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I have tried to give algebraic topology a better Idea-section.
brief category:people-entry for hyperlinking references at homotopy theory
the book Monoidal Functors, Species and Hopf Algebras is very good, but still being written. Clearly the current link under which it is found on the web is not going to be the permanent link. So I thought it is a bad idea to link to it directly. Instead I created that page now which we can reference then from nLab entries. When the pdf link changes, we only need to adapt it at that single page.
I came across the page internal category in a monoidal category, which was lacking even a definition, so I put one in.
Where the page has
The index theorem is supposed to have an interpretation in terms of the quantum field theory of the superparticle on the given space,
is the “is supposed to” necessary? Why not “has an interpretation”? Is it just the general issue of any translation from mathematics to physics?
started model category theory - contents and added this as floating toc to relevant entries
am giving this its own entry for ease of hyperlinking, split off from Delta-generated space
Gave proper reference for (Kieboom 1987).
stub for Blakers-Massey theorem. Need to add more references…
Did anyone ever write out on the Lab the proof that for locally compact and Hausdorff, then with the compact-open topology is an exponential object? (Many entries mention this, but I don’t find any that gets into details.)
I have tried to at least add a pointer in the entry to places where the proof is given. There is prop. 1.3.1 in
but of course there are more canonical references. I also added pointer to
added pointer to today’s
Is the Strøm model category left proper? I know that pushout along cofibrations of homotopy equivalences of the form are again homotopy equivalences. (e.g. Hatcher 0.17) Maybe the proof directly generalizes, haven’t checked.
following a suggestion by Zoran, I have created a stub (nothing more) for Kuiper’s theorem
started some remarks at physical unit. But I really need to stop with that now and do more urgent things…
The entry used to start out with the line “not to be confused with neutral element”. This was rather suboptimal. I have removed that sentence and instead expanded the Idea-section to read now as follows:
Considering a ring , then by the unit element one usually means the neutral element with respect to multiplication. This is the sense of “unit” in terms such as nonunital ring.
But more generally a unit element in a unital (!) ring is any element that has an inverse element under multiplication.
This concept generalizes beyond rings, and this is what is discussed in the following.
In view of discussion in another thread (here), I have added (here) the following warning:
Beware that section 4.4 claims a new proof of the Strøm model structure, but relying on a statement in
which later was noticed to be false, by Richard Williamson, for details see p. 2 and Rem 5.12 and Sec. 6.1 in:
It remains unclear, to me anyways, what this implies for the sliced generalization which is the core claim of May & Sigurdsson’s book.(?)
Initial version:
Thomason-type model categories provide simple 1-categorical models for (∞,1)-categorical objects.
The provide a particularly convenient setting for results like Quillen’s Theorem A and Theorem B.
|(∞,1)-categorical structure|1-categorical structure|model structure| |∞-groupoid|category|Thomason model structure| |∞-groupoid|poset|model structure on posets| |(∞,1)-category|relative category|Barwick–Kan model structure| |connective spectra|symmetric monoidal groupoid|Fuentes-Keuthan model structure|
[…]
At first Zoran's reply to my query at structured (infinity,1)-topos sounded as though he were saying "being idempotent-complete" were a structure on an (oo,1)-category rather than just a property of it. That had me worried for a while. It looks, though, like what he meant is that "being idempotent" is structure rather than a property, and that makes perfect sense. So I created idempotent complete (infinity,1)-category.
created (finally) lax monoidal functor (redirecting monoidal functor to that) and strong monoidal functor.
Hope I got the relation to 2-functors right. I remember there was some subtlety to be aware of, but I forget which one. I could look it up, but I guess you can easily tell me.
started an entry geometric realization of simplicial topological spaces.
I decided this is a topic big enough to justify splitting it off from geometric realization (of simplicial sets).
But not much there yet. I just wanted to record for the moment that this realization too, does preserve pullbacks.
am giving this its own entry, in order to record sufficient conditions on topological subgroup inclusions for the coset space coprodoction to admit local sections.
So far I have two original references here (Gleason 50, Mostert 53). One should add some textbook account, too.
Am also referencing this at closed subgroup, at coset space and maybe elsewhere.
for completeness, to go with U(ℋ), for the moment mainly in order to record references, such as:
I added two recent examples of enriched categories: tangent bundle categories and Lawvere theories.
brief category:people-entry for hyperlinking references at twisted K-theory
splitting off this definition from neighborhood retract, for ease of linking, and in order to record the characterization AR = ANR+contractible
Many additions and changes to Leibniz algebra. The purpose is to outline that the (co)homology and abelian and even nonabelian extensions of Leibniz algebras follow the same pattern as Lie algebras. One of the historical motivations was that the Lie algebra homology of matrices which lead Tsygan to the discovery of the (the parallel discovery by Connes was just a stroke of genius without an apparent calculational need) cyclic homology. Now, if one does the Leibniz homology instead then one is supposedly lead the same way toward the Leibniz homology (for me there are other motivations for Leibniz algebras, including the business of double derivations relevant for the study of integrable systems).
Matija and I have a proposal how to proceed toward candidates for Leibniz groups, that is an integration theory. But the proposal is going indirectly through an algebraic geometry of Lie algebras in Loday-Pirashvili category. Maybe Urs will come up with another path if it drags his interest.
added pointer to
added pointer to:
added a sentence to the Idea-section at Kan complex
brief category:people-entry for hyperlinking references at Curry’s pardox and axiom of full comprehension
brief category:people-entry for hyperliunking references at Haag-Łopuszański-Sohnius theorem
there were no references here. I have added one:
have added a tad more content to Stein manifold and cross-linked a bit more
have started simplicial topological group
a bare sub-section with a list of references – to be !included into relevant entries – mainly at confinement and at mass gap problem (where this list already used to live)
at monadicity theorem in the second formulation of the theorem, item 3, it said
has
I think it must be
has
and have changed it accordingly. But have a look.
I have expanded the Idea section at state on a star-algebra and added a bunch of references.
The entry used to be called “state on an operator algebra”, but I renamed it (keeping the redirect) because part of the whole point of the definition is that it makes sense without necessarily having represented the “abstract” star-algebra as a C*-algebra of linear operators.