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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.
I added a brief equivalence between two notions of characters of profinite groups that I spotted on MO.
looks like I started extremum
(I wanted minimum not to be gray at higher dimensional Chern-Simons theory…)
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.
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.
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.
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?
Have created a table of contents descent and locality - contents.
Am including this now as a “floating TOC” into the relevant entries.
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.
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.
started a stub entry Toposes on the category (2-category) of all toposes. But nothing much there yet.
I have put some content into the entry relative cohomology
stub for analytic manifold
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 started
Langlands 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.)
I noticed that there was some wild formatting at building. I have tried to tame it a bit.
a reference on non-commutative analytic spaces (Berkovich style)
started analytic spectrum (redirecting Berkovich spectrum)
I have started a table of contents measure theory - contents and started adding it as a floating toc to the relevant entries
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?
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.
A stub for the famous Solomon’s descent algebra.
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.
created Mayer-Vietoris sequence
added to diffeomorphism group statements and references for the case of 3-manifolds (Smale conjecture etc.)
I added four more references at twisted spin^c structure (aka “Freed-Witten anomaly cancellation”) after A clean formulation and review is provided in….
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.
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 is a reflective localization functor on a balanced accessible category such that the unit morphism is an epimorphism for all and the class of -local objects is defined by an absolute formula, then the existence of a suficciently large supercompact cardinal implies that 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?
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.
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 , be full is not equivalent in to being full. Moreover, in , this condition implies faithfulness. We will define (Definition 197) a notion of full arrow in a -category which, in and (symmetric 2-groups), gives back the ordinary full functors.» Note that this works only for some good groupoid enriched categories, not for , 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)?
created line object
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.
I am starting stubs
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...)
Stubs for fibrant object and acyclic fibration (also redirecting their duals, for now).
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.
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.
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”.
There is a query on category of representations. Basically asking for some references to be added???
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!
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!
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”.)
Added Thom-Federer and Gottlieb thorems to Eilenberg-MacLane space; added the remark “ in any (oo,1)-category with homotopy pullbacks” in loop space object.
have added pointers to Alex Hoffnung’s preprint to tetracategory, tricategory, span and (infinity,n)-category of spans.
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.
I noticed that cell complex was missing, so I created it
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.
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.
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.
placeholder for 1d WZW model, to be expanded
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.
Heard the rumours and wondered what it is? Now you know.
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.
I have expanded at DHR category the Idea-section and added more hyperlinks.
I have given the definition of localized endomorphism its own entry (it is otherwise sitting inside DHR superselection theory).
stub for Jacobian
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.
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).
added the recent Barwick/Schommer-Pries preprint to (infinity,n)Cat, together with a few more brief remarks.