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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|
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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.
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.
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.
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?
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
started to collect some references at Riemann hypothesis and physics. But just a puny start so far, have to quit now.
The cut rule for linear logic used to be stated as
If and , then .
I don’t think this is general enough, so I corrected it to
If and , then .
started G-CW complex.
updated 2012 webpage from
to
Adeel Khan created sheaf of meromorphic functions.
(He currently has problems logging into here, that’s why I am posting this for the moment.)
wrote a definition and short discussion of covariant derivative in the spirit of oo-Chern-Weil theory
edited classifying topos and added three bits to it. They are each marked with a comment "check the following".
This is in reaction to a discussion Mike and I are having with Richard Williamson by email.
Added a couple more papers.
looks interesting as a way to take linear algebra in a slightly alternative way to groupoidification.
Slighly adjusted the lead-in of this page: Added a line on what the book is actually about, and moved the line advertizing how great this is to after that.
Added the line:
The approach is echoed in Riehl & Verity 13 with Cat enhanced to the homotopy 2-category of (∞,1)-categories.
Also touched some wording further below (“is very difficult to read” “may be difficult to read”)
and added the missing cross-link with John Gray (1)
Finally, I made “formal category theory” a redirect to this page (this would deserve to point to a dedicated page, but as long as that doesn’t exit, it’s good to have it redirect here)