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I am being bombarded by questions by somebody who is desiring details on the proofs of the statements listed at regular monomorphism, e.g. that
in Grp all monos are regular;
in Top it’s precisely the embeddings
etc.
I realize that I would need to think about this. Does anyone have a nice quick proof for some of these?
This semester I have been asked to join Jaap with overlooking a handful of students who run a seminar on basic category theory.
In the course of that I will be re-looking at some nLab entries on basic stuff. Today I started looking at the cornerstone entry of the whole nLab: category theory.
I was very unhappy with that entry. Until a few minutes back. Now I am feeling a little better. That entry had consisted to a large extent (and still somewhat does) of lengthy lists of statements, all not exactly to the point, interspersed with lots of discussion with people like Todd and Toby continuously disagreeing with what somebody had written.
I think it is not sufficient to try to steer that somebody (who seems to have left us anyway). We need to rewrite this entry. If we can’t get a decent entry on category theory on the nLab, then we have no business making any claims about having a useful wiki focused on category theory.
So, I started reworking the entry:
I moved the historical remark from the very beginning to a dedicated section. An entry should start with explaining something, not with recounting how other people eventually understood that something.
After editing further the Idea section a bit, I inserted two new sections, in order to get to the main point of it all, and not bury that beneath various secondary aspects:
A section: “Basic constructions” namely universal constructions. That’s what category theory is all about, after all. There is not much to be said about the concept of category itself, that’s pretty trivial. The magic is in the fact that categories support universal constructions.
A section “Basic theorems”: a list of the half-dozen or so cornerstone theorems that rule category theory and mathematics as a whole. I want that nobody who glances at the entry can get away with the impression that its “just language”.
I haven’t edited much more beyond that, except that I did remove large chunks of old discussion that looked to me like mostly resolved, mostly about content that I didn’t find too exciting anyway. Should I have accidentally removed something of value, those who remember it will be able to find it in the entry’s history.
I am still not happy with the entry, but at least now I am feeling a bit better about its first third or so. I would wish a genuine category theory guru – you know who you are – would take an hour and set himself the task: here I have the chance to expose the beautiul power of category theory to the world.
in reply to a question that I received, I expanded the entry (infinity,1)-functor in various directions.
I’m confused by the definition of at circle n-bundle with connection. Is there a “modulo ” missing? and, if so, which sense we quotient by there?
Started a page at link. More to add, especially some nice pictures!, but have to go to parents’ evening now.
I’m reading Milnor’s paper “Link Groups” so shall add stuff as I read it. This should also serve as warning to a certain Prof Porter (assuming it’s the same one!) that his 1980 paper is on my list of “things to read really soon”.
Given all the discussion on the categories list, I decided it would be worth creating Street fibration. While writing it I had occasion to put up a stub at strict 2-equivalence of 2-categories.
I tentatively added the reference
to Top.
I have to admit, though, that I did not study it. Does anyone know more about this?
André Joyal left a comment at evil, presumably sparked by the debate raging on the categories mailing list.
(Don’t remember the exact message that sparked the “debate”, but the archives for the mailing list are here).
I will admit that I’m not too enamoured of the word “evil”, but I don’t find it particularly offensive and indeed it’s “shock” value is something that I would try to retain: if you do something that is “evil” you should be darned sure that you know that you’re doing it and convinced that the final outcome justifies the means. I’m also not convinced by Joyal’s arguments about “choosing a triangulation” or whatever. Sure, we choose a triangulation to compute homology groups, but the homology groups wouldn’t be worth a dime if they actually depended on the choice of triangulation.
I also think that the “subculture” argument is vacuous. Every group that has something in common could be called a “subculture” and every subculture is going to invent shortenings for referring to common terms. And of course there is great confusion when two subcultures choose the same word. My favourite story on this is when I was sitting in a garage whilst my car was being fixed. The mechanic yelled out, “You’ve got a crack in your manifold.”. I was a little confused as to what he meant! (The latest Dr Who puts a different spin on this, I believe).
The thing is not to avoid being a subculture, that’s impossible, but to avoid being a clique. The distinction that I intend to draw is that cliques are defined by who they don’t contain whereas subcultures are defined by who they do. Therefore anyone can join a subculture, but not anyone can join a clique.
Clashes of terminology are inevitable in such a broad subject. What does the word “category” conjure to a functional analyst? Someone not well versed in algebraic geometry might ponder the meaning of a “perverse sheaf”. And the connections between limits and limits seems, if not tenuous, at least to not be all that useful in conveying intuition.
created fundamental (infinity,1)-category
This is supposed to propose the evident definition. But have a critical look.
there are two different concepts both called “Weil algebra”. One is in Lie theory, the other is a term for duals of infinitesimally thickened points.
Promted by a question that I received, i have tried to make this state of affairs clearer on the nLab. I added a disambiguation sentence at the beginning of Weil algebra and then created infinitesimally thickened point for the other notion.
Created isotopy and circle, also a bit of housekeeping (adding redirects and drop-downs) at knot and knot invariants.
For circle, my thought was to present it as an example of … just about everything! But I’m sure that there’s things I’ve missed, so the intention is that it not be a boring page “the circle is the units in ” but rather “the circle is an example of all these different things”.
(On that thought, I’ve sometimes wondered how much of the undergraduate syllabus could be obtained by applying the centipede principle to .)
started floating TOC gravity contents and added it to the relevant pages
I am still not happy with my rudimentary understanding of the characteristic classes of homotopy algebras, e.g. A-infinity algebras as presented by Hamilton and Lazarev. Kontsevich had shown how to introduce graph complexes in that setup, almost 20 years ago, but in his application to Rozansky-Witten theory he has shown the relationship to the usual Gel’fand-Fuks cohomology and usual characteristic classes of foliations. On the other hand all the similar applications are now systematized in the kind of theory Lazarev-Hamilton present. Their construction however does not seem to directly overalp but is only analogous to the usual charactersitic classes. These two points of view I can not reconcile. So I started a stub for the new entry Feynman transform. The Feynman trasnform is an operation on twisted modular operads which is Feynman graph expansion-motivated construction at the level of operads and unifies variants of graph complexes which are natural recipients of various characteristic classes of homotopy algebras.
New entry relaxed multicategory. A relaxed multicategory is a special case of enriched multicategory which is accomodating for singularities like in the work of Richard Borcherds on G-vertex algebras.
Added a mention of more general change-of-enrichment to enriched category, and a reference to Geoff Cruttwell’s thesis.
Added some discussion of other ways to define doubly-weak double categories.
New entry suitable monad. More references at enriched category.
A recent question about Freyd categories on the mailing list has led me to write premonoidal category. (Freyd categories themselves are a little more obscure, and I haven’t written anything about them.)
Someone has left rubbish on several pages: Fort Worth Web Design : Essays : Digital Printing : Halloween Contacts : Whitetail Deer Hunting I will go and tidy up but it is worth checking where it came from.
New entries coideal, quotient bialgebra and various redirects like left coideal, right coideal, quotient Hopf algebra. And quantum flag variety. This complements earlier today reported stubs quantum homogeneous space and Ulrich Kraehmer.
started infinity-Lie algebroid valued differential forms , since that is needed all through our discussion of oo-Chern-Weil elsewhere. But right now the entry is stubby.
I wrote down a definition at pseudofunctor.
created Chern-Simons circle 7-bundle
created stubs classical Lie group and exceptional Lie group and linked to them from simple Lie group (all very stubby)
New article: opposite magma (including monoids, groups, rings, algebras, etc).
without really intending to do so and certainly without having the time to do so, I ended up creating some stubs for
following a public demand, I have
created inner derivation Lie 2-algebra,
cross-linked it with Cartan calculus and pointed out how they are related,
and started at Weil algebra a section As the CE-algebra of the L-oo algebra of inner derivations.
expanded algebraic Kan complex. Added Idea-section and Properties-section
When I told him about it today, Dmitry Roytenberg urged that we highlight a bit more visibly a fact that was mentioned on the nLab before, but not highlighted maybe sufficiently.
Namely by the general theory of infinity-Lie algebra cohomology we have for every -Lie algebroid equipped with an invariant polynomial the corresponding Chern-Simons wich exhibits the transgression to the corresponding -Lie algebroid cocycle.
If you apply this general theory to a Poisson Lie algebroid, then then Chern-Simons form that drops out is the action functional of the Poisson sigma-model.
I added this remark more visibly now to Poisson Lie algebroid, Poisson sigma-model and infinity-Lie algebroid cohomology.
Following sections 34 and 35 in Dwyer-Hirschhorn-Kan-Smith (DHKS), I have begun to write up a page on hammock localization and simplicial Grothendieck construction localization. simplicial localization of a homotopical category.
Mike has added to Georges Maltsiniotis a link to an English version of Maltsiniotis’ version of Grothendieck’s version of an oo-groupoid. I can’t believe I missed this on the arXiv yesterday! The French version (or rather, the French predecessor) of this paper is one of those things I wish I could sit down and read in detail and pull apart, but haven’t the time (and the language was a small, but psychological barrier).
Created stub homotopy algebra as this is traditional terminology unifying L, A, G, B etc infinity algebras for references and disambiguation and for operadic point of view.
For some bizarre reason I added a lot of information to the page Pontrjagin duality.
created coherence law
(was surprised to find that we didn’t have this already. Or do we?)
I clarified how this works in single-sorted definition of category
created stubs for (2,1)-presheaf and hom-groupoid, just because I needed to link to them
I separated binary digit from boolean domain.
the entry geometric definition of higher categories had been in a sorry state. I have edited it a bit in an attempt to bring it into shape
stub for 3-morphism
created infinity-anafunctor
created (or edited)
Here the first item is always defined as a coycle in, respectively, the
And I tried to establish the same kind of link pattern for
noticed that the entry curvature was in all its stubiness already a mess.
So I tried to write an Idea-section that indicates how the notion of curvature appears for embedded surfaces and then gradually generalizes to that of connections on bundles and further.
Eventually I would like to split off the section on extrinsic curvature to a separate entry extrinsic curvature and Gaussian curvature.
But not now, I need to be doind something else…
created Poisson Lie algebroid
New entries ribbon graph, mapping class group, Jakob Nielsen and just one reference at Mumford conjecture. Mainly to set some background for yesterday started entries graph homology, noncommutative symplectic geometry and related effort to understanding of the characteristic classes for infinity algebras in work of Andrey Lazarev and Alastair Hamilton who use variants of the Kontsevich’s construction involving graph (co)homology.
created vertical composition
Several new articles have been announced on the thread Entropy, and many don’t have much to do with that subject. So I’ll just list them all here. (Many of them already have discussions on the other thread, however, so do look there.)
stub for nonabelian bundle gerbe, since I need the link at infinity-Weil theory introduction
started a stub for pseudo-connection, in order to satisfy links. But now I am really too tired. More tomorrow.
New entry formal noncommutative symplectic geometry within the circle of entries related to graph homology.
I added some more variations, links, and references to string diagram. I’m sure there are a lot more references that ought to go there.
I expanded linear algebra and wrote linear operator.
started semisimple Lie algebra
trivial group, trivial ring, trivial Lie algebra, abelian Lie algebra (which may not be trivial exactly but by special dispensation is still too trival to be simple)
there is a bunch of things on my to-do list concerning write-up of stuff on the nLab in the big context of my “diff cohomology in an (oo,1)-topos”-writeup. I am lagging behind. I could use some more help!
Of course you all are busy with your own stuff. But maybe there is a tiny chance somebody reads this here, maybe somebody who has been lurking all along, somebody who would enjoy helping out. I could say: I offer 60 reputation points! More seriously, this might be a chance to dip your feet into the water and join in to some interesting research. Here is the deal:
I have a LaTeX writeup of a fairly long proof that establishes the weak equivalence of 3 different strict 2-group models of the string 2-group. It appears as the proof of prop. 5.25 on p. 94 of these notes here. The trouble (for me) is that this proof involves some diagrams that would require code-cogs for implementation on the nLab. I want these diagrams on the nLab!
I have the LaTeX source code, of course, so it’s not much work to make this run on the nLab! But a bit of work. A tad more work than I find leisure for right now.
Let me know if you are interested and I’ll send you the LaTeX source!
Best, Urs
I considerably expanded and reorganized the discussion at Chern-Simons 2-gerbe.
There is now a decvent Idea-section;
I created three subsections for three different kinds of constructions of this beast.
The first subsection contains now a detailed account of the consztruction by Brylinski-McLaughline of an explicit Cech-Deligne cocycle. I end this with saying that once the construction is there, proving that it is correct is easy. The mystery is maybe how one comes up with the construction in the first place!
So then in the next subsection I rederive this construction as a special case of the general methods described at infinity-Chern-Weil theory. So I show that from Lie integration of the underlying Lie algebra cocycle one gets a canonical lift to pseudo-connections with values in the Lie algebra, and turning the crank, out drops the Brylinski-McLaughlin construction. I’ll later see if I can streamline this discussion a bit more.
Then there is a third subsection which is supposed to deal with the construction of bundle 2-gerbe representatives. But here I am being lazy and just give the references so far. Even though the construction is actually simple.
@Urs: I answered your query over at homotopy category of an (infinity,1)-category. The Ho(SSet) enrichment is given by the lax monoidal functor SSet -> Ho(SSet).
Using codecogs recipe and ascii table I wrote short entries fork and split equalizer. For those who distinguish fork and cofork, I have hard time remembering which one is which one.
By the way, nForum is today having lots of problems on my computer, it asks for human recognition, it bails out my automatically remembered password many times out and resets the settings for markdown when writing etc. often.