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I renamed n-symplectic manifold into symplectic Lie n-algebroid and then expanded that entry
I also brushed up Courant algebroid a bit
created reference-entry Moduli Problems and DG-Lie Algebras
I added to Hochschild homology in the section Function algebra on derived loop space a statement and proof of the theorem that “the function complex C(ℒX) on the derived loop space C(ℒC) is the Hochschild homology complex of C(X)”.
There is a curious aspect to this: we are to compute the corresponding pushout in ∞-algebras. But in the literature on Hochschild homology, the pushout is of course taken not in algebras, but in modules
HHk(C(X)):=Tork(C(X),C(X))C(X)⊗C(X).So how is that HH-complex actually an derived algebra?
The solution of this little conundrum is remarkably trivial using Badzioch-Berger-Lurie’s result on homotopy T-algebras .This tells us that we may model the ∞-algebras as simplicial copresheaves on our syntactic category T, using the left Bousfield localizatoin of the injective model structure at maps that enforce the algebra property.
But since we are computing a pushout and since the traditional bar complex provides a cofibrant resolution of our pushout diagram already in the unlocalized structure, and since left Bousfield localization does not affect the cofibrations, due to all these reasons we may (or actually: have to) compute the pushout of ∞-algebras as just a pushout in simplicial copresheaves.
In particular it follows that the pushout of our product-preserving coproseheaves is not actually product-presrving itself. Instread, it is (the simplicial set underlying) the standard Hochschild complex. So everything comes together. We know that if we wanted to find the actual ∞-algebra structure on this, we’d have to form the fibrant replacement in the localized model structure. That would make a bit of machinery kick in and actually produce the ∞-algebra structure on the Hochschild complex for us.
But if we don’t feel like doing that, we don’t have to. The homotopy groups of our simplicial copresheaf won’t change by that replacement.
cretated concrete (infinity,1)-sheaf
finally: stub for topological infinity-groupoid
main point there: the reference! :-)
I split off ∞-connected (∞,1)-topos from locally ∞-connected (∞,1)-topos and added a proof that a locally ∞-connected (∞,1)-topos is ∞-connected iff the left adjoint Π preserves the terminal object, just as in the 1-categorical case. I also added a related remark to shape of an (∞,1)-topos saying that when H is locally ∞-connected, its shape is represented by Π(*).
I hope that these are correct, but it would be helpful if someone with a little more ∞-categorical confidence could make sure I’m not assuming something that doesn’t carry over from the 1-categorical world.
started a stub for cartesian closed (infinity,1)-category
stub for effective Lie groupoid
I put some very very basic notes down at Finite Topological Spaces-Project (timporter) trying to get the ball rolling.
created the obvious epimorphism in an (infinity,1)-category
Mike,
I have expanded your discussion of the sheaf topos on a locally connected site at locally connected site. Please check if you can live with what I did.
added to cohesive site an example in the section Examples – families of sets. It is intentionally simplistic. And depending on which axioms we settle on, it is a counter-example. But maybe still of some use.
I presume that
Definition
Let C be a category with pullbacks. Then the tangent category TC of C is the category whose
- objects are pairs (A,𝒜) with A∈Ob(C) and with 𝒜 an abelian group object in the overcategory C/B;
contained a minor typo, so I replaced C/B with C/A.
New entry internal diagram, generalizing internal functor.
added a stubby section on free operads (free on a "collection") to operad, but a bit example-less at the moment. Have to run...
Created totally connected geometric morphism.
Please check the statement of Reidemeister’s theorem at Reidemeister moves, I was not that happy with the precise wording of the previous version as it made everything look as if it was happening in the plane, rather than indicating that what was happening in the plane mirrored what was happening in 3-dimensions. (Note that there was a discussion on MO, [here], on the proof.)
A minor correction to fully formal ETCS was suggested to me on my Wikipedia talk page. So I implemented it.
wrote something at topological operad
I am being asked for a list of references on the little disks operad, their action on higher categories, their higher traces, higher centers, etc.
So I went and improved the entry little k-cubes operad a little. Copied over some theorems, and then created/expanded the list of references.
If you have a favorite reference not yet listed there, this would be a good chance to list it, as I wil now point the people who asked me to this list
This area is linked to cubical sets and I just came on a recent paper by Glynn Winskel and Sam Staton, that may be of interest as it links several of the models for concurrency with presheaves. The paper is here.
(Edit: I have also linked to another paper by Winskel, Events, causality, and symmetry, (online version), from 2009. This may be useful for various aspects of the Physics-Theoretical Computer Science/Logic interface. It is well written and reasonably chatty.)
in the course of last night’s events, I created a handful of stubs for some basic concepts:
I fixed the link on Template page where it reads
“You can look at its source code” to
“You can look at its source code” where this link actually works.
added stuff to 2-groupoid.
created – for our derived seminar – the entry homotopy T-algebra with the main result by Badzioch and interlinked it with model structure on simplicial T-algebras and (infinity,1)-algebraic theory
I expanded Levi-Civita connection:
moved the discussion in terms of Christoffel-symbol components that had been there to its own section “In terms of Christoffel symbols”;
stated the abstract definition clearly right at the beginning;
stated this more in detail in “first order formalism”, i.e. in terms of a compatible ISO-connection.
added various theorems about injectivity radius estimates and relevant literature to geodesic flow.
Important take-home message for everybody: every paracompact manifold admits a metric with positive injectivity radius.
New entry skew-simplicial set with redirects crossed simplicial group, crossed simplicial set, skew-simplicial set and plurals.
added a bit more to T-algebra, but still incomplete. Need to copy over propositions and proofs from Lawvere theory
created smooth structure
(also created differentiable manifold and added hyperlinks to the various notions in the Idea-section of manifold)
If you want to divert any young minds that you know (your own for example, or some offspring or cousin or sibling or whatever) you might like to look at the colorability entry. It is sort of ’for fun’ but not completely as I hope to get on to when I’ve done some other things. (@Eric. you will have something else to do on the train! Get out your colouring pencils and a piece of paper! Find the link between 3-colourability and the symmetric group S_3. (If you know don’t tell!) You only need three pencils at the moment and as those infuriating waiters in American style restaurants say : Enjoy! :p )
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 BnU(1)diff,simp at circle n-bundle with connection. Is there a “modulo Bnℤ” missing? and, if so, which sense we quotient by Bnℤ 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 ℂ” 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 ℝ.)
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