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.
Urs:
started adding comments on embeddings at geometric morphism, but ran out of time
following Mike’s suggestion I moved the material that was at sheafification in a Lawvere-Tierney topos over to Lawvere-Tierney topology
have (started) a discussion with David Roberts on the relation between internal anafunctors and localization in higher sheaf categories over at his private area comments on chapter 2
incorporated in part Zoran’s comment at sheafification of non-set-valued presheaves and created an entry on the IPC-property in the course of that
started Sheaves in Geometry and Logic, on the MacLane-Moerdijk book
created dense monomorphism (thanks to Mike for the relation to local isomorphism) and then sheafification in a Lawvere-Tierney topos
created (infinity,1)-essentially surjective functor and (infinity,1)-fully faithful functor?
Mike:
Mentioned at triangulated category that the definition is redundant. If I had time I would fix it myself.
We are having an interesting discussion at derived functor. Unfortunately I am leaving town for a couple of weeks (starting with PSSL88) so will be less active for a while.
Urs:
created category of open subsets and presheaf on open subsets to satisfy links at Categories and Sheaves
fixed and expanded local isomorphism and used that to create category of sheaves, (infinity,1)-category of (infinity,1)-sheaves and sheafification
created base change and cobase change
started listing Zoran’s latest entries and previous ones on examples/refinements of additive and abelian categories at additive and abelian categories.
Tim:
Zoran Škoda: Created suspended category and Quillen exact category. This is a continuation of our efforts to enter various classes of additive categories useful in homological algebra and K-theory.
Urs:
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…
agreed at homological algebra and filtrant category
created homology
expanded differential object and created differential just to satisfy links
created matrix calculus and mapping cone
Tim:
Urs:
concerning the entry homological algebra: Tim Porter and Zoran Skoda (or anyone else): please feel free to improve/revise the exposition
created triangulated category – have a question about dérivateurs there
created local epimorphism
created multiplicative system
Mike:
A grammatical suggestion at stuff, structure, property.
Did some work on chain complex and all the additive and abelian categories pages.
Urs:
began derived category before running out of time again…
did some layout-editing for Mike’s additions to homotopy category of an (infinity,1)-category
Finn Lawler: Continued experimenting with graphics for diagrams. Have a look and see what you think:
Added some PNGs at Kan extension
Replaced PNGs at adjunction with larger ones.
started preparing the ground for derived category and triangulated category by creating category with translation, chain complex and differential object before running out of time…
created homological algebra, mainly as a collection of links to the keywords listed there
thanks to Mike for his polishing of my original filtrant category at filtered category
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 pdfcrop
and convert
on the pdflatex
output.)
Edited linear logic in response to Mike’s question.
Created star-autonomous category.
Mike: Prompted by discussion with Zoran, created strict epimorphism and added a lengthier discussion of types of epimorphism to epimorphism.
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).
Mike:
Zoran and I are having a discussion about definitions of abelian category.
Redirected filtrant category to filtered category.
Moved the discussion about the word “bimorphism” from balanced category to bimorphism.
Redirected parallel morphism to parallel morphisms (which is in line with the naming conventions).
Urs:
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 $p^* := - \circ p : [C',D] \to [C,D]$ for precomposition with a functor $p : C \to C'$ (pullback notation) at Kan extension to $p_*$ (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
Tim: I have raised a query at Kan extension.
Zoran Škoda: Created abelian category with multiple equivalent definitions.
Mike:
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.
Added some comments on syntactic categories to internal logic, since Toby kindly saved me the work of defining them at context.
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.
Mike:
Split nice category of spaces from nice topological space and fixed all the links I could find.
Wrote universe in a topos by way of responding to the query at Grothendieck universe.
Focused adjunction on the internal version in a 2-category, to distinguish it from adjoint functor, which I reorganized and added a definition to. (The zig-zag identities are crying out for SVG!)
Corrected generalized element to distinguish it from global element.
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 a stub-entry on Stable Infinity-Categories (Lurie’s PhD part I) and advertized this little program of textbook $n$labification here at the blog
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
created fiber product, parallel morphism, zero morphism, kernel, added example to pointed object
good to see that Mike is back! Mike, there is a request for you at Grothendieck universe: can you say something about rephrasing that as a “topos object internal to the ETCS-version of $Set$”?
added the globular zig-zag diagrams to adjunction
started continuous functor
added a link to David’s new entry algebraic set theory at set theory. It would be nice to put it into context there, eventually.
started expanding limit (and also a bit colimit): more motivation, more details on definition, more examples
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 the illustrative diagram to over category and added a remark on over categories to subobject;
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.
followed Mike’s remark and moved the previous content in stable infinity-category to stable (infinity,1)-category, keeping just a general nonsense statement at stable infinity-category
added a remark on this and a link at spectrum
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.
Andrew: Added some more to Froelicher space. I’ve started on a new project on this: adapting topological notions to Froelicher spaces.
Urs:
created stable infinity-category
implemented Zoran’s remark below by rephrasing a bit at Categories and Sheaves
added a bit more to large category
Toby Bartels: Let specialization topology lead me to specialization order.
Mike:
Removed the discussion about accented characters from fundamental group of a topos and put it in the FAQ.
Displayed my happiness at quotient object by removing the discussion.
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.
Urs:
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)
in reaction to Toby’s discussion at large category I created entries for accessible category, locally presentable category and sketch. But very incomplete.
Toby Bartels: Started a discussion about large category.
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.
Mike:
Tim:
Mike:
Yes, homotopy theorists call a k-tuply groupal n-groupoid a grouplike $E_k$-space.
Continued discussion at Crans-Gray tensor product.
Tim: I have been trying to give an adequate categorical treatment of profinite completion of a group.
Urs:
made a little remark on Mike’s question at Crans-Gray tensor product and ask another question myself
added a remark to Waldhausen category on its relation to category of fibrant objects and ask a question about the precise statement to be made here over at the $n$Café
Toby Bartels: Combined proset into preorder, etc, as planned there.
Zoran Škoda created Waldhausen category and made a remark into entry cofibration category.
Toby Bartels: I created symmetric set out of material that Zoran Škoda added to FinSet.
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.).
Tim:
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 $D_0$ should not exist.
Mike: Created homotopy equivalence and weak homotopy equivalence.
Tim:
Bruce Bartlett has created nInsights.
Ronnie Brown: I’ve rewritten and expanded homotopy groups partly to clarify the operation of dimension 1 on higher dimensions and to emphasise the groupoid aspects.
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.
Tim: I have included a discussion of the nerve of an internal category at that entry.
Tim: I have changed the initial sentence of homotopy n-type. I think this is converging well thanks to the efforts of Mike and Toby.
Toby Bartels: Since we already have fundamental groupoid and even fundamental infinity-groupoid, I started fundamental group and homotopy group. But I only wrote #Idea# sections.
Mike:
Responded at homotopy n-type and proset.
Did a little more bettering of profinite group, and was inspired to create filtered category, pro-object, ind-object, and (as stubs) free completion and topological group.
Tim:
Mike Shulman: A question about the Crans-Gray tensor product.
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.
Made a short page on proset, since Toby seems to be using this as a synonym for preorder.
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.
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