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- 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 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.

- Discussion Type
- discussion topicAT category + Steve Lack
- Category Latest Changes
- Started by Tim_Porter
- Comments 4
- Last comment by Todd_Trimble
- Last Active Dec 5th 2011

Steve (Lack) has put a comment box on AT category. I have not been following that entry so am not able to reply to his point.

- Discussion Type
- discussion topic1d WZW model
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 5th 2011

placeholder for 1d WZW model, to be expanded

- Discussion Type
- discussion topicn-truncated object in an (oo,1)-category
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Dec 5th 2011

I filled in content at n-truncated object of an (infinity,1)-category.

to go with my discussion with David Roberts. I had planned to go further and also write the entry on Postnikov twoers, but got distracted all day.

Apart from that I just added this link to Higher Topos Theory and did some editing there, added a table of contents, expanded the floating toc.

- Discussion Type
- discussion topicDream mathematics
- Category Latest Changes
- Started by TobyBartels
- Comments 5
- Last comment by TobyBartels
- Last Active Dec 4th 2011

Heard the rumours and wondered what it is? Now you know.

- Discussion Type
- discussion topicReflexive Banach space
- Category Latest Changes
- Started by Yemon Choi
- Comments 9
- Last comment by TobyBartels
- Last Active Dec 4th 2011

I started editing the page on reflexive Banach spaces - in particular I corrected the definition and stuck in a mention of "James space". A link or reference is needed but I am currently a bit too frazzled/stressed to do further editing today.

- Discussion Type
- discussion topicDHR category
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Tim_Porter
- Last Active Dec 2nd 2011

I have expanded at DHR category the Idea-section and added more hyperlinks.

- Discussion Type
- discussion topiclocalized endomorphism
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 1st 2011

I have given the definition of localized endomorphism its own entry (it is otherwise sitting inside DHR superselection theory).

- Discussion Type
- discussion topicJacobian
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by zskoda
- Last Active Nov 30th 2011

stub for Jacobian

- Discussion Type
- discussion topicOutside, looking in
- Category Latest Changes
- Started by Tom Leinster
- Comments 59
- Last comment by Todd_Trimble
- Last Active Nov 30th 2011

Joel Hamkins and Andy Putman made some comments about the nLab on MathOverflow, beneath an answer by Andrew.

It’s interesting to know what people’s perceptions are, even if they’re wrong. (And I would think that Andy P’s perception

*is*wrong.) I don’t know what Andrew S has in mind when he says that Joel’s point is extremely easy to answer.

- Discussion Type
- discussion topicContractibility of embedding spaces
- Category Latest Changes
- Started by Andrew Stacey
- Comments 6
- Last comment by Andrew Stacey
- Last Active Nov 30th 2011

Following a discussion on the algebraic topology list, I’ve written a proof of the contractibility of the space of embeddings of a smooth manifold in a reasonably arbitrary locally convex topological vector space. The details are on embedding of smooth manifolds and it also led to me creating shift space (I checked on MO to see if there was an existing name for this, and Bill Johnson said he hadn’t heard of it).

- Discussion Type
- discussion topic(infinity,n)Cat
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 30th 2011

added the recent Barwick/Schommer-Pries preprint to (infinity,n)Cat, together with a few more brief remarks.

- Discussion Type
- discussion topictwisted bundle
- Category Latest Changes
- Started by Urs
- Comments 28
- Last comment by Urs
- Last Active Nov 28th 2011

I have been working on the entry twisted bundle.

Apart from more literature, etc. I have started typing something like a first-principles discussion: first a general abstract definition from twisted cohomology in any cohesive $\infty$-topos, then unwinding this in special cases to obtain the traditional cocycle formulas found in the literature.

Needs more polishing here and there, but I have to pause now.

- Discussion Type
- discussion topicMaurer-Cartan equation
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Nov 28th 2011

I have added to Maurer-Cartan equation a brief section MC elements in L-infinity algebras and then a Properties section As homomorphisms of L-infinity algebras where I explain how MC elements are secretly homs of dg/$L_\infty$-algebras.

- Discussion Type
- discussion topicLie bialgebra
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Nov 28th 2011

created Lie bialgebra, but so far just a comment on their quantization.

- Discussion Type
- discussion topiccooperad
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Nov 28th 2011

stub for cooperad

- Discussion Type
- discussion topicconnected object
- Category Latest Changes
- Started by Todd_Trimble
- Comments 2
- Last comment by Urs
- Last Active Nov 28th 2011

Added a new Properties section to connected object. Including a theorem which is a bit of a hack (where I leave it to others to decide if ’hack’ should be interpreted positively or negatively!).

- Discussion Type
- discussion topicsyntax and semantics
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 25th 2011

while polishing up type theory - contents I felt the need for entries called syntax and semantics. I have created these just so that the links to them are not grey, but I put in only something minimalistic . I could add some general blah-blah, but I’d rather hope some actual expert feels inspired to start with some decent paragraphs.

- Discussion Type
- discussion topicproofs as programs
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 24th 2011

created proofs as programs

- Discussion Type
- discussion topicbicategorical pasting
- Category Latest Changes
- Started by Todd_Trimble
- Comments 2
- Last comment by Todd_Trimble
- Last Active Nov 23rd 2011

Added to pasting diagram a reference to the bicategorical pasting theorem given by Verity in his thesis.

- Discussion Type
- discussion topic(hyperconnected,localic) factorization system
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 22nd 2011

just for completness, I have split off a brief entry (hyperconnected,localic) factorization system.

- Discussion Type
- discussion topicfactorization systems - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 20th 2011

I have created a list factorization systems - contents and included it as a “floating TOC” into the relevant entries.

- Discussion Type
- discussion topicnormal field etension
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Nov 20th 2011

Created a stub for normal field extension. (and missed out the x in the title of this page.)

- Discussion Type
- discussion topicnotions of subcategory
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Nov 19th 2011

have created a “floating TOC”

*notions of subcategory*and have included it in the relevant entries

- Discussion Type
- discussion topicindexed adjoint functor theorem
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 18th 2011

stub for indexed adjoint functor theorem

- Discussion Type
- discussion topicreflective factorization system
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 17th 2011

In the Definition-section at reflective factorization system I found the “$\Psi$” and “$\Phi$” used in the text oppositely to how they appear in the displayed diagram. I think I have fixed this.

- Discussion Type
- discussion topicStiefel-Whitney class
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by TobyBartels
- Last Active Nov 16th 2011

added to Stiefel-Whitney class briefly the definition/characterization.

- Discussion Type
- discussion topicUniform continuity
- Category Latest Changes
- Started by TobyBartels
- Comments 4
- Last comment by Mike Shulman
- Last Active Nov 13th 2011

I finally wrote uniformly continuous map. Pretty much just definitions.

- Discussion Type
- discussion topicmore on isometries on normed vector spaces
- Category Latest Changes
- Started by Todd_Trimble
- Comments 1
- Last comment by Todd_Trimble
- Last Active Nov 11th 2011

After contributing to the article on parallelogram identity, I added to isometry and created Mazur-Ulam theorem. The easy proof added at isometry, that shows an isometry $E \to F$ between normed vector spaces is affine if $F$ is strictly convex, might lead one to suspect that the proof under parallelogram identity was overkill, but I think that’s an illusion. Ultimately, I believe the parallelogram identity is secretly an expression of the perfect ’roundness’ of spheres, connected with the fact observed by Tom Leinster recently at the Café that the group of isometries for the $l_2$ norm is a continuum, whereas for other $p$ in the range $1 \lt p \lt \infty$, you get just a finite reflection group (this is for the finite-dimensional case, but there’s an analogue in the infinite-dimensional case as well).

The Mazur-Ulam theorem removes the strict convexity hypothesis, but adds the hypothesis that the isometry is surjective. The conclusion is generally false if this hypothesis is omitted.