Archive of changes made during March 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.
started adding comments on embeddings at geometric morphism, but ran out of time
started Sheaves in Geometry and Logic, on the MacLane-Moerdijk book
QUESTION: came across MacLane’s Foundations for Categories and Sets where it it argued that neither standared set/class set theory nor Grothendieck universes provide decent foundations for categories and a formalism of schools is introduced instead – can anyone comment on that in the light of our discussion at Grothendieck universe?
added an “Idea” section to derived functor and split off derived functor on a derived category from that in order to discuss the special homological algebra aspects of derived functors separately – but incomplete for the moment
created null system
created category of chain complexes, but then didn’t quite know where to go with this
Finn Lawler: Continued experimenting with graphics for diagrams. Have a look and see what you think:
created homological algebra, mainly as a collection of links to the keywords listed there
in a first attempt to clean up the entries surrounding abelian category I created the overview entry additive and abelian categories and branched off Ab-enriched category, made pre-additive category a commented redirect to that and “commented out” the respective discussion still to be found at additive category; also made pre-abelian category a separate entry, so that now there is in order of increasing structure/property
am all in favor of Finn’s graphics! The only reason I don’t include nice graphics myself a lot is that currently these take me longer to create than the MathML hacks
Finn Lawler: Uploaded PNG images of the zig-zag identities and added them to adjunction. They’re probably a bit too small, but what do people think of this approach as a work-around until there’s an easy way to convert TeX to SVG? Any other suggestions? (Note: I tried converting these diagrams to SVG as described here but the resulting files were huge and didn’t display anyway when inserted into the markdown source. Instead I used
convert on the
Zoran Škoda: Created etale space. The order of exposition is important, particularly in view of anticipated additional details. In Kan extension added a detailed paragraph on an example how left Kan extension pointwise formula has intuitive meaning in the case of constructing pullback for (pre)sheaves on topological spaces. Created torsor with structure category following the version in Moerdijk’s book.
Finn Lawler: Created linear logic – just a short stub with basic ideas on motivation and models, plus a couple of references. Comments effusively welcomed. (Edit: also removed Thursday’s query box from context).
Zoran and I are having a discussion about definitions of abelian category.
created exact functor
created filtrant category
added to Higher Topos Theory more introductory/overview remarks which are supposed to be helpful for the newbie
created Yoneda extension
added section to Kan extension on formulas in terms of limits and colimits over comma categories;
added a section on the “local” computation of adjoint functors at adjoint functor and point out how this induces the local/global dichotomy at limit, homotopy limit and Kan extension (see my previous modification below)
if I noticed correctly, Mike had changed my original notation for precomposition with a functor (pullback notation) at Kan extension to (pushforward notation). I have now added a section Remark on terminology: pushforward vs. pullback which is supposed to clarify this terminology issue.
Mike: That wasn’t me. I’m not sure that such a discussion belongs at Kan extension; it might belong somewhere but I would rather than the page Kan extension just pick one notation and possibly link to a discussion.
Zoran Škoda: It was me who changed, though I better did not. I am happy with the original notation as well. For as your discussion on pushfowards I am less happy. Namely, if one is not happy with the direction of maps between open sets, one just redefines what is a morphism of sites (opposite to the functor direction), so that the morphism of sites is always correct direction. So, unless one does not have strong feeling on the choice of pushfoward pullback meaning, what is not in this case, mayeb original notation just caring about covariant vs contravariant was better.
addressed Zoran’s and Tim’s remarks at Kan extension: I have added now to Kan extension as well as to limit – in analogy to what we already had at homotopy limit – an explicit discussion of the difference between local and global definitions of the universal constructions
created universal construction – but filled in just a question/query
I have a question about the meaning of “large” at Grothendieck universe.
I’d like to request that people not add new sub-bullets under their own names on a given day if other people have since listed more changes above; rather, add a new bullet point at the top with your name. If that didn’t make sense, it’s what I’m doing now, rather than (what I could have done) adding a new bullet point below the other copy of my name today.
Finn Lawler: (Hello all – long-time lurker, first-time editor.) For my first edit, I asked a silly question at context and then answered it myself a little later. I’ll delete the query box if nobody has any comments. Apologies for noise.
Made a terminological suggestion at set theory.
Commented about property-like structure at stuff, structure, property. It would be nice to move the examples earlier on this page.
started an entry Higher Topos Theory (on Lurie’s book) in a style analogous to Categories and Sheaves – I included a link to Mike’s personal page n-topos for large n; eventually it would be nice if we had an entry on the general idea and purpose of higher topos theory
started expanding Kan extension
added the globular zig-zag diagrams to adjunction
started continuous functor
created generalized element
more or less completed the hyperlinked keyword list of chapter one of Categories and Sheaves
filled in three equivalent definitions at adjoint functor
expanded a bit at natural transformation
added simple remarks to contravariant functor
added an illustrative diagram to comma category and added a section there on how a comma category is a pullback;
added a little bit of discussion that every presheaf is a colimit of representables to presheaf;
expanded a bit more at stuff, structure, property
Andrew Stacey: lifted the tangent/cotangent section from “Comparative Smootheology” to Froelicher space. I intend to remove this section from that paper and this seems like a good place to put and develop it.
Zoran Škoda: created algebraic monad, generalized ring, compact object, noncommutative algebraic geometry, spectrum (geometry), Pierce spectrum, filter (thanks Mike for an essential typographic correction), generator, cogenerator (the latter were prompted by editing Morita equivalence, paragraph on classical Morita). Zoran and Toby distributed the paragraph on ultrafilters from long growing entry filter partly to the new entry ultrafilter. Uploaded Warsaw circle with link within shape theory.
Zoran Škoda: moved the earlier material from entry algebra to new entry associative unital algebra, and put new material into algebra; one should have separate entry for any framework for algebras, and general entry algebra should have pointers to the major classes (like algebra over operad). Thanks Toby, we continue together on that: now there is an entry nonassociative algebra and so on. I have also addressed concerns of Mike in Connes' cyclic category which now has I hope correct definitions, plus more foundational issues and relevant literature and link to just uploaded file R. Krasauskas, Skew-simplicial groups, Lith. Math. J.
Zoran Škoda has created dense subcategory (intentionally organized different than the entry for the entry for slight generalization, dense functor); created shape theory but needs much more work; I copied here references from fundamental group of a topos (plus to a Batanin’s article) and in fundamental group of a topos I added the reference and link to Pataraia’s article important for the abstract notion of fundamental groupoid in internal contexts.
created an entry on the book Categories and Sheaves (comment you say that “sheaf condition is localization” there; well the category of sheaves is a localization of the category of presheaves, but the sheaf condition…you really meant what you say? – Zoran)
More done on Froelicher spaces. I think that I have finally figured out the relationship between Frölicher spaces and Isbell duality so if anyone else is interested in taking a look I’d appreciate your comments.
I also found the standard layout of the page a little hard to work with, in particular with regard to delimiting proofs and definitions (both of which could get quite long) so I’ve been experimenting with alternative ways of demarking them (on Froelicher space). Let me know if you like or dislike what you see.
Toby Bartels: Tired of writing
[[constructivism|constructive mathematics]], I moved constructivism to constructive mathematics and fixed links. Similarly, I moved predicativism to predicative mathematics. After some thought, I also moved finitism to finite mathematics and expanded it a bit to fit the new name better. To go with this, I finally created FinSet.
Zoran: Created Loday-Pirashvili category, dense functor and equivariant object. There are two different notions of dense subcategory, first of which has two different definitions and is related to colimits and nerve functor, and second which is related to pro-objects. In the entry Bousfield localization I added a paragraph on Bousfield localization for triangulated categories; made changes to nerve (more to be done: one needs to clarify the example of geometric realization etc.).
Zoran: Created (in last two days) several entries mainly related to co- Hopf- algebras and algebras in categories of chain complexes: Frechet-Uryson space, Hopf module, Hopf-Galois extension, Maurer-Cartan equation, category of elements, compactly generated space, coring, dg-algebra,distributive law, torsor (very unfinished!), twisted module of homomorphisms, twisted tensor product, twisting cochain and made changes to few other entries including many changes in entry Hopf algebra and some in A-infinity-algebra. With the (Fukaya) convention used there should not exist.
Toby Bartels: I've written sequence, net, multi-valued function, partial function, and the long-delayed surjection and injection. Those interested in foundations may be particularly interested in my proposed alternative definition of sequence.
Eventually we probably need a summary of some of the theory of algebraic homotopy that Baues has developed as if impinges on the homotopy hypothesis and on homotopical cohomology theory. To this end I have created a sort of historical entry on algebraic homotopy.
Created cofibration category as the first of the ‘Bauesian’ detailed entries.
Toby Bartels: I've written several more articles on very basic topics, such as those that used to be ‘?’-links below. You can see them on Recently Revised; I don't think that anything merits great attention.
Toby Bartels: I tried to clarify the difference between a preorder (a structure on a given set that satisfies certain properties) and a proset (a set equipped with such a structure). I need to finish that for partial order/poset and total order/toset, although I would also entertain the idea that these should all be redirected one way or the other. But I got sidetracked writing linear order and loset instead. (And then there's quasiorder; I don't think that quoset is necessary for reasons that I don't want to get into here.)
Attempted to answer Eric’s plea for a category-theoretic definition of ‘Hasse diagram’, in the discussion at the bottom of preorder. Unfortunately I don’t know the official definition of ‘Hasse diagram’ — though I know one when I see one.
Toby Bartels: Since Urs is using both ‘over category’ and ‘over-category’ (and not ‘slice category’), I tried to standardise things as ‘over category’ to diminish the temptation to slip further into ‘over-category’. Principally this means that I moved slice category to over category and over-category in quasi-categories to over quasi-category.