Archive of changes made during June 2009. The substantive content of this page should not be altered.
created Eilenberg-MacLane object prompted by the blog discussion here
replied at homotopy
created Quillen adjunction
added definition to Quillen equivalence
reorganized model structure on simplicial presheaves:
expanded the idea/introduction-section
moved the original material to
fixed (hopefully) the nonsense paragraph (due to me) that Toby had pointed out at sieve
Tim:
I have pointed out at simplicial homotopy that the combinatorial description has much greater validity than claimed there before. It works for simplicial objects in any finitely cocomplete category. (I think that is in Duskin’s Memoir volume, but is not difficult to check directly.) It depends on having the copower with $\Delta[1]$, which brings me to
We have two notations for simplices in $SSet$. We use both $\Delta[n]$ and $\Delta^n$. I do not like the second one. I have started a query on this at simplicial set.
prompted by David‘s remark at homotopy group I was led to all of the following:
I split homotopy into
and moved the material originally found at homotopy to homotopy (as a transformation), kept only two commented links at homotopy and then wrote homotopy (as an operation) in the spirit of David’s remark and in the light of the entry Eckmann-Hilton duality (but just some tentative abstract nonsense so far).
then finally I added to homotopy group a last section “some abstract nonsense” that indicates how the discussion of homtopy groups could be given analogous, just dual, to the discussion the way we have it (since recently) at cohomology group
expanded at cohomology group (standard abelian examples, standard nonabelian examples)
continued polishing and adding statements and proofs to category of fibrant objects
created a stub for simplicial skeleton – much more to be said here eventually
John Baez: I polished up the entry about the book Towards Higher Categories on my personal web, and modified the nLab entry accordingly. I expect someday to have lots of my papers and books on the $n$Lab. Somehow I want them to be easy to find from the main $n$Lab, while discouraging other people from changing them (yet perhaps not making this impossible). The search feature on the main portion of the $n$Lab doesn’t search people’s personal webs, does it?
Toby Bartels: People keep linking to left adjoint and right adjoint when talking about adjoint functors, even though these redirect to the general concept of adjunction. This is very natural, so I've now created those as pages in their own right. They consist of the definitions in a variety of contexts and then refer you to the other pages for more detail. While I was creating pages that collected information from other pages, I also wrote triangle identities and weak inverse. A picture in string diagrams would be very welcome at the former.
Urs Schreiber continued filling in propositions and detailed proofs at category of fibrant objects
Toby Bartels fleshed out discrete fibration a bit.
Eric Forgy created interval category.
Zoran Škoda: created discrete fibration, quasideterminant and made additions (please check for correctness) to orthogonal factorization system, added Duskin’s reference to gerbe (general idea) (btw it emphasises on internal point of view to gerbes – as “bouquets”, missing in division of gerbe entries in nlab).
Toby Bartels responded to Todd below.
Todd: piped in on a discussion between Bruce Bartlett and Urs Schreiber at sieve; put in a related two cents at subobject.
Todd: added more material to Trimble n-category, outlining Leinster and Cheng’s extension from $n$ to $\infty$.
Eric Forgy has begun some pages related to administration of the Lab:
created a section “Category theory for Trimble $n$-categories” at Trimble n-category and put in a query box with a list of questions – I’d be interested in whatever partial answers and comments
created Approaching Higher Category Theory (following blog discussion here and here)
added to point of a topos a section “enough points” with discussion about what it means for a topos to have “enough points” and two examples
created QFT with defects
added to cobordism hypothesis more details on formalization and proof by J. Lurie
reworked (infinity,n)-category of cobordisms
split off n-fold complete Segal space from (infinity,n)-category
created (infinity,2)-category
Urs:
created gerbe (in nonabelian cohomology) based on the blog discussion here
removed the former section “references” at cohomology and replaced it with a new section “History and references” by using the material of my blog comment here
added to Trimble n-category a section “basic idea” with a paragraph that John suggested to include (here)
added discussion that Kan complexes form a category of fibrant objects to category of fibrant objects that is slightly more direct than the argument using the full model category structure (crucial point being the theorem now at model structure on simplicial sets that acyclic fibrations of simplicial sets are characterized by a right lifting property).
added the example of a functor that is an equivalence of groupoids as inducing isomorphisms of simplicial homotopy groups under the nerve to simplicial homotopy group
created right proper model category
started filling in detailed proofs at category of fibrant objects – am thinking that maybe eventually now the entry should be broken apart into a brief overview and one containing all the details
Eric Forgy has a request at notation.
David Roberts: created fibration theory. Also a comment at gerbe (general idea) about the fibration theory axioms.
Todd: Recently loop space was created. Also, I added some examples to Chu construction and to star-autonomous category.
Urs:
added the example of principal bundles to fibration sequence
added an example section to group cohomology with details of how the abstract-nonsense definition reproduces the familiar formulas
created twisted cohomology
created n-group
created group cohomology
fiddled a bit with the entry higher category theory (added one more introductory sentence, created a hyperlinked list of definitions of higher categories) but I still feel that we should put more energy in this particular entry. It is sort of the single central entry one would expect an “$n$Lab” to be built around, but currently it doesn’t even come close to living up to playing such a pivotal role. I am imagining that it should carry some paragraphs that highlight the powerful recent developments in view of Pursuing Stacks, of the kind that I filled in today in the entry Carlos Simpson. Does any higher category theory expert out there feel like writing an expositional piece for the $n$Lab here?
created simplicial localization but was then too lazy to draw the hammock. But main point here is the link to an article by Tim Porter that nicely collects all the relevant definitions and references
created principal infinity-bundle, just for completeness
created principal 2-bundle – this is just the result of what came to mind while typing, I am sure to have forgotten and misrepresented crucial aspects. Toby Bartels should please have a critical look and modify as necessary, as should Igor Bakovic and Christoph Wockel in case they are reading this.
based on a reaction I received concerning my comment below on the entry gerbe I have split the material into entries gerbe (as a stack), gerbe (general idea), bundle gerbe and kept at gerbe only pointers to these entries – let me know what you think
created an entry for Carlos Simpson motivated by a link to a recent pdf note – that Zoran Skoda kindly pointed me to – where Simpson briefly sketches the topic of higher stacks, old and recent progress and putting his own contribution into context. I thought that was a nice short comprehensive collection of keywords and so I reproduce that text now at the entry Carlos Simpson with all the keywords hyperlinked
created fibration sequence
added an “Idea”-section to gerbe supposed to be read as “general idea”, where I try to describe the concept in a way independent of the notion of stack, relating it to princpal bundles, principal $\infty$-bundles, fibration sequences and cohomology . At the end I say “in the following we spell out concrete realization of this idea”. Then I made Tim Porter‘s material a section “Realization of gerbes as stacks”. Eventually I’d like to add similarly “Realization of gerbes as Stasheff-Wirth fibrations”, “Realization of gerbes as bundle gerbes” etc.
Tim:
I have added material to gerbe, although it is still a long way from explaining the link with cohomology with integer coefficients that was requested.
In the process of doing the above I have added a deconstruction section to torsor, and created trivial torsor.
I created a brief entry on Jean-Luc Brylinski, but this is really a stub with a link to Wikipedia.
Finn Lawler: Added a proof (sketch) of the ‘lax Yoneda lemma’ to lax natural transformation. Also replied re terminology at modification (thanks to Urs for a fantastic reply to my question there).
Eric: I’ve started the process of applying redirects for symbolic links. This maintains the original ascii titles and urls, but makes links look much better. I’ve also started adding redirects for plural nouns, e.g. you no longer have to type [[category|categories]]. Now you can simply type [[categories]] and will be redirected automatically to category. To see the changes in action, have a look at higher category theory. Feedback welcome!
Zoran Škoda: Created Hopf envelope, free Hopf algebra and matrix Hopf algebra simultaneously with posting a related comment to a discussion between Baez, Trimble and Vicary on free and cofree functors for bialgebras.
Tim: I have created a stub on gerbes. There was a request on the Café for en entry and I thought this was a good way to remind me to start one. … but feel free to do it for me!
Created class of adapted objects. It would be better to say class of objects adapted to a functor (we talk half exact additive functors between abelian categories). Created Grothendieck spectral sequence and spectral sequence. Both need more input/work…
I have added a general paragraph at the beginning of descent because far not all the cases of descent theory fit into the framework of sheaf/stack theory, Grothendieck topologies and homotopical methods. Namely something is a sheaf when it satisfies the descent for all covers, but there are cases when one considers just the descent problem for a single morphism, not all morphisms of anything like a Grothendieck topology. There is also a descent along families of noncommutative localizations, when one needs to resort to noncommutative generalizations of Grothendieck topologies. So descent theory is more general than the geometry of sheaves and stacks, though the latter is surely he most important part.
Urs:
created K-theory spectrum with just a question
created a stub for tmf based on a recent Category Theory mailing list contribution – am thinking that we should more generally try to move good stuff from the mailing list into $n$Lab entries
am offering a reply to Finn Lawler at modification
Toby Bartels: Just more questions, instead of answers, for Finn and Gavin.
Gavin Wraith wrote matrix theory, tensor product theory, and bimodel, with a question at the last.
Finn Lawler: created modification (and asked a question there) and lax natural transformation, including a statement of what I think is the Yoneda lemma for them. I have a proof (I think), which I’ll add a sketch of later, if no-one finds the statement obviously wrong in the meantime.
added to weak factorization system the full details of the (elementary) proof that morphisms defined by a right lifting property are stable under pullback –
added the theorem to Kan fibration that Kan fibrations between Kan complexes that are also weak equivalences are precisely those with the right lifting property with respect to simplex boundary inclusions
added example of fundamental $\infty$-groupoid = singular simplicial complex to Kan complex
edited Kan complex slightly and added two little propositions characterizing groupoids in terms of their Kan complex nerves
added further illustrations to Kan fibration
Toby Bartels: A section on measurable space about the constructive theory due to Cheng. (This is not a straightforward variation on the classical theory but distinctly a generalisation, so may be of interest even to classical mathematicians.)
Urs linked to normal complex of groups by a little lemma now added to Moore complex
Urs:
worked on simplicial group:
added section about the adjunction between simplicial groups and simplicial sets, in particular mentioning the free simplicial abelian group functor (since that plays a role in the proof at abelian sheaf cohomology, which now links to it)
moved statement and proof that every simplicial group is a Kan complex into a formal theorem-proof environment
added references to Goerss-Jardine, to Tim Porter‘s Crossed Menagerie and to Peter May’s book “Simplicial objects in algebraic topology”
added an “Idea” section
added to Moore complex
the theorem of how it relates to the other two complexes in the game (it is quasi-isomorphic to the alternating sum complex and isomorphic to that divided by degeneracies)
illustrations of cells in low degree
the reference to Tim Porter‘s The Crossed Menagerie notes
a brief comment on use of terminology (after checking with Tim Porter)
Toby Bartels: Started more articles that I'd linked earlier: exclusive disjunction (with a philosophical claim that I need to back up), countable choice (very much a stub), sigma-ideal, Hartog's number.
Urs:
added the detailed formal proof to abelian sheaf cohomology that shows how it is sitting as a special case inside the more conceptual nonabelian cohomology (= hom-sets of infinity-stacks, really)
Toby Bartels: Wrote Moore closure, since I linked it and it's a neat idea. As I was writing it, I realised that it was more abstract than what I'd been writing lately and must correspond to something very nice categorially; by the time I was done, I realised what it was: a special case of a monad. Then I saw that there were really no examples of monads on our page!, so I added a few, but the majority of examples of monads on the wiki now are Moore closures. (And believe me, Moore closures are everywhere.) Anyway, if you're trying to understand what monads are, why not try Moore closures first?
added an “Idea” section to Moore complex, edited the section headers a bit – and have a question onm terminology: isn’t this really the “normalized chain complex” whereas the Moore complex is the one on all cells with differential the alternating sum of face maps?
created organization of the nLab – please see there for what this is about (or in fact, see the corresponding nForum thread that is being linked to there)
added references to Goerss-Jardine’s Simplicial homotopy theory to Dold-Kan correspondence and Moore complex
reacted a bit at An Exercise in Kantization – behind the scenes this is being developed further, I’d be happy to provide more detailed replies in a while, when things have stabilized a bit more
added a mention of and a link to Kan lift to the “Idea” section of Kan extension
replied to the discussion about pullback notation at Kan extension: I originally had “$p^*$” there. After somebody changed that to “$p_*$” I wrote the section “note on terminology”. I’d be happy to have the $p^*$ reinstalled.
Todd:
I have a notational comment at Kan extension. Spurred by David Corfield‘s post at the Café, I hope to get started on Kan lift soon. (Done.)
There’s a running discussion between Toby Bartels and me at cyclic order, centering on whether Connes’ cycle category $\Lambda$ has been correctly characterized, or if not how to fix it. We agree now that a notion of “total cyclic order” is classically equivalent to the “linear cyclic order” notion used in the article, and equally feasible for purposes of trying to characterize $\Lambda$, but I’m currently perplexed by the apparent presence of a terminal object.
Tim: Finally I have created 2-crossed complex, and in the process needed to create normal complex of groups. What would be nice is to work out exactly what $\infty$/$\omega$-groupoids correspond to 2-crossed complexes. Any ideas? (It probably is obvious viewed from the right perspective but I fear I do not yet have the right perspective to say ‘aha!’) I can characterise these objects in homotopy theoretic language, but really would like some neat way of describing them in $\infty$-cat terms.
Todd Trimble: asked a question of Mike and Toby over at cyclic order.
Tim:
I have added more material to 2-crossed module including some exercises (at the foot of the ‘page’! (Have fun!) I will not get around to doing an entry on 2-crossed complexes today.
I have adjusted homotopy coherent nerve in an attempt to answer some of the points made there by Todd,
Urs:
almost missed Tim Porter‘s addition about the Dwyer-Kan loop groupoid to simplicial homotopy group – that sounds very good, I’d be happy if we make this the default point of view at that entry and derive the more traditional description only as a special case from that
added the example of the bar construction of a group $G$ as the nerve of $\mathbf{B} G$ to nerve
added illustration to Kan fibration
added illustrative diagrams to boundary of a simplex and horn
added a section with details on ordinary nerves of ordinary categories to nerve
Todd Trimble keeps creating analysis pages that I link to; today's were measurable space and topological vector space.
Tim:
I finally added some stuff into crossed complex giving more information on the link with simplicial group and Moore complex. I hope to get around to creating 2-crossed complex in a day or two!
The discussion on simplicial homotopy group possibly needs more opinions! I just added a bit more of my viewpoint to Urs’s thoughts box, but I do find that I am not sure what the idea / context / viewpoint (or whatever) of this entry should ‘optimally’ be!
Eric: In an attempt to understand Kan extension, I cooked up an example of a functor and added it to functor. Let me know if I made any mistake.
Toby Bartels: Instead of changing links to path-connected space (which doesn't yet exist) to connected space, I instead added “o connected space. That way, if we ever decide to separate out a new page path-connected space, we don't need to go through all of the links and fix them!
Urs: added the defintion of the product to simplicial homotopy group
Todd: wrote the beginnings of an apparently long-awaited article on weighted colimits.
filled in the definition at simplicial homotopy together with an intentionally pedestrian proof that simplicial homotopy in a Kan complex is an equivalence relation
slightly polished the “remarks” section at simplicial set
Todd:
wrote measurable space, including material on measures and basic integration theory.
responded to queries of David Roberts at covering space and locally path-connected space. David wondered whether some material at covering space ought to be moved to universal covering space. I think we’re covering similar material but with slightly different emphases; I’d like to see David put his imprimatur on the article on universal covering spaces, but leave covering space essentially intact while the material continues to settle into place; I’m thinking of adding more to it anyway (so that “to be continued” might still be appropriate).
Urs:
added a reference (and minor comments on references) to model structure on simplicial presheaves
added an opinion to simplicial homotopy group
Andrew Stacey: Changed the example in topological concrete category for generalized smooth space since I think that all the examples listed there are topological over set.
Edited covering space as requested by Todd. I still wonder whether some of the discussion would be better off at universal covering space.
Created stubs at semi-locally simply-connected space and locally path-connected space.
Toby Bartels: Fixed mistakes at topological concrete category, thanks to having a good online reference.
Todd:
Tim:
Todd:
Tim:
Urs: I have a comment and appeal at nInsights
in this context I also want to ask again everybody:
please don’t forget to drop latest changes logs here alerting us of which changes you made where, if it’s anything beyond fixing typos
if you feel something needs to have a place in the nLab, don’t feel hesitant to add it. I am getting the impression there is in parts some hesitation to insert material without checking with everybody else first. I’d say more efficient would be: add the material you would like to see, and if it should really turn out that there is serious disagreement with some other contributor, we can still roll back things and look for compromises.
Urs:
expanded inverse image: added an “Idea” section, restructured discussion into presheaf and sheaf and sheaf on top-space bits, added the crucial lemmas and theorems and provided detailed proofs of some of them.
replied again at simplicial homotopy group: Tim or Toby should please feel free to implement the remaining terminology adjustment
reacted and replied at simplicial homotopy group (also have a question) and added a few further bits
Tim:
created simplicial homotopy group
added to model structure on simplicial sets a puny beginning of a discussion of what is supposed to eventually become a discussion of the relation to model structures on strict $\infty$-groupoids
created omega-nerve and model structure on strict omega-groupoids
expanded a bit on the discussion at the beginning of model structure on simplicial sets
added detailed proof to geometric morphism of the fact that geometric morphisms $Sh(X) \to Sh(Y)$ are bijectively induced from continuous functions $X \to Y$
Tim: I have continued at homotopy coherent nerve. This ended up with several new links to not-yet-existing pages!
created homeomorphism and local homeomorphism just to satisfy links
added theorem about stalkwise characterization of epi/mono/isos of sheaves to stalk and included an example
added a bit more to string structure (discussion of description in terms of class on total space, also some references, but very incomplete) - but still not done
split off remaining Friday session Oberwolfach Workshop, June 2009 – Friday, June 12 for Oberwolfach Workshop, June 2009 – Strings, Fields, Topology following Toby
Todd Trimble created connected space.
Toby Bartels: Oberwolfach Workshop, June 2009 – Strings, Fields, Topology was getting too long for me to handle, so I removed the duplication of abstracts and broke the rest up by day.
Tim: I have had a go at homotopy coherent nerve. Still more to add though.
started creating string structure but had to quit before fully finished – will continue tomorrow
created point of a topos and stalk
created Oberwolfach
pasted notes on Thomas Schick’s talk on differential cohomology at Oberwolfach Workshop, June 2009 – Strings, Fields, Topology into differential cohomology
created
but more like stubs to get some material in place, didn’t find the leisure to really polish this
started adding links to keywords in the list of abstracts that Bruce added to Oberwolfach Workshop, June 2009 – Strings, Fields, Topology – am hoping that eventually we’ll fill these links with content, partly just by copy-and-pasting the relevant bits from the lecture notes (as a start)
Tim:
Mike, ditto from me!
Made some changes to simplex category and join of simplicial sets. A more explicit mention of ordinal sum has been made and some minor corrections done.
Congrats Mike!
Added lots of talk summaries and notes to Oberwolfach Workshop, June 2009 – Strings, Fields, Topology. Lots of editing help needed… what would be nice would be to have the links to the notes for the talks next to the talk titles — both my notes as well as those of Gabriel Drummond-Cole (by linking to his webpage).
Urs: I gather it is in order to express congratulations to Mike Shulman given the latest changes to his page
Toby Bartels: I think that the (unordered) notion of category with inverses at partially ordered category should be the same as that of dagger category.
Victor Porton: Some other newer terms added (notably partially ordered category) to Categories and Sheaves.
Todd Trimble: I finally found some time to write up more of what I had planned for covering spaces, and tried to link up with the comments made by David Roberts. David and Urs, please take a look.
Urs:
created tetracategory
added Thursday talks to Oberwolfach Workshop, June 2009 – Strings, Fields, Topology
replied to Tim Porter at join of simplicial sets
Bruce: Added some comments abotu query boxes in personal nLab pages in the HowTo page.
Tim: I have added a query to join of simplicial sets. Am I missing something, but I find the initial definition confusing as it does not mention the ordinal structure nor ordinal sum. (There may be a subtlety that I am missing so …)
New pages: doctrine, pretopological space.
New pages: Green-Schwarz mechanism, quasigroup.
Urs Schreiber added Wednesday talk notes to Oberwolfach Workshop, June 2009 – Strings, Fields, Topology
More new pages: Freudenthal suspension theorem, replete subcategory, strictly full subcategory.
Zoran Škoda: much expanded orbifold, created Lie group, compact Lie group, orbispace.
created orbifold and orientifold
added further talk notes to Oberwolfach Workshop, June 2009 – Strings, Fields, Topology
added notes on first part of lecture by Thomas Schick on differential cohomology to Oberwolfach Workshop, June 2009 – Strings, Fields, Topology – am hoping to eventually move a polished version of that to differential cohomology
New pages:
Urs Schreiber: created Oberwolfach Workshop, June 2009 – Strings, Fields, Topology, linked to from this blog entry (am hoping to get around to add lots of further keyword links to existing $n$Lab entries, but maybe won’t)
Todd Trimble: asked Alex a question at Alex Hoffnung.
Bruce: Added link to notes to the strings, fields and topology workshop at Oberwolfach. Also added a “How To” section for creating CSS gismos like query boxes on your personal nLab wiki space.
John: I created myself a new personal web since I lost the password for my old one — and besides, the name of my old one was nonstandard. So far the only thrilling feature of this new web is the introduction to a paper I’m writing with James Dolan, tentatively titled ‘Doctrines of Algebraic Geometry’.
Mike: Tried to distill a bit of the cafe discussion about the empty space.
Todd Trimble: gave a proof of Zorn's lemma. Wouldn’t mind expanding that entry to include the mutual equivalence between AC, Zorn, and well-ordering principle (assuming excluded middle). May get around to putting in something at Hausdorff maximality principle.
Mike: Inspired by gauge space, created prometric space.
Todd Trimble: Added some details on categorical operations on the category of Banach spaces.
Andrew Stacey: replied to Mike Shulman at Froelicher space
Andrew proved a theorem about Hausdorff Frölicher spaces and the relationship to limits and colimits of manifolds.
Bruce added some stuff to Section 4 geometric infinity-function theory (with about query $QC(X)$ when $X$ is an $\omega$-groupoid internal to dg-manifolds) and ticked some things.
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