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- 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 topicscissors congruence
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Jan 19th 2012

New entry scissors congruence.

- 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 topicboundary of a simplex
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 1
- Last comment by Stephan A Spahn
- Last Active Jan 8th 2012

I added the coequalizer description of a boundary of a simplex. The horn has such a description, too…

- Discussion Type
- discussion topicnerve and realization
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 2
- Last comment by Urs
- Last Active Jan 8th 2012

I inserted two links in nerve and realization: simplicial topological space (redirecting cosimplicial topological space) and geometric realization of simplicial topological spaces. I would like to add some remarks to the internal case.

- 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 topiccategory with weak equivalences
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Mike Shulman
- Last Active Jan 3rd 2012

I have a further comment/question in the query box at category with weak equivalences

- 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 topictwisted spin^c structure
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 30th 2011

I added four more references at twisted spin^c structure (aka “Freed-Witten anomaly cancellation”) after A clean formulation and review is provided in….

- Discussion Type
- discussion topicfiltered (infinity,1)-category
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 3
- Last comment by zskoda
- Last Active Dec 24th 2011

I added the definition of a filtered (infinity,1)-category from HTT. Since this is performed in a simplicial model which is supposedly not to be emphasized from the nPov and I felt that the below proposition should center this article I added a sentence indicating this in the ”Idea”.

- 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 topicintuitionistic logic
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 22nd 2011

I have added the Tarski reference to intuitionistic logic and internal logic

- 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 topicclassical field theory
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Dec 20th 2011

created stub for classical field theory

- Discussion Type
- discussion topichomotopy level
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by TobyBartels
- Last Active Dec 18th 2011

created homotopy level

However, the instiki-table does not come out correctly yet. It did before I added the third column. These tables are the most delicate things. I never know why sometimes they display correctly and sometimes not.

- 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 topiclocally cartesian closed model category
- Category Latest Changes
- Started by Mike Shulman
- Comments 6
- Last comment by Urs
- Last Active Dec 16th 2011

I’ve added a definition to locally cartesian closed model category, although I’m open to debate about whether this is the right definition. This definition is more or less exactly what one needs to interpret dependent products in type theory with function extensionality (I plan to add a proof of this). But it’s certainly less obviously correct from a pure model-categorical viewpoint. For one thing, it doesn’t imply that we have a cartesian closed model category, which one would naively expect a notion of “locally cartesian closed model category” to do.

- 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 topicinterval type
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 9th 2011

created interval type

- 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 topicstrong adjoint functor
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by zskoda
- Last Active Dec 8th 2011

created strong adjoint functor

- 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.

- Discussion Type
- discussion topicSimple homotopy theory
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Dec 6th 2011

Partially spurred on by an MO question, I have started an entry on simple homotopy theory. I am also intrigued as to whether there is a constructive simple homotopy theory that may apply in homotopy type theory, but know so little (as yet) about that subject that this may be far fetched.

- Discussion Type
- discussion topicexact functor
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Mike Shulman
- Last Active Dec 6th 2011

added to exact functor the characterization of left exact functors as those preserving terminal object and pullbacks. This was previously stated only at finitely complete category.