nLab 2009 May changes

Archive of changes made during May 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.

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2009-05-30

• Toby Bartels: I don't understand $diag$ at Atiyah Lie groupoid.

• Urs:

  • started connection on a bundle in reply to discussion here

  • added to distributor the description in terms of colimit preserving functors on presheaf categories and added corresponding remarks at the beginning of symmetric monoidal (infinity,1)-category of presentable (infinity,1)-categories

  • created Atiyah Lie-groupoid and principal bundle to link to from the blog discussion here

• Toby Bartels:

  • A new query at sieve.
  • Wrote finitely cocomplete category by copying finitely complete category and cutting it down.
  • Responded to Bruce at coverage, to Urs at sieve, and to Eric at category of elements.

2009-05-29

• Todd Trimble kindly created quasitopos.

• Urs

  • reacted to Bruce at sieve

• Bruce

  • added a question at coverage

  • added comments to sieve

• Urs

  • added a long section “A detailed description of what’s going on” to sieve, meant to be pedagogical, veery small-stepped. I’d be interested to hear from non–experts to which extent they find this bit helpful

  • replied (to Toby, I presume?) at sieve

  • created Dijkgraaf-Witten theory

2009-05-28

• Urs

  • created On the Classification of Topological Field Theories, factorization algebra

  • created category of generalized elements as a place for the material that was at Exploding a Category – I am voting for not using that explosive terminology

  • expanded path category and created Moore path category in the process

• David Roberts: Added comment to plus construction re disambiguation (recall the Quillen plus construction).

2009-05-27

• Toby Bartels: Last night, I made some lists of entries relevant to constructive mathematics, starting with the back links and classifying them. (Hopefully the third list will become so large that we can't really use it, but for now I want to keep track of which entries have that material.)

• Urs

  • created Quantization as a Kan Extension

  • created plus construction – the material that should go there is currently at sheaf; should be extracted from there eventually

  • added a link to the great Catsters video material to limits and colimits by example and created The Catsters and Eugenia Cheng

• David created extended natural numbers, corecursion, and coalgebra for an endofunctor. Moved proof of terminal coalgebra being fixed point to terminal coalgebra. ‘Coalgebra’ still needs disambiguation.

2009-05-26

• Bruce filled in a small thing at semisimple category.

• Urs:

  • created path integral

  • added to sieve a – potentially long-winded – lemma with detailed proof that spells out details of the fact that every subfunctor is the coimage of a morphism out of a coproduct of representables

  • created separated presheaf

  • further worked on the section “in terms of geometric embedding” at sheaf

2009-05-25

• Toby Bartels: Copied comments on Google and page names to the Café here.

• Urs

  • expanded and restructured sieve – introduced the explicit distinction between the notion of sieve and that of subfunctor and made their bijection explicit: this seems to be the standard way that “sieve” is used in the literature: authors don’t seem to say “sieve” synonymously with “subfunctor”, but say “sieve” if they explicitly mean the set of elements of the subfunctor. There is a bijection, but not an equality of concepts.

  • further expanded image and coimage

  • edited and replied at semisimple category

• Bruce responded to comments at semisimple category…and then got stuck. Can anyone help? Update from Future Bruce to Past Bruce: solved.

• Zoran Skoda: as far as google (and other search machine’s) counterarguments of Toby (see down 22. may) I still disagree. I do care if our stuff is well indexed and hence better used by anybody looking for answer at the google including us; and I do not care about vanity issues of pro or contra google movements. If somebody gets directed to a less relevant page this is creating noise, and showing a less convenient side of our work. If effectivity is not important why are we doing this ?

• Urs

  • further developed the section “In terms of geometric embedding” at sheaf, but will continue later. I am lacking (at least) one argument in one of the proofs currently…

  • created Bertrand Toën (and Bertrand Toen as a redirect)

2009-05-24

• Toby Bartels:

  • Cleaned up Cauchy space a bit.
  • An answer for Mike Shulman at subsequential space.
  • A question for Bruce Bartlett at semisimple category.

• Todd: Added a section “morphisms of manifolds” to manifold which attempts to enlarge the notion of pseudogroup, so as to get general notions of map for general notions of manifold. However, this is “original research” and should be vetted – I don’t see the enlarged notion (or an equivalent) in any of the literature I’ve read.

• Bruce: started/added material to some pages related to fusion categories, like fusion category, Pivotal symbols (this name needs to be changed to fit the naming conventions), semisimple category and rigid monoidal category.

• Urs:

  • started expanding sheaf – my goal is (as now indicated) to give a full derivation of how a geometric morphism into a presheaf category defines a Grothendieck topology and sheaves with respect to that – but my machine’s battery runs low and will leave me offline any minute now, so will have to continue later (darn technology…)

  • many thanks to Bruce for expanding fusion category and creating the relevant further entries

  • created a stub for fusion category

  • created a stub for String field theory

  • created A Survey of Cohomological Physics

  • added cross-links to monoidal category and manifold

2009-05-23

• Toby Bartels:

  • Created Banach spaces. I need to at least describe products and coproducts in $Ban$.

  • Another example, Banach spaces, at internal hom.

  • Looking at other things that Urs has done, the problem may be this: If you want to put an equation within a bullet list, then you either have to put no blank lines between the equation and the surrounding text

    $$like this$$

    or put enough spacing before the equation to match the indentation of the list

    $$like this$$

    or, of course, both

    $$like this$$

    but it will fall outside of the bullet list if you do neither

$$like this$$

• Bruce:
  • Added example of the internal hom in super vector spaces to internal hom.
  • Urs: I tried to understand what went wrong with your query box example for Tychonoff theorem but I think the roll-back has removed the original problem from the records.

2009-05-22

• Toby Bartels:

  • Expanded centipede mathematics and negative thinking, which David Corfield created yesterday.
  • In response to Zoran immediately below, I don't think that we should change what works for us to fit Google. It's Google's job to give good search results, not ours to optimise Google's finding us, which Google tries to prevent anyway. That said, I agree with you about the page names, for other reasons. (Note to all: There is discussion here.)

• Zoran Škoda: created comorphism (in sheaf theory); there is hard to find a sensible and comprehensive account in the literature. Note that the treatment is more general than in the usage for the case of ringed spaces. By the way, the server is very erratic tonight, and having sometimes responses delayed by 5-10 minutes. On the other hand I strongly disagree with the changes of the names of entries massively being done today by Eric: he moves infinity-category into $\infty$-category. Though this is graphically appealing, google and others put higher in search results items which have search name in the title, and the entries they index are index basically by the ascii. I want to see nlab entries high in the google search, this makes our effort more useful. Fancy graphics can be WITHIN the entry, and prefereably in this decade still not in the title. I would do the redirects in the symbol variant of the title instead!! What the others think ?

• Urs

  • I tried to fix the query box layout at Tychonoff theorem only to find that there is apparently some software clash between the query box syntax and some symbols used. So I “rolled back”. Hopefully the entry is now again in the form that Todd and Body left it. Please check.

  • started filling something into n-categorical physics just so the page looks less blank – but didn’t really have the energy or intention to produce anything of more than vaguely suggestive nature so far

• Zoran Škoda made changes to pure motives. Please do not use defined term algebraic spaces when it is not appropriate.

• David added exposition references to pure motive

• Urs

  • added explicit description of colimits in $Set$ to limits and colimits by example – I have a dumb question: frequently I want to use displayed math equations in a bullet list item. But more often than not when I try this the parser gets very mixed up and produces weirly formatted output – what am I doing wrong??

2009-05-21

• Bruce added link to Evan Jenkins’ notes of the Northwestern TFT Conference 2009, and changed text+math font on the nLab to a serif theme by editing the CSS. This will only be for one day, just to create awareness of what’s possible. Discussion on the nForum.

2009-05-20

• Chris started work on section 5 of BZFN

  for the journal club on geometric $\infty$-function theory.

• Todd wrote a longish reply to a query of David Roberts about tame topology over at nice topological space.

• Toby Bartels wrote Hausdorff space and Zorn's lemma.

• Todd Trimble wrote Tychonoff theorem.

2009-05-19

• Todd added a bit more to compact space, in particular linking back to quasicompact. There is so much one could say about compact spaces, it’s quite the embarrassment of riches.

• Zoran Škoda: created algebraic geometry, algebraic stack, topological stack; updated quasicompact with the definition and facts about quasicompact morphisms.

• Urs:

  • created Northwestern TFT Conference 2009

  • added cross-links between compact object and compact space

• Todd Trimble created compact space

• Urs:

  • added a short section Basic idea in three words to the beginning of geometric infinity-function theory, as I noticed that this simple introductory statement was not sufficiently amplified in the existing introductory section

  • thanks to Toby for (in particular) his editorial work! it doesn’t go unnoticed, is much appreciated

2009-05-18

• Todd Trimble has also created uniform space and ultrafilter theorem.

• Chris:

  • Replied to Urs on fiber product, suggesting that fiber products be computed using a cobar resolution and that we should eventually make a section about derived fiber product of spaces.

• Urs:

  • added an “Idea”-section to homotopy category and added the precise definition of the universality condition on $Q : C \to Ho(C)$

  • replied to Chris at fiber product

  • expanded local isomorphism: more about sieves, more about pullbacks – but needs polishing

  • created commutativity of limits and colimits as a place to list results such as commutativity of filtered colimits with finite limits;

  • added a further definition to image, created coimage and strict morphism

2009-05-17

• Todd Trimble wrote metric space, and there was much rejoicing.

• Urs:

  • restructured compact object

2009-05-16

• Toby Bartels:
  • Created algebra for an endofunctor and monoid object just to be linked to from algebra, but they are very much stubs; there is related discussion at coalgebra.
  • Changed the category of Anders Kock from people to biography, although frankly I can't remember why we keep these separate.
  • Noted exactly which limits are needed at power object.
  • Noted on direct product just how trivial the direct coproduct is.
  • Answered Mike's question at algebraic theory by incorporating actual theorems from an actual reference instead of using vague handwaving in the query box, for once.

2009-05-15

• Bruce:

  • I created a Stylish extension for Firefox to make the edit box bigger and wider when editing pages (I always found this annoying). To install it, install Stylish and then click here.

  • Started adding Todd’s comments about compact objects in categories to the compact object page.

• David:

  • began synthetic differential geometry and analytic versus synthetic

• Urs:

  • further refined the discussion of the relation to localization at geometric embedding

2009-05-14

• Urs:

  • added some bits to subobject classifier: a short intro, the realization in presheaf topoi and a paragraph on $n$-catgorical generalizations and the interpretation of $true$ as a $-1$-category.

  • created power object, which is mentioned in the alternative definition at topos – guessed the definition from the Axiom of power sets at Trimble on ETCS I, but I am not really sure – somebody please have a look and check

  • expanded and restructured Grothendieck topos

  • added what should be a detailed proof to geometric embedding that for every geometric embedding $f_* : F \hookrightarrow E$ the category $F$ is equivalent to the localization of $E$ at the morphisms that are sent by $f^*$ to isos

    • there are two possibilities here: either I am mixed up or this is somewhere in the literature. If the latter, can anyone give me a pointer to a reference that mentions this explicitly?

  • created geometric embedding – am in the process of filling in a detailed discussion of the relation to localization

  • added a paragraph to localization describing and emphasizing the point that lots of localizations one runs into in practice are reflective subcategories or actually geometric embeddings. In particular Bousfield localization presents the corresponding reflective (infinity,1)-subcategory. I am still not really satisfied with the entry localization, though, I am hoping we can eventually present the conceptual basis here more clearly. After all, the perspective on sheaves and sheafification in terms of localization/geometric embedding has been shown to be the workable road to $\infty$-stacks, which, as tradition has it, are well worth pursuing in general and on an $n$Lab in particular :-) It is curious that there seems to be a cultural divide in the literature here: the book by Kashiwara-Schapira for instance amplifies sheafification as localization, which paves the road for $\infty$-stacks presented by the model structure on simplicial presheaves, while the book by MacLane-Moerdijk amplifies sheafification as geometric embedding, which paves the road for the simple definition of (infinity,1)-sheafification by reflective (infinity,1)-subcategory. Lurie’s book effectively gives the unified perspective, which I think is worthwhile presenting very clearly here on the $n$Lab, since it is coceptually so simple and transparent and in practice so powerful. Over.

  • added example to localization of a simplicial model category?

  • added a little proposition to dependent product and a little example to geometric morphism

• Tim: I have given a partial reply to Urs question at homotopy coherent nerve.

2009-05-12

• Toby Bartels: An extensive rewrite and expansion of predicative mathematics, including material on the non-constructive school (which rejects function sets), thanks to reading sparked by one of Jonathan’s posts to the Café.

• Urs

  • created derived smooth manifold, but will have to continue tomorrow

  • created category of local models and locally modeled monoid

  • expanded a bit more on the “good limits”-motivation at derived stack, alongside the blog discussion

  • added a few more details to generalized smooth algebra – and have a question on that on the blog here

  • added a few bits and pieces to derived stack, ind-object in an (infinity,1)-category, perfect infinity-stack

• David

  • began Eckmann-Hilton duality. Referred to it from duality, but maybe that page needs some adjustment. By the way, I still don’t think I have answers to questions posed here and below.

  • began initial algebra and terminal coalgebra.

• Urs

  • added the coherence diagrams to braided monoidal category

  • added the enriched versions to representable functor and opposite category

2009-05-11

• Toby Bartels: I moved all of the SVG out of article pages and into included subpages as described at HowTo. (Note that this is supposed to be a temporary fix until we get a working automated system to include somehthing like TikZ; see the discussion.)

• Urs:

  • added more details to and slightly restructured exact functor

    • in particular added a remark on and a link to “flat functor”, which overlaps in content

    • I changed at flat functor the condition that $(d/F)$ be filtered to the condition that it be cofiltered. Please check that I am not hallucinating! But see for instance KashSch prop. 3.3.2

  • after reading it I got the idea that Toby‘s latest addition to stuff, structure, property is really about higher subobject classifiers as discussed at generalized universal bundle. I added corresponding remarks, but didn’t find the time to look into this very carefully.

• David: Started locally finitely presentable category

• Toby Bartels:

  • Added a sort of logical interpretation to stuff, structure, property. (This has some links that ought to be filled out.)
  • There's more work going on at SVG Sandbox? and being discussed in a Forum post. See also the new Inclusion Sandbox.
  • There's been some work recently at generalized element, in response to a reader's question.

2009-05-08

• Toby Bartels: Fixed the last ‘heuristic’, at local system, among other such fixes. (You gotta open them up for editing and look at the source!)

• Urs:

  • tied up the loose end at descent using the expected result that Dominic Verity was kind enough to proof on request and then confirm by private communication

• Andrew: (I forget whether I’m supposed to add this to my earlier comment, or here, or add a word of this to each of the other entries of the day). I’ve created SVG Sandbox? expressly for the purpose of mucking about with SVGs to get them to look right. My rationale is explained at the top of that page, together with some suggestions on how it might work. The point is that one SVG can be rather large and I think that putting them in the regular Sandbox to test stuff is a bit anti-social. Hopefully we can get stuff in the FAQ and HowTo on good ways of importing SVGs as there seem to be a few “special features”. I’ve shifted the recent SVG-related stuff from the original Sandbox to the SVG Sandbox?, but I shifted the discussion that Bruce started to the nForum.

  • Urs: I believe Mike had requested that we announce each change here always on the very top of the list, even if we had earlier logs the same day with other people’s logs already on top of them. Seems to be a reasonable practice if the point of this page here is to alert others of changes. Which it is. So, yes, the way you did it should be the preferred way.

• Urs:

  • recalled Tim Porter‘s comment at simplicial set and moved simplicial nerve of simplicial categories to homotopy coherent nerve;

  • started expanding, polishing and reorganizing simplicial set – still not satisfied, though.

• Bruce: Added discussion to Sandbox about Andrew’s TikZ->SVG method. Maybe we can get good and easy graphics going soon in Instiki!

• Urs:

  • added lots of further stuff to simplex category

• Andrew: Finished off the heuristic shift. Bizarrely, one page, local system, comes up in the Instiki search as having “heuristic” in it but I can’t find it. The other instances of “heuristic” that are left are either correct or within discussions.

• Urs:

  • added explicitly the description in terms of weighted (co)limits and in terms of (co)ends to Kan extension; also reorganized the existing material somewhat.

  • added the definition of composition to enriched functor category;

  • thanks to Andrew for the thing about “heuristic”. I’ll be aware of that in the future.

    • following Toby I made heuristic intro… a redirection to motivation for sheaves, cohomology and higher stacks

• Toby Bartels:

  • I split identity into identity morphism and identity element. Maybe I'll do the same to inverse later, but not now.
  • More details at extensional relation.

2009-05-07

• Toby Bartels: Reply to Andrew below:

  • Yeah, that's pretty much how to do it. Except that you mark the old page as category: redirect rather than as category: delete, since it contains edit history that we want to preserve. (Instiki has no cool page-move feature like MediaWiki does.) As you did, I also usually decline to change links from discussion.
  • I would move heuristic introduction to sheaves, cohomology and higher stacks to motivation for sheaves, cohomology and higher stacks, since Urs doesn't like introduction to sheaves, cohomology and higher stacks.

• Andrew

  • Heuristically altered records containing the word “heuristic”. The original list of records matching the search term “heuristic” is below (so anyone worried can check what I’ve done). Most actually were references to heuristic introduction to sheaves, cohomology and higher stacks which obviously needs to be renamed before the other pages are changed (what’s the best way to rename a page? Is it: create a new one, copy over the content, put in a redirect, change all referring pages, mark old one for delete? Or is there a simpler way?). In a couple of the others the word was used in a discussion so I didn’t change those. One I even left as I thought it was (almost) correctly used!

    • 2009 April changes
    • Categories and Sheaves
    • General Discussion
    • Grothendieck's Galois theory
    • cohomology
    • full functor
    • geometric infinity-function theory
    • heuristic introduction to sheaves, cohomology and higher stacks
    • hyperstructure
    • induced representation
    • limit
    • local system
    • sheaf
    • sheaf cohomology
    • space and quantity
    • stuff, structure, property
    • weighted limit
    • why (infinity,1)-categories?

• Urs

  • as a reaction to the discussion taking place there I expanded the text at simplicial model for weak omega-categories – also added references

2009-05-06

• Urs

  • reacted to the discussion at Day convolution and provided the requested further details in the main entry

  • corrected the statement about the relation of $SetMod$ to $Span(Set)$ at distributor by adding the missing clause about discrete categories

  • created structured generalized space

  • created string theory

2009-05-05

• David:

  • began things to be categorified

• Toby Bartels:

  • Fixed up extensional relation, although there seems to be a problem with one of the counterexamples.
  • I also have an opinion at simplicial model for weak omega-categories.

• Urs:

  • added a pedagogical example to enriched functor category

• David:

  • began microcosm principle

• Urs:

  • added details to enriched functor

  • added a bit more detail to closed monoidal structure on presheaves

  • replied at cohomology

2009-05-04

• Urs

  • worked on end: added a section “End as equalizer” where I try to motivate the formula for the end over $V$-valued functors from the equalizer formula for limits, then give the equalizer formula for ends – personally I find this a bit more helpful than dinatural transformations, but that is certainly just my ignorance – I added an “Idea” and a “References” and an “Examples” section

  • added the definition to enriched category (yes, that was still “left as an exercise”)

2009-05-03

• Toby Bartels:

  • Wrote FOLDS (and Michael Makkai), identity functor, identity natural transformation, anafunctor category, identity anafunctor, anabicategory, and ananatural transformation to be linked from the below.
  • Rewrote equivalence of categories, moving one paragraph to Cat and otherwise expanding.
  • Urs and I would like some advice from Tim Porter at simplicial nerve.

• Urs:

  • further expanded section 2 of geometric infinity-function theory, which I am to “report” on next Monday in our Journal Club.

  • finally created simplicial nerve of simplicial categories, but more details necessary…

  • am trying to find out, for A-infinity ring and E-infinity ring to which extent the two definitions “algebra over an $\infty$-operad” and “algebra object in a monoidal $(\infty,1)$-category” are equivalent, as one would expect…

  • created symmetric monoidal (infinity,1)-category, commutative algebra in an (infinity,1)-category, smash product of spectra, associative ring spectrum, A-infinity ring, commutative ring spectrum, E-infinity ring

  • expanded the intro at higher algebra a bit

  • thanks, Toby, for your editorial help! Yes, right, “higher algebra” should be lower case, true.

2009-05-02

• Toby Bartels:
  • Disambiguated simplicial category, hopefully correctly, at (infinity,1)-category, derived stack, and pretriangulated dg-category.
  • Wrote proper class.
  • Moved Higher Algebra to higher algebra because it seemed to be about the subject rather than about just Lurie's two papers, but I may have been wrong about that.
  • Formatted theorems at descent for simplicial presheaves and descent.
  • Answered questions at biproduct, comma category, exponential object, and context.

2009-05-01

• Finn Lawler

  • Created lax 2-adjunction, complete with spiffy PNGs.

  • Added some links at Gray-adjointness-for-2-categories to entries on concepts in the book. Some of these are to non-existent pages like lax natural transformation. Do pages on ‘elementary’ stuff like this belong here on nLab, or should we add them to Wikipedia and link there?

    Mike: If we have a page on categories, we can certainly have a page on lax natural transformations!

    Jonas Frey: The problem about lax natural transformations is that there is no consensus about the orientation of constraints in the literature. Johnstone in the elephant and Gurski in his PhD-thesis for example use an other orientation than Leinster and Borceux in their books.

    Toby: All the more reason to write an article to explain the differences!

• Urs

  • created new top-level link list Higher Algebra – to be filled with content…

  • considerably expanded (infinity,1)-category – motivation was really to write relation between quasi-categories and simplicial categories, but then ran out of steam

  • removed all existing logs here and archived them as 2009 April changes

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