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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.
Page created. Idempotent monoids should be to monoids as idempotent monads are to monads.
I’ve added the examples of idempotent elements in (ordinary) monoids (1), idempotent morphisms in categories (2), solid rings (3), idempotent monads (4), idempotent -morphisms in bicategories (5), and “solid ring spectra” (6) ―What are other examples?
Also, should idempotent monoids have a unit? The examples 1 and 2 I mentioned above don’t, but 3, 4, and 6 do, while whether 5 does or doesn’t seems to vary a bit among the literature (AFAIU).
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|
[…]
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.
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
Gave proper reference for (Kieboom 1987).
splitting off this definition from neighborhood retract, for ease of linking, and in order to record the characterization AR = ANR+contractible
gave core of a ring some minimum content
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
I have tried to give algebraic topology a better Idea-section.
added pointer to:
I have deleted an old out of date query box from homotopy theory.
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:
started an entry on the Borel construction, indicating its relation to the nerve of the action groupoid.
have added a tad more content to Stein manifold and cross-linked a bit more
added pointer to Bredon 72. Will add this pointer also to various related entries on equivariant homotopy theory
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.
I worked on brushing up (infinity,1)-category a little
mostly I added in a section on homotopical categories, using some paragraphs from Andre Joyal's message to the CatTheory mailing list.
in this context I also rearranged the order of the subsections
I removed in the introduction the link to the page "Why (oo,1)-categories" and instead expanded the Idea section a bit.
added a paragraph to the beginning of the subsection on model categories
added the new Dugger/Spivak references on the relation between quasi-cats and SSet-cats (added that also to quasi-category and to relation between quasi-categories and simplicial categories)
there had been no references at Hilbert space, I have added the following, focusing on the origin and application in quantum mechanics:
John von Neumann, Mathematische Grundlagen der Quantenmechanik. (German) Mathematical Foundations of Quantum Mechanics. Berlin, Germany: Springer Verlag, 1932.
George Mackey, The Mathematical Foundations of Quamtum Mechanics A Lecture-note Volume, ser. The mathematical physics monograph series. Princeton university, 1963
E. Prugovecki, Quantum mechanics in Hilbert Space. Academic Press, 1971.
There’s a paper out characterising the category of continuous linear functions between Hilbert spaces
But Hilb concerns short linear maps between Hilbert spaces. Should we have a page for the former category?
added pointer to:
Stub for topological string with redirect topological string theory.
I created homotopy extension property and homotopy lifting property. If somebody wonders why I made identical copy of one of them on my personal nlab part is because there I want to keep conservative page for students and here in the main nlab I expect more vigorous extensions by others. On the other hand, I would like to have under homotopy lifting property mention of various variants like "soft map" homotopy lifting property, the homotopical variant of Dold etc. all in one place.
brief category:people-entry for hyperlinking references at flavour anomaly
(Hi, I’m new)
I added some examples relating too simple to be simple to the idea of unbiased definitions. The point is that we often define things to be simple whenever they are not a non-trivial (co)product of two objects, and we can extend this definition to cover the “to simple to be simple case” by removing the word “two”. The trivial object is often the empty (co)product. If we had been using an unbiased definition we would have automatically covered this case from the beginning.
I also noticed that the page about the empty space referred to the naive definition of connectedness as being
“a space is connected if it cannot be partitioned into disjoint nonempty open subsets”
but this misses out the word “two” and so is accidentally giving the sophisticated definition! I’ve now corrected it to make it wrong (as it were).
added a second equivalent definition at quasi-category , one that may be easier to motivate
am finally giving this its own entry (this used to be treated within the entry on Elmendorf’s theorem)
but just a stub for the moment