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- Discussion Type
- discussion topicidentity component
- Category Latest Changes
- Started by Todd_Trimble
- Comments 1
- Last comment by Todd_Trimble
- Last Active Feb 3rd 2012

Created identity component, and added some little remarks to open map and quotient space.

- Discussion Type
- discussion topicHigher doctrines
- Category Latest Changes
- Started by fpaugam
- Comments 29
- Last comment by Mike Shulman
- Last Active Feb 3rd 2012

- A new contribution

- Discussion Type
- discussion topicalmost connected topological group
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Feb 2nd 2012

just for completeness, I have created an entry

*almost connected topological group*.

- Discussion Type
- discussion topiccubical set
- Category Latest Changes
- Started by Tim_Porter
- Comments 25
- Last comment by Todd_Trimble
- Last Active Feb 2nd 2012

There is a strange glitch on this page: the geometric realization of a cubical set (see geometric realizationealization) below) tends to have the wrong homotopy type:

That is what appears but is not a t all what the source looks like:

the geometric realization of a cubical set (see [geometric realization](#geometric realization) below) tends to have the wrong homotopy type:

What is going wrong and how can it be fixed?

Another point : does anyone know anything about symmetric cubical sets?

- Discussion Type
- discussion topicnormal morphism of infinity-groups
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Feb 2nd 2012

Added to

*normal subgroup*a section*Normal morphisms of ∞-groups*.

- Discussion Type
- discussion topicexceptional geometry
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 1st 2012

I am slowly starting to add stuff at

*exceptional geometry*

- Discussion Type
- discussion topicclassifying topos
- Category Latest Changes
- Started by Todd_Trimble
- Comments 25
- Last comment by Mike Shulman
- Last Active Feb 1st 2012

I made some edits at classifying topos to correct what I thought were some inaccuracies. One is that simplicial sets classify interval objects, but offhand I didn’t see the exact notion of linear interval over at interval object that would make this a correct sentence. In any event, I went ahead and defined the notion of linear interval as a model of a specified geometric theory.

The other is for local rings. I think when algebraic geometers refer to a sheaf of local rings, they refer to a sheaf of rings over a (sober) space all of whose stalks are local. I wasn’t sure that description would be kosher for a general (Grothendieck) topos $E$ since there may not be any “stalks” (i.e., points $Set \to E$) to refer to. In any case, it seems to me safer to give the geometric theory directly.

- Discussion Type
- discussion topicmodel structure on cosimplicial simplicial sets
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 31st 2012

Matan Prezma kindly pointed out the he has an article with a correction to what used to be prop. 2 at

*model structure on cosimplicial simplicial sets*. (One has to use restricted totalization instead of ordinary totalization.)I have corrected this and added the reference.

I also added to the entry a remark that makes the relation to descent objects explicit. Right now this is remark 2.

- Discussion Type
- discussion topictrivial and mixed model structures
- Category Latest Changes
- Started by Mike Shulman
- Comments 3
- Last comment by Mike Shulman
- Last Active Jan 28th 2012

New pages: 2-trivial model structure and mixed model structure.

- Discussion Type
- discussion topicbialgebra vs. GS cohomology
- Category Latest Changes
- Started by fpaugam
- Comments 17
- Last comment by zskoda
- Last Active Jan 27th 2012

- How is Majid's bialgebra cohomology related to GS cohomology for bialgebras?

I saw the recent paper by Shoikhet where GS cohomology is k-monoidally understood.

Is there a similar understanding of Majid's cohomology?

- Discussion Type
- discussion topicfactorization system over a subcategory
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Jan 27th 2012

New page factorization system over a subcategory.

- Discussion Type
- discussion topicMeasurable stuff
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Jan 22nd 2012

Split measurable subset and measurable function (unfinished) from sigma-algebra and measurable space. Also added to sigma-ideal. Still need to split some things from measure space.

- Discussion Type
- discussion topicSuslin
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Tim_Porter
- Last Active Jan 21st 2012

I have created a stub entry for A. Suslin. Can someone add in the Russian original of his name please, as I do not know if the Wikipedia version is correct?

- Discussion Type
- discussion topicCheng measurable spaces
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by TobyBartels
- Last Active Jan 21st 2012

Some time ago, I split Cheng space from measurable space, but I never announced it here (nor removed if from the list of things to do at the latter). Note: Henry Cheng, not Eugenia Cheng.

- Discussion Type
- discussion topicCharacters of profinite groups
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Jan 20th 2012

I added a brief equivalence between two notions of characters of profinite groups that I spotted on MO.

- Discussion Type
- discussion topicextremum
- Category Latest Changes
- Started by Urs
- Comments 20
- Last comment by TobyBartels
- Last Active Jan 20th 2012

looks like I started

*extremum*(I wanted

*minimum*not to be gray at higher dimensional Chern-Simons theory…)

- Discussion Type
- discussion topicsome classifying toposes
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Jan 17th 2012

I added a section to simplicial set remarking on their status as a classifying topos, and a sentence to classifying topos (in the section “For objects”) recording the answer to this question.

- Discussion Type
- discussion topicWu class
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 16th 2012

I have created an entry

*Wu class*.At the end I have also included an “Applications”-section with comments on Wu classes in the definition of higher Chern-Simons functionals. That eventually deserves to go in a dedicated entry, but for the moment I think it is good to have it there, as it is a major source of discussion of Wu structures in the maths literature, quite indepently of its role in physics.

- Discussion Type
- discussion topic''(-) object vs. internal (-)''
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 11
- Last comment by Stephan A Spahn
- Last Active Jan 15th 2012

I added links to the horizontal categorifications in group object and created groupoid object.

In groupoid object in an (infinity,1)-category I read the conspicious statement: ”an internal ∞-group or internal ∞-groupoid may be defined as a group(oid) object internal to an (∞,1)-category C with pullbacks” - but this terminology seems to hinder distinguishing between them and ∞-groupoid objects in (∞,1)-categories.

- Discussion Type
- discussion topicScone
- Category Latest Changes
- Started by David_Corfield
- Comments 11
- Last comment by Mike Shulman
- Last Active Jan 13th 2012

After Mike’s post, scone was created. But I see at Freyd cover it says

The Freyd cover of a category – sometimes known as the Sierpinski cone or “scone” – is a special case of Artin gluing

Are they synonyms?

- Discussion Type
- discussion topicdescent and locality - contents
- Category Latest Changes
- Started by Urs
- Comments 20
- Last comment by Urs
- Last Active Jan 13th 2012

Have created a table of contents

*descent and locality - contents*.Am including this now as a “floating TOC” into the relevant entries.

- Discussion Type
- discussion topicAlfsen-Shultz theorem
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 13th 2012

I am just hearing about the Alfsen-Shultz theorem about states on C*-algebras, so I started a stub to remind me. Still need to track down the reference and the details.

- Discussion Type
- discussion topicLocales
- Category Latest Changes
- Started by TobyBartels
- Comments 4
- Last comment by Mike Shulman
- Last Active Jan 11th 2012

I went through locale and made some of the language consistent throughout the article. Also I added a new section, Subsidiary notions, to which I intend to add.

- Discussion Type
- discussion topicToposes
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jan 11th 2012

started a stub entry Toposes on the category (2-category) of all toposes. But nothing much there yet.

- Discussion Type
- discussion topicrelative cohomology
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 9th 2012

I have put some content into the entry

*relative cohomology*

- Discussion Type
- discussion topicanalytic manifold
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Jan 9th 2012

stub for analytic manifold

- Discussion Type
- discussion topicLanglands correspondence -- contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 7th 2012

when creating a stub entry

*local Langlands conjectures*I noticed that it has already become hard to know which entries on the Langlands program we already have. I always take this as a sign that a summary table of contents is called for. So I startedLanglands correspondence – contents

and added it as a “floating table of contents” to the relevant entries.

(Even though all of these entries are still more or less stubs.)

- Discussion Type
- discussion topicbuilding
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 6th 2012

I noticed that there was some wild formatting at

*building*. I have tried to tame it a bit.

- Discussion Type
- discussion topicnon-commutative analytic space
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 6th 2012

a reference on non-commutative analytic spaces (Berkovich style)

- Discussion Type
- discussion topicanalytic spectrum
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 4th 2012

started analytic spectrum (redirecting Berkovich spectrum)

- Discussion Type
- discussion topicmeasure theory - contents
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Jan 4th 2012

I have started a table of contents measure theory - contents and started adding it as a floating toc to the relevant entries

- Discussion Type
- discussion topicArchimedean field
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 4th 2012

- Discussion Type
- discussion topicNori motives
- Category Latest Changes
- Started by zskoda
- Comments 4
- Last comment by zskoda
- Last Active Jan 4th 2012

- Discussion Type
- discussion topicanalytic geometry
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by zskoda
- Last Active Jan 4th 2012

currently the bulk of the entry analytic geometry is occupied by a long section on “Holomorphic functions of several complex variables”. Should that not better be moved to some dedicated entry of its own? Any opinions?

- Discussion Type
- discussion topicbranched manifold
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 4
- Last comment by Urs
- Last Active Jan 3rd 2012

I created branched manifold -linked from orbifold- with a definition from ”expanding attractors” by Robert F. Williams (1974) quoted in wikipedia. This description is -as it stands- not precisely compatible to that given in Dusa McDuffs ”Groupoids, Branched Manifolds and Multisections” which I am rather interested in. So I plan to comment on this as a side note in the -yet to be written-article orbifold groupoid.

- Discussion Type
- discussion topicdescent algebra
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Jan 3rd 2012

A stub for the famous Solomon’s descent algebra.

- Discussion Type
- discussion topicCofinality
- Category Latest Changes
- Started by TobyBartels
- Comments 3
- Last comment by TobyBartels
- Last Active Jan 1st 2012

New article: cofinality, with the basic case being the cofinality of a quasiorder as a collection of cardinal numbers, a variation as a collection of ordinal numbers (or equivalently an ordinal number), and an apparently separate case of the cofinality of a collection of cardinal numbers, all of these tied together and interpreted as a single cardinal number if one assumes the axiom of choice.

- Discussion Type
- discussion topicMayer-Vietoris sequence
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Dec 31st 2011

created Mayer-Vietoris sequence

- Discussion Type
- discussion topicdiffemorphism group
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 31st 2011

added to diffeomorphism group statements and references for the case of 3-manifolds (Smale conjecture etc.)

- Discussion Type
- discussion topicfinite (infinity,1)-limit
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Dec 23rd 2011

I was surprised to discover that we had no page finite (infinity,1)-limit yet, especially given that they are slightly subtle in relation to the 1-categorical version. So I made one.

- Discussion Type
- discussion topicsupercompact cardinal
- Category Latest Changes
- Started by DavidRoberts
- Comments 1
- Last comment by DavidRoberts
- Last Active Dec 22nd 2011

From supercompact cardinal:

**Theorem:**The existence of arbitrarily large supercompact cardinals implies the statement:Every absolute epireflective class of objects in a balanced accessible category is a small-orthogonality class.

In other words, if $L$ is a reflective localization functor on a balanced accessible category such that the unit morphism $X \to L X$ is an epimorphism for all $X$ and the class of $L$-local objects is defined by an absolute formula, then the existence of a suficciently large supercompact cardinal implies that $L$ is a localization with respect to some set of morphisms.

This is in

*BagariaCasacubertaMathias*Urs Schreiber: I am being told in prvivate communication that the assumption of epis can actually be dropped. A refined result is due out soon.

does anyone know about this refined result?

- Discussion Type
- discussion topiclocalizing subcategory
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by zskoda
- Last Active Dec 21st 2011

this is a message to Zoran:

I have tried to refine the section-outline at localizing subcategory a bit. Can you live with the result? Let me know.

- Discussion Type
- discussion topicfull functor
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 21st 2011

discovered the following remnant discussion at full functor, which hereby I move from there to here

*Mathieu says:*I agree that, for functors, there is no reason to say “fully faithful” rather than “full and faithful”. But for arrows in a 2-category (like in the new version of the entry on subcategories), there are reasons. I quote myself (from my thesis): «Remark: we say*fully faithful*and not*full and faithful*, because the condition that, for all $X:\C$, $C(X,f)$ be full is not equivalent in $\Grpd$ to $f$ being full. Moreover, in $\Grpd$, this condition implies faithfulness. We will define (Definition 197) a notion of full arrow in a $\Grpd$-category which, in $\Grpd$ and $\Symm2\Grp$ (symmetric 2-groups), gives back the ordinary full functors.» Note that this works only for some good groupoid enriched categories, not for $\Cat$, for example.*Mike*says: Do you have a reason to care about full functors which are not also faithful? I’ve never seen a very compelling one. (Maybe I should just read your thesis…) I agree that “full morphism” (in the representable sense) is not really a useful/correct concept in a general 2-category, and that therefore “full and faithful” is not entirely appropriate, so I usually use “ff” in that context. I’ve changed the entry above a bit to reflect your comment; is it satisfactory now? Maybe all this should actually go at full and faithful functor (and/or fully faithful functor)?

- Discussion Type
- discussion topicline object
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Dec 21st 2011

created line object

- Discussion Type
- discussion topichigher U(1)-gauge field
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 20th 2011

I have finally started – long overdue – an entry higher U(1)-gauge theory. What I really needed right now (as an entry) is the subentry higher electric background charge coupling, that I also started.

I am not done yet with what I wanted to do, but need to quit for tonight.

- Discussion Type
- discussion topicinductive type
- Category Latest Changes
- Started by Urs
- Comments 25
- Last comment by Mike Shulman
- Last Active Dec 16th 2011

I am starting stubs

- Discussion Type
- discussion topicmapping cocone
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Mike Shulman
- Last Active Dec 16th 2011

created an entry mapping cocone, following a suggestion by Zoran, that this is the right technical term for what is discussed in more detail at generalized universal bundle.

(the examples section needs more attention, though...)

- Discussion Type
- discussion topicmodel category stubs
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Dec 15th 2011

Stubs for fibrant object and acyclic fibration (also redirecting their duals, for now).

- Discussion Type
- discussion topicfinal lifts and adjoint triples
- Category Latest Changes
- Started by Mike Shulman
- Comments 4
- Last comment by Mike Shulman
- Last Active Dec 15th 2011

I have created final lift, and added to adjoint triple a proof that in a fully faithful adjoint triple between cocomplete categories, the middle functor admits final lifts of

*small*structured sinks (and dually). This means that it is kind of like a topological concrete category, except that the forgetful functor need not be faithful.I find this interesting because it means that in the situation of axiomatic cohesion, where the forgetful functor from “spaces” is not necessarily faithful, we can still construct such “spaces” in “initial” and “final” ways, as long as we restrict to small sources and sinks.

- Discussion Type
- discussion topictruncated object
- Category Latest Changes
- Started by Mike Shulman
- Comments 4
- Last comment by TobyBartels
- Last Active Dec 13th 2011

I figured it was high time we had a general page on truncated objects. I think some number of links, and perhaps redirects, which currently point to n-truncated object of an (infinity,1)-category would more usefully point here, but I haven’t updated any yet.

- Discussion Type
- discussion topicdiscrete object/morphism
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Mike Shulman
- Last Active Dec 13th 2011

I noticed that discrete object used to redirect to discrete morphism, where I expected it to take me at least to discrete space, if not to its own entry.

We should eventually disambiguate here and add some comments. For the moment I made it redirect to discrete space and added there a remark “to be merged with discrete morphism”.

- Discussion Type
- discussion topiccategory of representations
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Urs
- Last Active Dec 12th 2011

There is a query on category of representations. Basically asking for some references to be added???

- Discussion Type
- discussion topicevil
- Category Latest Changes
- Started by TobyBartels
- Comments 33
- Last comment by Urs
- Last Active Dec 12th 2011

If you're not following the

`categories`

mailing list, then you're missing out on a great discussion of evil. Peter Selinger has come from the list to the Lab to discuss it here too!

- Discussion Type
- discussion topicBanach bundle
- Category Latest Changes
- Started by zskoda
- Comments 15
- Last comment by zskoda
- Last Active Dec 9th 2011

New entry Banach bundle covering for now also more special notion of Hilbert bundle and a

*different*notion of Banach algebraic bundle. Sanity check is welcome!

- Discussion Type
- discussion topicSymmetric product of circles
- Category Latest Changes
- Started by Andrew Stacey
- Comments 6
- Last comment by jim_stasheff
- Last Active Dec 9th 2011

Thought I’d write up some old notes at symmetric product of circles (linked from unitary group, explanation to come on symmetric product of circles). Not finished yet, but have to leave it for now.

(I was incensed to discover that to look at the source article for the material for this to check that I’m remembering it right - I last looked at it about 10 years ago - I have to pay 30 UKP. The article is 3 pages long. That’s 10UKP per page! So I’m going from vague memories and “working it out afresh”.)

- Discussion Type
- discussion topicDomenico: latest changes
- Category Latest Changes
- Started by domenico_fiorenza
- Comments 12
- Last comment by domenico_fiorenza
- Last Active Dec 8th 2011

Added Thom-Federer and Gottlieb thorems to Eilenberg-MacLane space; added the remark “$\Omega\mathbf{C}(X,Y)\simeq \mathbf{C}(X,\Omega Y)$ in any (oo,1)-category with homotopy pullbacks” in loop space object.

- Discussion Type
- discussion topictetracategory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 7th 2011

have added pointers to Alex Hoffnung’s preprint to tetracategory, tricategory, span and (infinity,n)-category of spans.

- Discussion Type
- discussion topic"inhabited" and "(-1)-connected"
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 7th 2011

At inhabited and (-1)-connected I have added cross-links pointing out the synonyms.

At local topos I have added a Properties-section stating that here every inhabited object is globally inhabited, which is a shadow of the homotopy dimension 0 of local (infinity,1)-toposes.

- Discussion Type
- discussion topiccell complex
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 7th 2011

I noticed that cell complex was missing, so I created it

- Discussion Type
- discussion topicfundamental vector field
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Dec 6th 2011

New entry fundamental vector field which covers also the somewhat dual notion fundamental differentiable form. Redirecting also fundamental form. The entry is partly intended to support the content in the entry Ehresmann connection. Please check the content.