” we really eventually need to point to an entry that discusses this sufficient structure in detail.In lack of a better idea, I named that entry

*presentation of homotopy type theory*. Feel free to make better suggestions.

- Discussion Type
- discussion topicinduction - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 12th 2012

time for a new topic-cluster table of contents:

*induction - contents*

- Discussion Type
- discussion topicinitial algebra
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Mike Shulman
- Last Active May 12th 2012

I have tidied up the entry

*initial algebra*and then made sure that it is cross-linked with*inductive type*(which it wasn’t!).We really need to rename this entry to

*initial algebra for an endofunctor*. But since I would have to fight the cache bug if I did it now, I decide not to be responsible for that at the moment.

- Discussion Type
- discussion topicfield with one element
- Category Latest Changes
- Started by zskoda
- Comments 12
- Last comment by zskoda
- Last Active May 11th 2012

More at field with one element, after creating person entry Christophe Soulé about the creator. By the way the Soulé has different encoding in n-Forum than in nlab so the link does not access the right page from here. See redirect Christophe Soule.

- Discussion Type
- discussion topicfiltered colimit
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by TobyBartels
- Last Active May 11th 2012

I have added various basic technical details to filtered colimit and flat functor.

- Discussion Type
- discussion topicseparated geometric morphism
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by zskoda
- Last Active May 9th 2012

created stub for

*separated geometric morphism*There is room to go through the Lab and interlink all the various entries on separated schemes, Hausdorff spaces etc. pp. and explain how these are all examples of a single notion. But I don’t have the energy for it right now.

- Discussion Type
- discussion topicJouanolou cover
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active May 7th 2012

New entry Jouanolou cover (prompted by its use in Van den Bergh’s version of a proof that every projective variety is a quiver Grassmanian, which JOhn posts about in cafe). Let me mention also the earlier entry Jean-Pierre Jouanolou.

- Discussion Type
- discussion topic(infinity,1)-vector bundle
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active May 4th 2012

started (infinity,1)-vector bundle with a bit of discussion of the Ando-Blumberg-Gepner-Hopkins-Rezk theory of (discrete) $\infty$-ring module $\infty$-bundles.

- Discussion Type
- discussion topicexhaustive category
- Category Latest Changes
- Started by Mike Shulman
- Comments 8
- Last comment by Urs
- Last Active May 3rd 2012

I have created exhaustive category — not just the page, but the terminology. No one at MO seemed to know a name for this exactness property, so I made one up. The adjective “exhaustive” seems harmonious with “extensive” and “adhesive”, and expresses the idea that the subobjects in a transfinite union “exhaust” the colimit. But I would welcome other opinions and suggestions.

- Discussion Type
- discussion topicabsolute colimit
- Category Latest Changes
- Started by Mike Shulman
- Comments 4
- Last comment by Mike Shulman
- Last Active May 2nd 2012

I added a characterization, reference, and some more examples to absolute colimit.

- Discussion Type
- discussion topicadhesive categories
- Category Latest Changes
- Started by Mike Shulman
- Comments 7
- Last comment by Urs
- Last Active May 2nd 2012

Created adhesive category.

- Discussion Type
- discussion topicstrongly compact topological space
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active May 2nd 2012

of course there is also the notion going by the name

*strongly compact topological space*.

- Discussion Type
- discussion topiccurved dg-algebra
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by zskoda
- Last Active May 1st 2012

started curved dg-algebra

- Discussion Type
- discussion topicepipresheaf - ''minus construction''
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 1
- Last comment by Stephan A Spahn
- Last Active Apr 30th 2012

I added a definition to epipresheaf. I am wondering if there is a ”minus construction” turning a presheaf into an epipresheaf.

- Discussion Type
- discussion topictopological M-theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 30th 2012

in reaction to Zoran’s remark here: stub for

*topological M-theory*

- Discussion Type
- discussion topicJames Wallbridge on (oo,1)-Tannakian theory
- Category Latest Changes
- Started by DavidRoberts
- Comments 8
- Last comment by Tim_Porter
- Last Active Apr 27th 2012

James Wallbridge put on the arXiv a paper derived from his thesis. I’ve linked to both from his page here. Urs, in particular, was interested in seeing a copy

- Discussion Type
- discussion topiccompact object in an (infinity,1)-category
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Apr 27th 2012

at compact object in an (infinity,1)-category I have added the definition and stated the examples: the $\kappa$-compact objects in $(\infty,1)Cat$/$\infty Grpd$ are the essentially $\kappa$-small $(\infty,1)$-categories/groupoids.

- Discussion Type
- discussion topicKan complexes as ∞-groupoids
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 14
- Last comment by Urs
- Last Active Apr 26th 2012

I would like to rearrange Kan complexes as ∞-groupoids to something like

general description

2-dimensional example

In particular I think the word oriental should occur more prominently in the beginning of this section.

- Discussion Type
- discussion topic[[line integral]]
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Apr 26th 2012

New page: line integral (also redirects from contour integral). Too damn long; somebody should edit this down.

- Discussion Type
- discussion topicinverse limit
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by TobyBartels
- Last Active Apr 26th 2012

added an illustrating diagram to

*inverse limit*, just so that one sees at one glance what the variance of the arrows is, since following through the “directed/codirected”-terminology and entries – if one really is in need of the $\mathbb{Z}_2$-orientation – can be a bit of a pain.

- Discussion Type
- discussion topicaction infinity-groupoid
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 1
- Last comment by Stephan A Spahn
- Last Active Apr 25th 2012

In need a definition of an action of a groupoid object $G$ in an ($\infty$,1)-category (actually in an ($\infty$,1)-topos) on an object $X$ - so I created one but I’m not yet sure if it coincides with the existing one if $X$ is pointed.

- Discussion Type
- discussion topicergodic theory
- Category Latest Changes
- Started by zskoda
- Comments 4
- Last comment by Urs
- Last Active Apr 25th 2012

New stub ergodic theory wanted at measure theory.

- Discussion Type
- discussion topicteleparallel gravity
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by SridharRamesh
- Last Active Apr 22nd 2012

started

*teleparallel gravity*

- Discussion Type
- discussion topicWeitzenböck connection
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 21st 2012

stub for

*Weitzenböck connection*

- Discussion Type
- discussion topicComplexes of groups
- Category Latest Changes
- Started by Tim_Porter
- Comments 4
- Last comment by Urs
- Last Active Apr 19th 2012

I have started a new entry on complexes of groups, the higher dimensional version of graphs of groups (in the bass-Serre theory). These are related to orbifolds and topological stacks, but as yet there is just a stub. I have put some stuff in the Menagerie so will transfer more across in a short while (I hope!).

- Discussion Type
- discussion topicEilenberg subcomplex
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 19th 2012

started

*Eilenberg subcomplex*

- Discussion Type
- discussion topicmodel structure for dendroidal complete Segal spaces
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Apr 18th 2012

Have been adding material to

*model structure for dendroidal complete Segal spaces*.

- Discussion Type
- discussion topicReedy model structure over the simplex category
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Apr 15th 2012

I added to Reedy model structure in the section Over the simplex category a bunch of basic useful lemmas and proofs. It works up to a proof of the Bousfield-Kan map.

- Discussion Type
- discussion topicderived geometry
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Apr 15th 2012

for a seminar that we will be running I need a dedicated entry

So I created it.

I inserted a disclaimer on top that there are variants to what people understand under “derived geometry” and point the reader to the entry higher geometry for more details. I would be grateful if we could keep this entry titled this way and discuss variants elsewhere.

I would also be grateful if anyone who feels like making non-controversial edits (typos, references, etc. ) to for the moment do them not on this nLab page, but on this page here on my personal web:

Because currently the content of both pages is identical – except that the latter also has a seminar schedule which is omitted in the former – and until the entry has stabilized a bit more I would like to make edits just in

*one*place and update the other one by copy-and-paste.

- Discussion Type
- discussion topiclocal fibration
- Category Latest Changes
- Started by Urs
- Comments 19
- Last comment by Urs
- Last Active Apr 14th 2012

quick note

*local fibration*

- Discussion Type
- discussion topicreduced simplicial set
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 14th 2012

created

*reduced simplicial set*, just for completeness

- Discussion Type
- discussion topicF-finite sets
- Category Latest Changes
- Started by TobyBartels
- Comments 19
- Last comment by Mike Shulman
- Last Active Apr 10th 2012

Although there is a standard meaning of ‘finite’ in constructive mathematics, it’s helpful to have a way to indicate that one really means this and is not just sloppily writing ‘finite’ in a situation where it is correct classically, without having to make a circumlocution like ‘finite (even in constructive mathematics)’. Based on Mike’s notation at finite set and drawing an analogy with ‘$K$-finite’, I’ve invented the term ‘$F$-finite’. (So now the circumlocution is simply ‘finite ($F$-finite)’ or ‘finite (F-finite)’, assuming that one wishes to relegate constructivism to parenthetical remarks.)

I’ve added this to finite set, added redirects, and used the new abbreviated circumlocution at dual vector space.

- Discussion Type
- discussion topicIndroduction to a simplicial model of homotopy type theory
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 3
- Last comment by Stephan A Spahn
- Last Active Apr 5th 2012

I created T. Streicher - a model of type theory in simplicial sets - a brief introduction to Voevodsky’ s homotopy type theory with a summary of that article and linked it from homotopy type theory. Maybe this article can serve as a base for some pedagogical nlab-entry providing some technical details concerning this simplicial model which are omitted in homotopy type theory.

- Discussion Type
- discussion topicLambek and Scott
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Apr 4th 2012

Started entries on Jim Lambek and Phil Scott. These are stubby.

- Discussion Type
- discussion topicdeformation retract
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Apr 4th 2012

Added to

*deformation retract*the general definition. Moved the previous content to a section*Examples - In topological spaces*.

- Discussion Type
- discussion topicmarked simplicial sets
- Category Latest Changes
- Started by Mike Shulman
- Comments 3
- Last comment by Mike Shulman
- Last Active Apr 4th 2012

It seems that the page marked simplicial set uses $X^#$ and $X^\sharp$ where Lurie uses $X^\sharp$ and $X^\natural$. That seems gratuitously confusing to me; is there a reason for it?

- Discussion Type
- discussion topicHNN-extension
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Apr 3rd 2012

I have created a stub HNN-extension. I have been wondering how to link in the connection between homotopy colimits and graphs of groups (see Fiore, Luck and Sauer), any ideas? Perhpas it will have to wait until there is a graphs of groups and a complexes of groups entry

- Discussion Type
- discussion topicdiscussions from preorder
- Category Latest Changes
- Started by Mike Shulman
- Comments 8
- Last comment by Mike Shulman
- Last Active Apr 3rd 2012

Moving to here some very old discussions from preorder:

*Todd says:*It’s not clear to me how one avoids the axiom of choice. For example, any equivalence relation $E$ on a set $X$ defines a preorder whose posetal reflection is the quotient $X \to X/E$, and it seems to me you need to split that quotient to get the equivalence between the preorder and the poset.*Toby says:*In the absence of the axiom of choice, the correct definition of an equivalence of categories $C$ and $D$ is a span $C \leftarrow X \rightarrow D$ of full, faithful, essentially surjective functors. Or equivalently, a pair $C \leftrightarrow D$ of anafunctors (with the usual natural transformations making them inverses).*Todd says:*Thanks, Toby. So if I understand you aright, the notion of equivalence you have in mind here is not the one used at the top of the entry equivalence, but is more subtle. May I suggest amplifying a little on the above, to point readers to the intended definition, since this point could be confusing to those inexperienced in these matters?*Urs says*: as indicated at anafunctor an equivalence in terms of anafunctors can be understood as a span representing an isomorphism in the homotopy category of $Cat$ induced by the folk model structure on $Cat$.*Toby says*: I think that this should go on equivalence, so I'll make sure that it's there. People that don't know what ’equivalence’ means without choice should look there.*Mike*: Wait a minute; I see why every preorder is equivalent to a poset without choice, but I don’t see how to show that every preorder has a skeleton without choice. So unless I’m missing something, the statement that every preorder is equivalent to a poset isn’t, in the absence of choice, a special case of categories having skeletons.*Toby*: Given the definition there that a skeleton must be a subcategory (not merely any equivalent skeletal category), that depends on what subcategory means, doesn't it? If a subcategory can be any category equipped with a pseudomonic functor and if functor means anafunctor in choice-free category theory, then it is still true. On the other hand, since we decided not to formally define ’subcategory’, we really shouldn't use it to define ’skeleton’ (or anything else), in which case ‹equivalent skeletal category› is the guaranteed non-evil option. You*still*need choice to define a skeleton of an arbitrary category, but not of a proset.*Mike*: We decided not to formally define a non-evil version of “subcategory,” but subcategory currently is defined to mean the evil version. However, I see that you edited skeleton to allow any equivalent skeletal category, and I can’t really argue that that is a more reasonable definition in the absence of choice.

- Discussion Type
- discussion topicStrict category theory vs preorder theory
- Category Latest Changes
- Started by TobyBartels
- Comments 7
- Last comment by Mike Shulman
- Last Active Apr 2nd 2012

The thread Category theory vs order theory quickly really became Topological spaces vs locales, so I’m putting this in a new thread.

At category theory vs order theory, I had originally put in the analogy with category : poset :: strict category : proset. Mike changed this to to category : proset :: skeletal category : poset. I disagree. A proset has two notions of equivalence: the equality of the underlying set, and the symmetrisation of the order relation; a poset has only one. Similarly, a strict category has two notions of equivalence: the equality of the set of objects, and the isomorphism relation; a category has only one. I’m OK with using skeletal categories to compare with posets, since this will make sense to people who only know the evil notion of strict category, but I insist on using strict categories to compare with prosets. So now its strict category : proset :: skeletal category : poset.

- Discussion Type
- discussion topicGaussian numbers
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Mar 31st 2012

Just a stub, for the record: Gaussian number.

- Discussion Type
- discussion topicproduct type
- Category Latest Changes
- Started by Mike Shulman
- Comments 5
- Last comment by Urs
- Last Active Mar 26th 2012

I have added details to product type on both positive and negative definitions, with the corresponding beta and eta reduction rules.

- Discussion Type
- discussion topicCompact support
- Category Latest Changes
- Started by Andrew Stacey
- Comments 5
- Last comment by TobyBartels
- Last Active Mar 24th 2012

The statement at compact support was that $f^{-1}(0)$ should be compact. I’ve corrected this.

- Discussion Type
- discussion topicprincipal oo-bundle
- Category Latest Changes
- Started by Urs
- Comments 49
- Last comment by Urs
- Last Active Mar 22nd 2012

I expanded the discussion at principal infinity-bundle to go along with the discussion with Mike over at the blog

- Discussion Type
- discussion topicThomas Hale(s)
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Urs
- Last Active Mar 22nd 2012

Urs, I noted you started a new entry on Thomas Hale. Can you check Hale(s) name as his website gives it with an s on the end? I do not know of him so hesitate to change it. (homepages on university websites are not unknown to get things wrong!)

- Discussion Type
- discussion topicTheorems
- Category Latest Changes
- Started by TobyBartels
- Comments 46
- Last comment by Todd_Trimble
- Last Active Mar 21st 2012

I put an actual definition at theorem. It is still quite the stub, however!

- Discussion Type
- discussion topicmodel structure on dendroidal sets
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Mar 21st 2012

created model structure on dendroidal sets (stubby)

expanded dendroidal set (still stubby, though)

- Discussion Type
- discussion topicLocalic reflection of the Zariski topos
- Category Latest Changes
- Started by Zhen Lin
- Comments 9
- Last comment by Zhen Lin
- Last Active Mar 20th 2012

Let $\mathcal{Z}$ be the Zariski topos, in the sense of the classifying topos for local rings. I was wondering whether there might be any connection between $\mathbf{Sh}(\operatorname{Spec} \mathbb{Z})$ and $\mathcal{Z}$. Certainly, there is a geometric morphism $\mathcal{Z} \to \mathbf{Sh}(\operatorname{Spec} \mathbb{Z})$, and there’s also a geometric inclusion $\mathbf{Sh}(\operatorname{Spec} \mathbb{Z}) \to \mathcal{Z}$. On the other hand, there’s no chance of $\mathcal{Z}$ itself being localic, since it has a proper class of (isomorphism classes of) points. Let’s write $L \mathcal{Z}$ for the localic reflection of $\mathcal{Z}$; the first geometric morphism I mentioned then corresponds to a locale map $L \mathcal{Z} \to \operatorname{Spec} \mathbb{Z}$. But what is $L \mathcal{Z}$ itself?

The open objects in $\mathcal{Z}$ can be identified with certain saturated cosieves on $\mathcal{Z}$ in the category of finitely-presented commutative rings, and so may be identified with certain sets of isomorphism classes of finitely-presented commutative rings. If I’m not mistaken, every finitely-presented commutative ring gives rise to an open object in $\mathcal{Z}$. This suggests that $L \mathcal{Z}$ might be some kind of (non-spatial) union of all isomorphism classes of affine schemes of finite type over $\mathbb{Z}$, which is an incredibly mind-boggling thing to think about. It’s not clear to me whether other kinds of open objects exist. For example, does every not-necessarily-affine open subset of $\operatorname{Spec} A$, for every finitely-presented ring $A$, also show up…?

- Discussion Type
- discussion topicetale morphisms - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 19th 2012

I have created a global table of contents

*etale morphisms - contents*and added it as a “floating table of contents” to relevant entries.

- Discussion Type
- discussion topicregular elements in a Heyting algebra
- Category Latest Changes
- Started by Todd_Trimble
- Comments 22
- Last comment by eparejatobes
- Last Active Mar 17th 2012

I’ve inserted some proofs of statements made at Heyting algebra, particular on the “regular element” left adjoint to the full inclusion $Bool \to Heyt$.

The proof that $L \to L_{\neg\neg}$ preserves implication seemed harder than I was expecting it to be. Or maybe my proof is a clumsy one? If anyone knows a shorter route to this result, I’d be interested.

- Discussion Type
- discussion topiccircuitoid
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Mar 17th 2012

by chance I noticed that two days ago somebody created an entry

*Circuitoids*. I am not sure what to do about it…

- Discussion Type
- discussion topic2-adjunction
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by David_Corfield
- Last Active Mar 16th 2012

at 2-adjunction I would like to list a bunch of 2-category theoretic analogs of standard facts about ordinary adjunctions. Such as: a right adjoint is a full and faithful 2-functor precisely if the counit of the 2-adjunction is an equivalence, etc.

But I haven’t really thought deeply about 2-adjunctions myself yet. Is there some reference where we could take such a list of properties from?

- Discussion Type
- discussion topic(0,1)-category theory - contents
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Mar 15th 2012

I wanted to add some stuff about completely distributive lattices, when I got annoyed by the fact that few of the entries on lattices, frames, etc, carried a table of contents, and that I kept being surprised by which related entries already existed and which not.

That’s a clear case for a context floating table of contents, so I started one

Just a start. Please feel free to expand.

- Discussion Type
- discussion topicalgebraic lattices
- Category Latest Changes
- Started by SridharRamesh
- Comments 2
- Last comment by Urs
- Last Active Mar 15th 2012

- I've made some small additions to the article on algebraic lattices (including fixing the languishing typo in the introduction, "An alegbraic lattice is...").

- Discussion Type
- discussion topicAxioms of choice in topology
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Mar 15th 2012

I stumbled across a nice reference while looking for something else, so added it to axiom of choice so I might read it all later.

- Discussion Type
- discussion topicspine
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 13th 2012

I have added a little bit to

*spine*.(Will maybe write out the proof of the proposition there in a little while.)