Archive of changes made during April 2009. The substantive content of this page should not be altered. For past versions of this page beyond its own history, start here and work backwards.
Urs:
continued filling in material at geometric infinity-function theory – am hoping that my co-journalists will eventually start helping me there
replied to David at why (infinity,1)-categories?
added
Urs:
Urs:
Urs:
Mike:
Urs:
based on Zoran’s references at enhanced triangulated category I created pretriangulated dg-category and twisted complex, but then ran out of steam
moved a bit of material from derived infinity-stack to derived stack and then made derived infinity-stack a redirect to derived stack
finally wrote at least a blurb at stack, only to make it look less orphaned in between sheaf and (infinity,1)-sheaf.
created Verdier site
further fine-tuned the DHI-review at descent: now I dropped the discussion of homotopy limits entirely, as it’s not really necessary; but I did include for a smoother presentation the assumption that we are on a Verdier site, so that hypercovers “split” (section 9) which happens to be a Reedy fibrancy kind of condition after all (page 11)
I browsed a bit through Dominic Verity’s work and created entries on stratified simplicial set, complicial set, weak complicial set, simplicial weak omega-category and Verity-Gray tensor product – my main motivation was the claim now recounted at stratified simplicial set that the $\omega$-nerve on strict $\omega$-categories with values in $Strat$ has a strong monoidal left adjoint
Urs:
filled the “details” section at descent for simplicial presheaves with the relevant material copy-and-pasted from descent.
keep polishing, expanding and rearranging descent – when Mike comes back online I am hoping to discuss a bit more the relation between Street’s descent for $Str \omega Cat$-valued presheaves and the standard descent for their SSet-valued image under the $\omega$-nerve. As the new version indicates: the homotopy limit may be a red herring and the lack of monoidalness of the left adjoint of the nerve might be fixed by recourse to stratified simplicial sets using Verity’s results (?)
added a pullback description to double comma object
Mike:
Mike:
Urs:
created Bousfield-Kan map
created category of simplices
following Toby’s suggestion I moved descent and codescent to descent – then I entriely rewrote it! Now it starts with very general nonsense on localization of $(\infty,1)$-presheaves and then derives descent conditions as concrete realizations of that localization. Currently where it ends I am planning to add discussion about how to further get from descent to gluing conditions (i.e. to $\Delta$- and oriental-weighted limits) following discussion that I am having with Mike on the blog here
added more details to ind-object, relating the two different definitions
added standard examples of presheaves on open subsets to inverse image
started adding a list “properties” to colimit analogous to the one at limit
Mike:
Added a couple new examples, and tried to uniformize the descriptions, at A-infinity-operad.
Created indiscrete category.
Urs:
added a discussion of hom-objects in terms of homotopy limits at simplicial presheaf (in a new section “Properties”)
created reflective (infinity,1)-subcategory and localization of an (infinity,1)-category and local object
created hypercompletion and descent for simplicial presheaves
created associahedron (see the discussion with Jim Stasheff over at the blog, here)
expanded A-infinity-operad: a bit about associahedra, but mostly more detailed links to references
Urs:
came across the useful interrelation diagram and associated literature list on “enhancements of triangulated categories” here and added this to stable (infinity,1)-category, together with some links it suggests, to entries still to be created – I am hoping we’ll eventually be able to accumulate a good collection of material on this topic
added Lie $\infty$-links to rational homotopy theory
added missing links back and forth between Yoneda extension and free cocompletion
added to generalized universal bundle the remark that it is a means to compute the “lax pullback” (really: comma object) of a point.
added at category of elements the equivalent definition in terms of comma category and in terms of pullbacks of the universal Set-bundle – and in terms of “lax pullback” (comma object) of the point
added the discussion and diagram at comma category characterizing it as a comma object
made the comma category explicit on which a simplicial local system is a functor
Tim: I have created a few entries relating to the interaction of local system with ideas from rational homotopy theory, especially algebras of differential forms on simplicial sets, based on Sullivan and further back Thom and Whitney. These included simplicial local system, see Urs comment below, to which I have started replying. Perhaps I will be able to add more shortly. These entries are not yet finished and do not yet deal with the Sullivan-Thom-Whitney stuff.
Urs: have some questions at simplicial local system
Mike: Since no one objected to my proposal on how to resolve the duplication between category of fractions and multiplicative system, I implemented it. The relevant material is now at calculus of fractions. I deleted a bit of the material about derived functors because it was not really specific to calculi of fractions, belonging more at derived functor.
Toby Bartels: A question (not a dispute!, what do you know?) on terminology at exponential object.
Mike:
Urs:
created path groupoid – other realizations of that idea should be stated there, too
started creating a random list of some examples at limits and colimits by example – but not in the intended detailed form yet
started reworking local system as we discussed there – still lots of room for improvement left, of course!, in particular many references could use more details and links
Urs:
have a reply at local system
found time to work a bit more on Lie infinity-algebroid representation
Toby Bartels: Please note that there are no actual links to Differential Nonabelian Cohomology (except this one just now). That there appear to be is actually a bug in Instiki (which I haven't bothered to report to Jacques yet).
Tim:
I have added a request at local system. Basically the current entry reads as if it related to a relatively recent idea. I suggest we look at the origins of the idea, at least as old as ‘Steenrod (1943)’ if not before. It is central to much of the nLab work. Probably we need to be much less restrictive in the motivation of this entry.
I have added some historical and motivational perspective in twisting function and would suggest that a similar section is needed for twisting cochain. The two threads of twisting the fibre and deforming the local structure of a ‘product’ are at the origin of both concepts.
Urs: created Lie infinity-algebroid representation – but ran out of time before done with polishing
Mike: Created cyclic order in order to propose a clean definition of the cycle category.
Zoran Škoda: created quasicoherent sheaf,kernel functor, Gabriel composition of filters, Gabriel filter, uniform filter, Serre subcategory. Corrected Gabriel multiplication, thanks Toby. Created ringed space differing from ringed site.
Urs: edited geometric infinity-function theory to go along with this blog message
Urs:
have two questions on examples at semi-abelian category
created dependent product just to satisfy the link from universe in a topos
added a list with a handful of general properties to adjoint functor
moved the old discussion at representable functor to the bottom of the page
Zoran Škoda: I have made D-module and local system somewhat more precise; actually I have put lots of more precise statements; the subtleties on wheather we work over a complex manifold, variety, variety in char zero, or nonsingular variety in char 0, may affect some of the statements. To suplement this I was forced to create a comprehensive entry regular differential operator. I see that for some reason people continue talking connections and avoid going down to sheaves, resolutions of diagonal, de Rham site and regular differential operators, which are all necessary to cover this subject properly in my view; there are missing related items like holonomic D-module, treatment of costratification, crystals and so on. I created coreflective subcategory, just giving the definition and saying that the rest of abstract preoprteis are dual to reflective subcategory where more is written. But one should write specific examples which call specifically for coreflective subcategories. Created topologizing subcategory, thick subcategory and Gabriel multiplication in the generality of abelian categories (one should add the proper discussion of thick, topologizing in triangulated and suspended categories, and Gabriel mult. for filters, but I run out of energy for today); one needs to add entry on Serre subcategory which is easy in module and Grothendieck categories, but more subtle in general abelian categories. There is a query under fibered n category, I think we should have both entry fibered n category and n-fibration; the first entry dedicated to STRICT n-categories and consequently strict universal properties and the name due Grothendieck, Gabriel, Gray and Hermida; and the latter in weak version and with homotopy style nomenclature accordingly. My praise for creative expansion to Mike.
Urs:
created stub for local system in the context of a comment I left here
created derived stack to go along with our Journal Club activity
Toby Bartels: I think that the naming discussion at ind-object is still current until Eric is happy.
Finn Lawler: Replied to Toby at minimal logic, and slightly expanded paraconsistent logic to incorporate some of our discussion.
Urs:
Toby Bartels: Put simulations everywhere. I need to think about this (and read Aczel) some more, but I'm fairly sure that Mike is right about them.
Urs:
started polishing the typesetting at bundle gerbe, but there is still plenty of room for further improvement
added a summary list to the section “Example: universes in SET” at universe in a topos
Mike:
Added the correct equivalent definition to semi-abelian category and deleted the discussion about what it might be.
Added another version of Cantor's theorem.
Expanded extensional relation to discuss several possible notions of extensionality, reserving the term ‘extensional’ for the one that I think is most important (but feel free to disagree).
Commented on the right notions of morphism for relation, well-founded relation, well-order, and extensional relation.
Urs:
Finn: No, you haven’t – I was wrong.
Urs:
Zoran Škoda: created comodule, flat module, cotensor product.
Toby Bartels: Added information on morphisms to relation, extensional relation, well-founded relation, and well-order. I hope that all of my claimed theorems are true, in which case I'm sure that all of my proposed definitions are good. (^_^)
made a request at Mike’s categorified logic
created D-module
created coherent logic
added John’s blog exposition to induced representation. Is the style OK?
Urs:
expanded hom-set a bit
remarked at semi-abelian category that the now deprecated second equivalent definition was taken directly from the (single) reference we give. I think instead of just removing it we should try to correct it.
Zoran Škoda: created Tohoku, quasi-pointed category, made changes to sheafification,additive and abelian categories, torsion, torsion subgroup
Urs:
added some links to Tim’s latest addition to rational homotopy theory – in that context I created torsion and torsion subgroup
finally created exact sequence which was requested by a bunch of entries
added to the “Idea” section of Grothendieck category a statement suggesting that these are precisely those additive categories for which sheafification exists. Is that right?
Zoran kindly looked up the definition of AB6-category which was still missing at additive and abelian categories, I put it in now
Tim:
I created the next entry in the rational homotopy lexicon series with the ungainly title differential graded algebras and differential graded Lie algebras-relationships.
I added another viewpoint to rational homotopy theory which is more in keeping with Quillen’s 1969 paper.
Urs:
created GUT and induced representation as places for collection of material currently discussed on the blog
included Todd’s proof of MacLane’s co-Yoneda into co-Yoneda lemma
tried to bring A-infinity-category into some shape by adding more introductory discussion and ordering the references a bit – also have a question
Tim:
Replied to Urs at bar and cobar construction (at least I hope the reply goes some way to answering the query).
Created reduced suspension as I needed it for my ‘reply’ above.
Urs:
replied to David at ind-object
have a request and a question at bar and cobar construction
Zoran Škoda: created fibered n category, Karoubian category, pseudo-abelian category (redirect), Koszul duality, pure motive, Voevodsky motive, motives and dg-categories (there needs to be a separate big entry on mixed motive, then motivic complexes, standard conjectures, Hodge filtration etc.), element in abelian category; created book-page Gray-adjointness-for-2-categories; additions and references to A-infinity category, dg-category, twisting cochain.
Urs:
created cofinal functor and cofinally small category, but just the bare defintion so far
expanded ind-object: added more motivation in “Idea” section, added examples, added list of properties
Urs:
added a discussion to universe in a topos with more details on how to get back the Grothendieck universe axioms in $SET$. Please check. I’d be grateful for improvement.
to Andrew: I think we want here on the $n$Lab as much detail as we can get hold of – if an entry becomes too long, though, it might be an option to split off entries from it “more details on xyz” or the like and link to them
Andrew Stacey: Added the basic definitions to Chen space and some of the other variants of generalised smooth space. How much detail do we want on these pages?
The English Pedant: I’m not convinced about the use of the word “heuristic” on the n-Lab. I’ve started a discussion on the n-Forum rather than force my views on the English language on the n-Lab. I realise that this is a little against the Wiki-spirit but I figured that if I went through changing all occurrences of the word “heuristic” then someone would object and we’d have a discussion about it; so to save a bit of agro, I’m instigating the discussion first.
Toby Bartels: I started a new category, foundational axiom
, in which I put the pages that contain axioms that one might (or might not) want in one foundations of mathematics. That way, anyone with opinions on the matter can check them to see that one's views are represented. (A few don't really have much in the way of an axiom right now, but one could be added or noted there.) This does not include things like the axioms for a group, but rather axioms for set theory (or other foundational theory). (Although set theory itself I don't think should really be included, but ETCS is in there.)
Mike: Did some massaging of well-order, well-ordering theorem, well-founded relation, choice object, and axiom of choice, and created extensional relation.
Zoran Škoda: completed the definition of congruence.
Mike: Thanks Finn! It was great to meet you too. I sliced up your paragraph mentioning fibrations and incorporated the material in other places where I thought it fit better.
Zoran Škoda: Created Frobenius category; added material in dg-category on dg-modules, Yoneda functor and “pre-triangulated”. One should also explain the pre-triangulated envelope functor.
Created minimal logic and intuitionistic logic, as very small stubs. Introduced yet more broken links there, which I’ll fill in later.
Slightly corrected Mike’s new entry type theory, and added a paragraph mentioning fibrations. I’m not sure it’s in the right place, though. (PS. Mike: it was great to meet you in Cambridge! Also, Bruce: thanks for finding me a seat in the restaurant!)
Toby Bartels: I aksed an idle question at combinatorial spectrum about Kan complexes and $\mathbf{Z}$-groupoids.
Mike:
Created type theory with an introduction for category-theorists. Additions and corrections are welcome.
Some improvements and corrections to cardinal number, ordinal, well-order (what an ugly noun!), transitive set, and inaccessible cardinal, and created von Neumann hierarchy. In particular, I added the structural point of view to complement the naive and material ones.
Urs:
Zoran Škoda: I have made additions to cardinal number: Urs uses a naive set definition where cardinals are equipotence classes of sets hence his cardinals are proper classes; I follow a choice of representative among ordinals (well-ordered transitive sets); to this aim I created transitive set, ordinal, successor, inaccessible cardinal. In doing this I used some intro parts of my lectures on sheaf theory which I currently teach in Zagreb in Croatian (and which initially started with cardinals, universes and categories). I created twisting function.
Urs:
added a remark on the $(\infty,1)$-version at Dold-Kan correspondence
created accessible (infinity,1)-category and accessible (infinity,1)-category, but incomplete
created compact object in an (infinity,1)-category, but incomplete
created cardinal number and well-order, but experts should please check these
created sigma-model
created an entry geometric infinity-function theory to go along with the $n$Café entry Journal Club – Geometric Infinity-Function Theory
created natural numbers object just to saturate links (and it should indeed be “natural numbers object”, not “natural number object”, agreed?)
worked comment by David Ben-Zvi into why (infinity,1)-categories?, but more needs to be done
Urs:
I had been asked by students to say something about why they should care about learning about $(\infty,1)$-categories. I thought that would be a good thing to try to answer in an $n$Lab entry, so I started an entry why (infinity,1)-categories?. This is just a first attempt. Maybe somebody would enjoy adding his or her own points of views of correcting/improving mine.
created Connes fusion, but filled in only pointers to further references
Zoran Škoda: created von Neumann algebra emphasising on sources of relations to category theory and low dimensional topology (particularly G. Segal’s program on relations between CFT and elliptic cohomology). There is a good wikipedia entry on von Neumann algebras with lost of references and details, but neglecting the connection to the above topics which should be expanded on. Moreover somebody should mayve write entres on related topics as Connes fusion, modular functor etc. as those are relevant for some of us.
Toby Bartels: I came to some sort of decision at direct sum.
Urs:
attempted a (long-winded) reply to Davids question “What does it mean” at Coyoneda lemma (and would anyone mind if we renamed that to co-Yoneda lemma?)
Mike:
Following Toby’s suggestion, moved subsequential space to sequential convergence space. Split convergence space and Cauchy space off from filter, and added some stuff about pseudotopological spaces to convergence space.
Created Reedy category.
David: Created Coyoneda lemma. What does it mean?
Mike: Created subsequential space with a bit of propaganda.
Zoran Škoda: I created entries orbit category, Dold-Thom theorem and satellite with most basic definitions, properties and references, but quickly run out of energy; one should at least add the definition of the morphism part of the satellite functors, the connecting morphism for long exact sequence of satellites, and connection to the Kan extensions. Made some additions to Dold-Kan correspondence.
Toby Bartels: I have a terminological question at direct sum. (It's a rather elementary question in universal algebra.)
Urs:
created Deligne cohomology
reacted to Bruce’s question at heuristic introduction to sheaves, cohomology and higher stacks by expanding further on the notion of morphisms of sheaves
expanded slightly at AQFT – still just a stub entry, though
touched combinatorial spectrum: replied to Mike, expanded the discussion of examples and changed the notation there a bit. But please check. I’ll send a request about this to the blog.
replied to Tim’s comments on Zoran’s comment at differential graded Lie algebra
replied to the questions that were at restriction and extension of sheaves (on notation and existence) by adding more details
Tim:
Urs:
added a few links to examples at space and quantity (we have a general problem that many entries created eraly on don’t currently point to entries created more recently which de facto they should point to)
touched combinatorial spectrum: replied to Mike, added a list of examples and have further questions
Tim:
I have put another of the Lexicon series of entries up. It is bar and cobar construction. This looks at the differential algebra behind those constructions, and sketches the bar-cobar adjunction.
I have tried to provide more links to and from this series of ‘lexicon’ entries. (soon will be finished!)
Urs:
further polished nonabelian cohomology
created combinatorial spectrum
expanded abelian sheaf cohomology
created cohomology
created heuristic introduction to sheaves, cohomology and higher stacks
Urs:
added a bit more details to abelian sheaf cohomology
added discussion of the sheaf version to Dold-Kan correspondence
started an entry abelian sheaf cohomology, but have just the “Idea”-section so far (aiming to provide the right $\infty$-categorical perspective)
provided, using Todd’s help, the details on the relations between the two definitions at closed monoidal structure on presheaves and created a supplementary entry functors and comma categories on properties of, well, functors on comma categories
Tim:
Urs:
reorganized the entry mathematics a bit – I am hoping that eventually this becomes a useful top of a small hierarchy of link-list entries which allow the reader to get an idea of the scope of topics covered (and not yet covered) by the $n$Lab, and possibly to facilitate searches by topic rather than by keyword
as discussed with Timothy Porter, I created a stub for rational homotopy theory whose main purpose at the moment is to contain the link list to his lexicon entries on concepts in differential graded algebra
created closed monoidal structure on presheaves and closed monoidal structure on sheaves, but am being dense: have a question at the former
created direct image and inverse image and restriction and extension of sheaves
moved discussion from semi-abelian category to Dold-Kan correspondence and added references
added explicit formulas to Yoneda extension (not the end-yoga, though)
added a question to Mike’s question at semi-abelian category (probably for Tim)
polished infinity-topos
Urs:
Tim: (I seem to remember a request to put more recent changes at the top, even if you have one on today’s page so … .)
created infinity-stackification
added a section “Idea” to abelian sheaf
added a section “Idea” to Dold-Kan correspondence
created injective object
created complex
created Grothendieck category – it feels like this should make me say something about that axiom list at additive and abelian categories…
tried to resolve/incorporate parts of the discussion at localization by reworking the entry a bit – also left a comment there
noticed that we have considerable overlap now between multiplicative system and category of fractions. Left a comment there to remind us. Somebody who knows the precise status of these two terms in the math community should please go ahead and merge the material in one entry, keeping a redirect page for the respective other term
Tim:
I have added a comment on the terminology localization. Perhaps an algebraic geometric historical perspective could be useful here to help explain the terminology. (I’m not sure that I am competent to provide this however!)
I have put another of the Lexicon series of entries up. It is differential graded coalgebra.
Added bicategory of fractions, category of fractions and wide subcategory.
Started adding the construction of the localization of a category, as well as a speculative comment at that page on computing this as a fundamental category.
Migrated March changes
Continued discussion with Urs at my private page comments on chapter 2.
First list — Previous list — Next list — Current list